Non-affine Landau-Ginzburg models and intersection cohomology
NNon-affine Landau-Ginzburg models and intersection cohomology
Thomas Reichelt and Christian SevenheckJune 22, 2016
Abstract
We construct Landau-Ginzburg models for numerically effective complete intersections in toricmanifolds as partial compactifications of families of Laurent polynomials. We show a mirror state-ment saying that the quantum D -module of the ambient part of the cohomology of the submanifoldis isomorphic to an intersection cohomology D -module defined from this partial compactification andwe deduce Hodge properties of these differential systems. Contents D -modules and the Radon transformation . . . . . 172.3 Intersection cohomology D -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 The equivariant setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 D -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Toric geometry of complete intersection subvarieties . . . . . . . . . . . . . . . . . . . . . 46 Mathematics Subject Classification. D -module, toric variety, intersection cohomology, Radon transformationDuring the preparation of this paper, Th.R. was supported by a postdoctoral fellowship of the “Fondation sciencesmath´ematiques Paris” and by the DFG grant He 2287/2-2; Ch.S. was supported by a DFG Heisenberg fellowship (Se1114/2-1/2). Both authors acknowledge partial support by the ANR grant ANR-08-BLAN-0317-01 (SEDIGA). Introduction
The aim of this paper is the construction of a mirror model for complete intersections in smooth toricvarieties. We consider the case where these subvarieties have a numerically effective anticanonical bundle.This includes in particular toric Fano manifolds, whose mirror is usually described by oscillating integralsdefined by a family of Laurent polynomials and also the most prominent and classical example of mirrorsymmetry, namely, that of Calabi-Yau hypersurfaces in toric Fano manifolds. Here the mirror is a familyof Calabi-Yau manifolds and the mirror correspondence involves the variation of Hodge structures definedby this family. One interesting feature of our results is that these apparently rather different situationsoccur as special cases of a general mirror construction, called non-affine Landau-Ginzburg model .It is well-known that quantum cohomology theories admit expressions in terms of certain differentialsystems, called quantum D -modules. This yields a convenient framework in which mirror symmetry isstated as an equivalence of such systems. Moreover, Hodge theoretic aspects of mirror correspondencescan be incorporated using the machinery of (mixed) Hodge modules. However, quantum D -modules haveusually irregular singularities, except in the Calabi-Yau case. In our mirror construction, this correspondsto the fact that we let the Fourier-Laplace functor act on various regular D -modules obtained from theLandau-Ginzburg model.The quantum cohomology of a smooth complete intersection (which in our case is given as the zero locusof a generic section of a vector bundle) can be computed using the so-called Euler-twisted Gromov-Witteninvariants . Basically, these are integrals over moduli spaces of stable maps of pull-backs of cohomologyclasses on the variety and of the Euler class of the vector bundle. It is well known (see [Kon95, Pan98,Giv98b] and also [Iri11] as well as [MM11] for more recent accounts) that the ambient part of the quantumcohomology of the subvariety (consisting of those classes which are induced from cohomology classes ofthe ambient variety), is given as a quotient of the Euler-twisted quantum cohomology.From the combinatorial toric data of this vector bundle, we construct in a rather straightforward manneran affine Landau-Ginzburg model , which is a family of Laurent polynomials. The Euler-twisted quantum D -module (which encodes the above mentioned Euler-twisted Gromov-Witten invariants) can then beshown to be isomorphic a certain proper FL-transformed Gauß-Manin system , namely, the Fourier-Laplace transformation of the top cohomology group of the compactly supported direct image complex(in the sense of D -modules) of this affine Landau-Ginzburg model. On the other hand we show thatthe Euler − -twisted quantum D -module which encodes the so-called local Gromov-Witten invariants isisomorphic to the usual FL-transformed Gauß-Manin system.The actual non-affine Landau Ginzburg model is constructed by a certain partial compactification ofthe affine one, which yields a family of projective varieties. Our main result is Theorem 6.13 (whichalso contains the above mirror statements on twisted resp. local quantum D -modules), it states thatthe ambient quantum D -module is isomorphic to a Fourier-Laplace transform of the direct image of theintersection cohomology D -module of the total space of this family, notice that this total space is usuallynot smooth.One of the big advantages of using this singular variety together with the intersection cohomology D -module is the fact that we do not need any kind of resolutions. In particular, we do not need to construct(or suppose the existence of) crepant resolutions like in [Bat94]. Notice also that [Iri11] discusses Landau-Ginzburg models of a more special class of subvarieties in toric orbifolds (the so-called nef partitions).In that paper, a mirror statement is shown in terms of A- resp. B-periods, but this construction needsa hypothesis on the smoothness of a certain complete intersection (given as the intersection of fibres ofseveral Laurent polynomials, see section 5.2 of loc.cit.). Some more remarks on the nef-partition modeland how it relates to our construction can be found in subsection 1.5 below.We will show that the direct image of the intersection cohomology D -module of the total space is itself(modulo some irrelevant free O -modules) an intersection cohomology D -module with respect to a localsystem measuring the intersection cohomology of the fibers of the projective family. An important pointin our paper is that this intersection cohomology D -module admits a hypergeometric description, thatis, it can be derived from so-called GKZ-systems (as defined and studied by Gelfand’, Kapranov andZelevinsky). More precisely, it appears as the image of a morphism between two such GKZ-systems(Theorem 2.16). This result is interesting in its own, as in general there are only very few cases wheregeometrically interesting intersection cohomology D -modules have an explicit description by differentialoperators.Notice that the intersection cohomology D -module mentioned above underlies a pure Hodge module.2rom this we can deduce a Hodge-type property of the reduced quantum D -module (see Corollary 6.14).As already mentioned above, it cannot underly itself a Hodge module, as in general it acquires irregularsingularities (this never happens for D -modules coming from variation of Hodge structures resp. Hodgemodules due to Schmid’s theorem). Rather, it is part of a non-commutative Hodge (ncHodge) structuredue to a key result by Sabbah ([Sab08]).There is another important aspect in the paper that has not yet been mentioned. The various quantum D -modules are actually not D -modules in the proper sense, rather, they are families of vector bundleson P together with a connection operator with poles along zero and infinity. This is reflected in thefact that we are looking at Fourier-Laplace transforms of certain regular D -modules (like Gauß-Maninsystems) together with a given filtration. The filtration induces a lattice structure on the FL-tranformed D -module (i.e., it yields a coherent O -submodule generating the FL-transformed D -module). Theselattices can be reconstructed by a twisted logarithmic de Rham complex (in the sense of log geometry) ofan intermediate compactification of the family of Laurent polynomials. We show in Corollary 3.20 thatthis twisted logarithmic de Rham complex can also be explicitly described by hypergeometric equations.Notice that for this result to hold true, we have to restrict to an open subspace of the parameter space,where certain singularities at infinity of these Laurent polynomials are allowed, but not all of them. Thissituation is different to the one in our earlier paper [RS15] where we had to exclude any singularity atinfinity.The remaining part of this introduction is a rather detailed synopsis of the content of the paper. It canbe read as a warm-up, where the main playing characters are introduced together with some exampleswhich illustrates the constructions done later.Our main case of interest is the following: Let X Σ be an n -dimensional smooth projective toric variety.Suppose that L = O X Σ ( L ) , . . . , L c = O X Σ ( L c ) are ample line bundles on X Σ such that − K X Σ − (cid:80) cj =1 L j is nef (for many intermediate results, we can actually relax both assumptions and suppose only that theindividual bundles L , . . . , L c are nef). Put E := ⊕ cj =1 L j , then E is a convex vector bundle. We willbe interested in several quantum D -modules, which correspond to twisted Gromov-Witten invariants of( X Σ , E ) as well as to Gromov-Witten invariants on the ambient cohomology of the complete intersection Y := s − (0) defined by a generic section s ∈ Γ( X Σ , E ). Let us consider the total space V ( E ∨ ), whichis a quasi-projective toric variety with defining fan Σ (cid:48) . We set Σ (cid:48) (1) = { R ≥ b , . . . , R ≥ b t } , wherethe vectors b i are the primitive integral generators of the rays of Σ (cid:48) . From this set of data one canconstruct Lefschetz fibrations, that is, family of hyperplane sections of some projective toric varieties.The actual Landau-Ginzburg models of the above toric variety (resp. of the complete intersection Y )will be obtained by restricting the base of such families to a certain sub-parameter space which is anopen subset of the K¨ahler moduli space of X Σ . Actually, we do consider two different situations: Eitherwe start with the data of a toric variety and some line bundles satisfying the above positivity conditions,and construct the set { b i } as sketched, or we consider only such a set of vectors, in which case we do nothave a K¨ahler moduli space, and the reduction of the parameter space of the Lefschetz fibration is doneusing an equivariance property of this fibration with respect to a natural torus action. Nevertheless,many of our constructions also make sense in this more general setup, therefore, the material in sections2 and 3 below only depend on vectors { b i } and do not suppose the existence of X Σ , L , . . . , L c . We consider the following situation: Let B be s × t -matrix of integer numbers, written as B = ( b , . . . , b t ).The only assumption we make is that (cid:80) ti =1 Z b i = Z s . As just explained, the example the reader shouldhave in mind is when these vectors are the primitive integral generators of the rays of a possibly non-compact toric variety, but most of the constructions below do not depend on this assumption. As aconcrete and easy though non-trivial example which will be considered throughout this introduction, let X Σ = P , H ⊂ P a hyperplane, and take the bundles L = O P (2 H ) and L = O P (3 H ). They areobviously ample, and we have O P ( − K P − H − H ) = O P (6 − −
3) = O P (1), which is also ample.The defining fan Σ (cid:48) of the total space V ( L ∨ ⊕ L ∨ ) has rays b , . . . , b , and the matrix B = ( b , . . . , b )3s given by B = − − − − − Let us return to the general setup of a matrix B ∈ M ( s × t, Z ) of rank s . Put S := ( C ∗ ) s , and considerthe following map g : S −→ P t , ( y , . . . , y s ) (cid:55)→ (1 : y b , . . . , y b t ) , which is an embedding due to the assumption on the rank of B . Here we write y b i for the product (cid:81) sk =1 y b ki k , b ki being the entries of B . The map g is only locally closed, so we denote by X its closure in P t . We are interested in a family of hyperplane sections of X , constructed in the following way: Considerthe incidence variety Z := (cid:110)(cid:80) ti =0 λ i · w i (cid:111) ⊂ P t × C t +1 , where λ , . . . , λ t are coordinates on C t +1 andwhere w : . . . : w t are homogenous coordinates on P t . The situation is visualized in the followingdiagram where p resp. p is the restriction of the projection to the first resp. second factor Z p (cid:121) (cid:121) p (cid:33) (cid:33) S ⊂ X ⊂ P t C t +1 (1)Here we have identified S with its image under g . The family of hyperplane sections is by definition themorphism Φ := ( p ) | p − ( X ) : p − ( X ) → C t +1 . It is a projective map, and its restriction ϕ := Φ | p − ( S ) isnothing but the family of Laurent polynomials S × C t −→ C λ × C t ( y , . . . , y s , λ , . . . , λ t ) (cid:55)−→ (cid:16) − (cid:80) ti =1 λ t · y b i , λ , . . . , λ t (cid:17) (2)For the concrete example from above, the first component of ϕ is given by( y , . . . , y , λ , . . . , λ ) (cid:55)→ − λ · y − λ · y y − λ · y y − λ · y y − λ · y y − λ y y · . . . · y − λ y − λ y The partial compactification Φ of this family is easy to calculate as the closure X of im ( g ) is a hyper-surface in P , namely, it is given by the binomial equation w w w − w w w w w w = 0. Hence Z isthe codimension 2 subvariety of P × C cut out by the two equations w w w − w w w w w w = 0 and λ w + . . . + λ w = 0 , and Φ is the projection from this variety to the space C with coordinates λ , . . . , λ .For various reasons, we will also need to work with the family Φ U , where in the above diagram (1) theincidence variety Z is replaced by its complement U := ( P t × C t +1 ) \ Z . Although geometrically thetwo morphisms Φ and Φ U behave differently (e.g. Φ U is no longer proper), they are strongly related onthe cohomological level. The transformation corresponding in cohomology to the geometrical operationof taking the inverse image of X under p followed by the projection by p is the so-called Radon transformation for D -modules (see subsection 2.2 for more details).The morphism ϕ resp. Φ can be considered as the maximal family of hyperplane sections of S resp. of itscompactification X . However, in applications like those presented in section 6 of this paper, we need torestrict these families to some subspace of the parameter space C t which is called KM ◦ in the main partof this article (see the discussion before Definition 6.3). We will not give the precise definition of KM ◦ here, let us only mention that the torus S acts on ( C ∗ ) t by ( y, λ ) (cid:55)→ (cid:0) y − b , . . . , y − b t (cid:1) · λ (see formula (26)4n subsection 2.4 below). Then we consider the orbit space of this action, which is a torus of dimension t − s . The parameter subspace KM ◦ is a certain open subvariety of this orbit space. We will actuallychose an embedding KM ◦ (cid:44) → C t , so that we always see C λ × KM ◦ as a locally closed subspace of C t +1 .In the case where our matrix B is defined by a toric variety X Σ together with a set of line bundles, KM ◦ is not just an open subset of an abstract torus, but of ( C ∗ ) t − s , i.e., it comes with a set of coordinatescalled q , . . . , q t − s . Notice however that the choice of these coordinates is not unique, it depends on thechoice of a basis of H ( X Σ , Z ) with good properties. Definition 1.1 (see Definition 6.3) . Let X Σ be smooth, toric and projective. Let L = O X Σ ( L ) , . . . , L c = O X Σ ( L c ) be ample line bundles on X Σ such that − K X Σ − (cid:80) cj =1 L j is nef. Let Σ (cid:48) be the defining fan of thetotal space V ( E ∨ ) , where E := ⊕ cj =1 L j is a convex vector bundle on X Σ . Let Σ (cid:48) (1) = { R ≥ b , . . . , R ≥ b t } ,where b i are the primitive integral generators of the rays of Σ (cid:48) . Let KM ◦ be the parameter space describedabove. Then the restrictions Π := Φ |Z ◦ X : Z ◦ X := Z ∩ p − ( X ) ∩ (cid:0) P t × C λ × KM ◦ (cid:1) −→ C λ × KM ◦ resp. π := ϕ | Z ∩ p − ( S ) ∩ ( P t × C λ ×KM ◦ ) : Z ∩ p − ( S ) ∩ (cid:0) P t × C λ × KM ◦ (cid:1) −→ C λ × KM ◦ are called the non-affine resp. affine Landau-Ginzburg model of ( X Σ , L , . . . , L c ) . Let us notice that in the main body of this text, the affine Landau-Ginzburg model appears in twoversions, called π and π . Actually, π is an intermediate partial compactification of π (i.e., the fibres of π contain those of π and are contained in those of Π).To illustrate this definition, we discuss the parameter subspace KM ◦ for the above example of completeintersections of degree (2 ,
3) in P . As B is a 7 × KM ◦ must be an open subset of C ∗ . We can choose the embedding C ∗ (cid:44) → C , q (cid:55)→ (1 , , , , q, , q ∈ C ∗ to be in KM ◦ is then simply that the family ϕ , whenrestricted to Z ∩ p − ( S ) ∩ ( P t × C λ × { q } ) yields a non-degenerate Laurent polynomial, i.e., has nosingularities at infinity (see Definition 3.8). One can easily show that the condition that − K P − L − L is ample (and not only nef) implies that this is the case for all q ∈ C ∗ (one has to argue along the linesof [RS15, Lemma 2.8]). Hence in this example, we have KM ◦ = C ∗ , and therefore the affine and thenon-affine Landau-Ginzburg model of ( P , O P (2) , O P (3)) are given as π : ( C ∗ ) × C ∗ −→ C λ × C ∗ ( y , . . . , y , q ) (cid:55)−→ (cid:16) − y − y y − y y − y y − y y − q y y · ... · y − y − y , q (cid:17) Π : Z ◦ X −→ C λ × C ∗ ( w : . . . : w t , λ , q ) (cid:55)−→ ( λ , q )where the quasi-projetive subvariety Z ◦ X of P t × C λ × C ∗ is given by Z ◦ X = (cid:8) w w w − w w w w w w = 0 , λ w + w + . . . + w + qw + w + w = 0 (cid:9) ⊂ P t × C λ × C ∗ . The main idea of this paper is that mirror correspondences can be expressed using the language of(filtered) D -modules. For the toric varieties (and possibly non-toric subvarieties of them) that we areconcerned with here, these D -modules are of special type, namely they are constructed from the GKZ-system. Let us therefore start by recalling their definition (see Definition 2.7 below). We only treat here aspecial case which leads to a regular holonomic system. Let as before a matrix B ∈ M ( s × t, Z ) be given.Here we do not even need the condition rank( B ) = s . Consider the matrix (cid:101) B ∈ M (( s + 1) × ( t + 1) , Z ))defined by . . . B efinition 1.2. Let (cid:101) B be as above. Moreover, let (cid:101) β = ( β , . . . , β s ) be an element in C s +1 . Write L forthe module of relations among the columns of (cid:101) B , i.e. the kernel of the linear mapping Z t +1 → Z s +1 givenby (cid:101) B . Let D C t +1 be the Weyl algebra in t + 1 variables, i.e., D C t +1 := C [ λ , λ , . . . , λ t ] (cid:104) λ , λ , . . . , λ t (cid:105) Define M (cid:101) β (cid:101) B := D C t +1 / (cid:0) ( (cid:3) l ) l ∈ L + ( E k − β k ) k =0 ,...s (cid:1) , where (cid:3) l := (cid:81) i : l i < ∂ − l i λ i − (cid:81) i : l i > ∂ l i λ i , l ∈ L E k := (cid:80) ti =0 (cid:101) b ki λ i ∂ λ i , k ∈ { , . . . , s } where (cid:101) B = (cid:16)(cid:101) b ki (cid:17) . Then M β (cid:101) B is called a GKZ-system. We will quite often work with the corresponding sheaf of D C t +1 -modules, denoted by M (cid:101) β (cid:101) B . It is wellknown (see, e.g., [Ado94, Hot98]) that M (cid:101) β (cid:101) B is a regular holonomic D C t +1 -module.Given the matrix B from the example of the last section (i.e for (2 , P ), wehave M (cid:101) β (cid:101) B = D C t +1 /I , with I = (cid:16) ∂ λ ∂ λ ∂ λ − ∂ λ ∂ λ ∂ λ ∂ λ ∂ λ ∂ λ , λ ∂ λ + λ ∂ λ + . . . + λ ∂ λ − β ,λ ∂ λ − λ ∂ λ − β , λ ∂ λ − λ ∂ λ − β , . . . , λ ∂ λ − λ ∂ λ − β ,λ ∂ λ + λ ∂ λ + λ ∂ λ − β , λ ∂ λ + λ ∂ λ + λ ∂ λ + λ ∂ λ − β (cid:17) , where ˜ β = ( β , β , . . . , β ).Let us describe a basic result from [Rei14] that shows how these D -modules enter into the study ofLandau-Ginzburg models. It uses the notion of Gauß-Manin systems, which are differential systemsassociated to any morphism between smooth algebraic (or analytic) varieties. Intuitivly, solutions ofsuch systems are given by period integrals (at least on the smooth locus of the map). The formaldefinition requires the notion of direct images of D -modules and is recalled in subsection 2.1 below.With these remarks in mind, we can state the result as follows (in the main part of the text it appearsin a more precise version as Theorem 2.11). For simplicity, we also impose the additional assumption ofnormality, which is discussed in detail in section 5. We write N (cid:101) B for the semi-group associated to (cid:101) B ,that is, N (cid:101) B := (cid:80) ti =0 N (cid:101) b i ⊂ Z s . Theorem 1.3.
Let the matrices B and (cid:101) B be as above. Suppose moreover that the associated semi-groupring C [ N (cid:101) B ] is normal. Consider the family of Laurent polynomials ϕ : S × C t → C t +1 defined in equation (2) . Then we have an exact sequence of regular holonomic D C t +1 -modules −→ H s − ( S, C ) ⊗ O C t +1 −→ H ϕ + O S × C t +1 −→ M (cid:101) B −→ H s ( S, C ) ⊗ O C t +1 −→ Here the left- and the rightmost terms are vector bundles on C t +1 together with the trivial connectionoperator which annihilates sections in H s − ( S, C ) resp. H s ( S, C ) , and H ϕ + O S × C t +1 is the Gauß-Maninsystem of ϕ alluded to above. An important aspect of the construction in [Rei14] that yields this result is that all the above D C t +1 -modules underly mixed Hodge modules and that the exact sequence exists in the abelian category MHM ( C t +1 ). Although Hodge theoretic considerations are one of the main motivations of this paper,we will not use this fact directly, and results on Hodge modules will not come into play until Corollary6.14.The theorem above shows that there is a tight connection between the Gauß-Manin system of themorphism ϕ and the GKZ-system associated to the matrix (cid:101) B . However, they are not equal, but theirdifference (i.e., kernel and cokernel of the morphism H ( ϕ + O S × C t +1 ) → M (cid:101) B ) are relativly simple.The next construction has the effect of erasing this difference and yields an isomorphism of the two D -modules we are interested in. First we need a certain variant of the Fourier-Laplace transformationfor holonomic D -modules. Again we present a simplified version, the actual definition can be found inthe next subsection as Definition 2.4. 6 efinition 1.4. Let Y be a smooth affine variety and let D C λ × Y the ring of global algebraic differentialoperators on C λ × Y . If M is a D C × Y -module, we denote by FL Y ( M ) the object which is equal to M as a module over D Y and where the new variable τ acts as ∂ λ from the left and where ∂ τ acts as leftmultiplication by − λ . In this way FL Y ( M ) becomes a left module over D C τ × Y . Then we define FL locY ( M ) := FL Y ( M )[ τ − ] to be the localized Fourier-Laplace transformation of M . Again we will denote by the same symbol thecorresponding functor acting on sheaves of left D C λ × Y -modules. With this definition at hand, we have the following easy consequence of Theorem 1.3.
Corollary 1.5.
Let B and (cid:101) B be as above. Write (cid:98) C t +1 for the affine space Spec C [ τ, λ , . . . , λ t ] . Thenthere is an isomorphism of D (cid:98) C t +1 -modules FL loc C t (cid:0) H ϕ + O S × C t +1 (cid:1) ∼ = FL loc C t ( M (cid:101) B )In the above example, we have FL loc C t ( M (cid:101) B ) = D (cid:98) C t +1 / (cid:98) I , where (cid:98) I = (cid:16) τ ∂ λ ∂ λ − ∂ λ ∂ λ ∂ λ ∂ λ ∂ λ ∂ λ , − τ ∂ τ + λ ∂ λ + . . . + λ ∂ λ − , λ ∂ λ − λ ∂ λ ,λ ∂ λ − λ ∂ λ , . . . , λ ∂ λ − λ ∂ λ , λ ∂ λ + λ ∂ λ + λ ∂ λ , λ ∂ λ + λ ∂ λ + λ ∂ λ + λ ∂ λ (cid:17) , The partial compactification Φ of ϕ is a projective morphism, but its source space p − ( X ) is usuallysingular. For that reason, we are more interested in the direct image of the corresponding intersectioncohomology D -module. More precisely, consider the regular holonomic D P t -module M IC ( X ) whichcorrespondes to the intersection complex IC X of the variety X (recall that X was defined as the closurein P t of the image of the embedding g : S (cid:44) → P t ) under the Riemann-Hilbert correspondence. Formally, M IC ( X ) can be defined as the image of the natural morphism g † O S → g + O S , where g † is the “directimage with proper support”-functor for holonomic D -modules. It is the minimal (also called intermediate)extension of its restriction to the smooth part of X , and as such is an irreducible D P t -module. Moreimportant, it underlies a pure polarizable algebraic Hodge module, i.e., an object of the category MH p ( P t )(see [Sai88]). This last property will play a key role in Hodge theoretic application of our mirror statement(see Corollary 6.14).In general it is quite hard to describe such intersection cohomology D -modules explicitly, however, thisis possible in the current situation. We have the following result (which we state directly in a forminvolving the functor FL loc C t since this is the result that will be used later) Theorem 1.6 (see Theorem 3.6 below) . Suppose that C [ N (cid:101) B ] is normal, then there is some parameter (cid:101) γ = ( γ , γ , . . . , γ s ) ∈ Z s +1 such that FL loc C t (cid:0) H p p +1 M IC ( X ) (cid:1) ∼ = im (cid:16) FL loc C t ( M (cid:101) γ (cid:101) B ) D −→ FL loc C t ( M (cid:101) B ) (cid:17) (3) Here D is the morphism induced from right multiplication by τ − γ · ∂ g λ · . . . · ∂ g t λ t , where g = ( g , . . . , g t ) is any element in Z t such that B · g tr = − ( γ , . . . , γ s ) . The object on the left hand side of the above isomorphism should be seen (up to the action of thefunctor FL loc C t ) as a D -module extending a local system the fibres of which are itself intersection co-homology groups, namely those of the fibres of the morphism Φ. We could also replace the object p +1 M IC ( X ) by M IC ( Z ◦ X ), by which we mean the regular holonomic D P t × C t +1 -module correspond-ing to the intersection complex IC Z ◦ X via the Riemann-Hilbert correspondence (so that the complex p p +1 M IC ( X ) ∼ = p M IC ( Z ◦ X ) corresponds to the topological direct image complex R Π ∗ IC Z ◦ X ).Notice that the functors p +1 and H p exist in MH p , hence the object occurring in the last theoremis the Fourier-Laplace transform of a D -module underlying a pure polarizable Hodge module (this isbasically the proof of Corollary 6.14)We would like to explain in an informal way the reason for this theorem to hold true. The main point isthat GKZ-systems behave quite well with respect to the duality functor for holonomic D -modules. Moreprecisely, we have the following very nice result of Walther (see [Wal07]).7 heorem 1.7. Let B and (cid:101) B be as above. Suppose again for simplicity that the semi-group ring C [ N (cid:101) B ] is normal. Then there is a parameter (cid:101) γ ∈ Z s +1 such that D M (cid:101) B ∼ = M (cid:101) γ (cid:101) B From this statement we see that the above morphism D can actually be seen (up to some shifts andnotational conventions) as a morphism D FL loc C t ( M (cid:101) B ) → FL loc C t ( M (cid:101) B ). As mentioned above, M IC ( X )is the image of g † O S → g + O S , notice further that these two D -modules are also dual to each other.Applying the Radon transformation functor to them yields precisely the two GKZ-systems on the righthand side of equation (3) (see Theorem 2.11 below for more details), hence it is plausible that theintersection cohomology module (resp. its Fourier-Laplace transform) on the left hand side of equation(3) can be identified with the image of the morphisms D between these two GKZ-systems.For our purposes, we need actually a stronger duality statement: We consider the object ( M (cid:101) B , F ord • )consisting of the regular holonomic D C t +1 -module M (cid:101) B together with the good filtration by coherent O C t +1 -submodules induced from the filtration by the order of differential operators on D C t +1 . This is anobject of M. Saito’s category MF ( D C t +1 ) (see [Sai88, section 2.4]), and there is duality functor on thiscategory extending the duality functor for holonomic D -modules. Then we have (see Theorem 5.4) that D ( M (cid:101) B , F ord • ) ∼ = ( M (cid:101) γ (cid:101) B , F ord • + k ) for some integer k .For our guiding example, a parameter (cid:101) γ such that D M (cid:101) B = M (cid:101) γ (cid:101) B can be chosen as (cid:101) γ = ( − , , , , , , − , − D is induced by right multiplication with τ · ∂ λ · ∂ λ .Similarly to the considerations of Lefschetz families above, we will need to restrict these D -modules tothe parameter subspace KM ◦ ⊂ C t . We will not explain here how to do this in detail, since it is a bittechnical (see the presentation in subsection 2.4 and section 6 below). Instead, let us consider again theabove example and the embedding C τ × KM ◦ (cid:44) → C τ × C ( τ, q ) (cid:55)−→ ( τ, , , , , , q, , M (cid:101) B and M (cid:101) γ (cid:101) B as well as the morphism D . Forsimplicity, we will also set τ = 1, more precisely, we will consider the inverse image under the map q (cid:55)→ (1 , q ). We will also twist the restriction of M (cid:101) γ (cid:101) B by some invertible map (see Definition 6.1) Thenthe (restriction of the) morphism D is given as C [ q ± ] (cid:104) ∂ q (cid:105) / ( P ) −→ C [ q ± ] (cid:104) ∂ q (cid:105) / ( P ) Q (cid:55)−→ Q · ( q∂ q ) (4)where P = q · (3 q∂ q + 1)(3 q∂ q + 2)(3 q∂ q + 3)(2 q∂ q + 1)(2 q∂ q + 2) + ( q∂ q ) = ( q∂ q ) · (cid:0) q · (3 q∂ q + 1)(3 q∂ q + 2)(2 q∂ q + 1) + ( q∂ q ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) Q (2 , =: ( q∂ q ) · Q (2 , P = q · (3 q∂ q )(3 q∂ q + 1)(3 q∂ q + 2)(2 q∂ q )(2 q∂ q + 1) + ( q∂ q ) = (cid:0) q · (3 q∂ q + 1)(3 q∂ q + 2)(2 q∂ q + 1) + ( q∂ q ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) Q (2 , · ( q∂ q ) =: Q (2 , · ( q∂ q ) The map D is obviously well defined, its kernel is generated by Q (2 , and we see that im ( D ) ∼ = C [ q ± ] (cid:104) ∂ q (cid:105) / ( P ) ker ( D ) ∼ = C [ q ± ] (cid:104) ∂ q (cid:105) / ( Q (2 , ) .Q (2 , is an inhomogenous hypergeometric operator with a regular singularity at q = 0 and irregularsingularity at q = ∞ .The following statement summarizes the above calculation and can be seen as an illustration of Theorem1.6 8 roposition 1.8. Consider the example from above. Then we have an isomorphism of left C [ q ± ] (cid:104) ∂ q (cid:105) -modules (cid:16) FL loc C t (cid:0) H Π + M IC ( Z ◦ X )) (cid:1)(cid:17) | τ =1 ∼ = C [ q ± ] (cid:104) ∂ q (cid:105) / ( Q (2 , ) . Let us finish this discussion with some remarks on the case of Calabi-Yau complete intersections in toricmanifolds. Suppose that instead of the above example we had considered a (2 , P , i.e., the matrix B = − − − − − then the same arguments as above would lead to the operator Q (2 , := 8 q · (2 q∂ q + 1)(4 q∂ q + 1)(4 q∂ q + 2)(4 q∂ q + 3) − ( q∂ q ) which is regular and homogeneous (with singularities at q = 0 , − , ∞ ). In that case, the above statementcan be sharpened in the following way. Proposition 1.9.
Consider a (2 , -complete intersection in P , then we have the isomorphism of left C [ q ± ] (cid:104) ∂ q (cid:105) -modules (cid:0) H Π + M IC ( Z ◦ X ) (cid:1) | λ =1 ∼ = C [ q ± ] (cid:104) ∂ q (cid:105) / ( Q (2 , )The reason for this to be true is that in the Calabi-Yau case, Fourier-Laplace transformation togetherwith restriction to τ = 1 has basically no effect, i.e., can be identified with restriction to λ = 1. Inparticular, the object thus obtained still underlies a pure polarizable Hodge module (whereas in general,we obtain a variation on non-commutative Hodge structures, see 6.14 and Conjecture 6.15 below). This isconsistent with classical results on mirror symmetry for Calabi-Yau hypersurfaces like the quintic in P .Notice however that in these constructions, one uses certain crepant desingularizations in order to workwith ordinary cohomology together with its Hodge structures instead of intersection cohomology as inthe present paper. This may introduces a new difference when compared to our construction, which willhowever disappear when using the functor FL loc . This should basically follow from the decompositiontheorem (say, for pure Hodge modules, see [Sai88, Corollaire 3]) when applied to the desingularizationmap. We will make some more remarks on how our construction is related to known mirror models forcomplete intersections in the later subsections 1.4 and 1.5 below. However, a thorough treatment of thiscomparison issue is delicate and will be postponed to a subsequent paper. We would like to state here in a slightly informal way the main results of this paper. They can beexpressed as isomorphism of two D C τ ×KM ◦ -modules, one obtained as sketched above (i.e., direct image D -modules under the morphisms Π resp. π ), the other one derived from Gromov-Witten theory of thevariety X Σ resp. from its subvarieties. The actual picture is considerably more complicated, in the sensethat we do not just look at D -modules over C τ × KM ◦ , but at modules over D P ×KM ◦ together with astructure of R C τ − ×KM ◦ -modules where R C τ − ×KM ◦ is the sheaf of Rees rings for the filtration by orderson differential operators on D KM ◦ . This corresponds to the fact that the Gauß-Manin-systems as wellas the direct image modules of intersection cohomology modules occurring do carry Hodge filtrations,i.e., underly objects of the category MHM of (algebraic) mixed Hodge modules (see [Sai90]). This veryimportant additional information can be reformulated as the structure of an R -module, and the latter isconserved by the functor FL loc . Hence our actual statements in section 6 are considerably stronger thanwhat is announced here. In particular, the simplified statement below is basically only an identificationof local systems and hence does not take into account the fact that these D -modules have irregularsingularities in general. Nevertheless, we think that it is still instructive. It should be seen as a snapshotof what the actual result looks like. 9e consider the situation described above, that is, we let X Σ be a smooth, projective toric variety,and L = O X Σ ( L ) , . . . , O X Σ ( L c ) ample line bundles such that the class − K X Σ − (cid:80) cj =1 L j is nef. Thenfor a generic section s ∈ Γ( X Σ , E ), the zero locus Y := s − (0) ⊂ X Σ is a complete intersection with nefanticanonical class. Put E = ⊕ cj =1 L j , then E is a convex vector bundle on X Σ , and we can consider twistedGromov-Witten invariants which give rise to the (small) twisted quantum- D -modules QDM( X Σ , E ), i.e.,a vector bundle on P × KM ◦ with fibre H ∗ ( X Σ , C ) with connection operator defined by the twistedquantum product. Moreover, we have the endomorphism c top ( E ) of H ∗ ( X Σ , C ) given by cup product withthe Euler class of E , and we put H ∗ ( X Σ , C ) := H ∗ ( X Σ , C ) /ker ( c top ( E )). Then the reduced or ambientquantum D -modules, denoted by QDM( X Σ , E ), is a vector bundle on P × KM ◦ with fibres H ∗ ( X Σ , C ),and the connection is defined via the quantum product on ambient cohomology classes, i.e., classes inthe image of the morphism H ∗ ( X Σ , C ) → H ∗ ( Y, C ) (notice that this image is isomorphic to the quotient H ∗ ( X Σ , C )). Finally, we can also consider moduli spaces of stable maps into the total space V ( E ∨ ) ofthe vector bundle dual to E (then E ∨ is concave), and this yields the so-called local Gromov-Witteninvariants. The corresponding quantum- D -module is denoted by QDM( E ∨ ). We refer to the sections 4and 6 for precise definitions of the various quantum D -modules.With these notions at hand, we have the following results. Theorem 1.10 (see theorem 6.13 and theorem 6.16) . Given X Σ , L = O X Σ ( L ) , . . . , O X Σ ( L c ) as above,but suppose moreover that L , . . . , L c are ample (but − K X Σ − L − . . . − L c is still only required to be nef ).Consider the matrix B constructed from the rays of Σ (cid:48) . Let π resp. Π the affine resp. the non-affineLandau-Ginzburg models of ( X Σ , L , . . . , L c ) . Then we have FL loc C t (cid:0) H π † O S ×KM ◦ (cid:1) | C ∗ τ × B ∗ ε ∼ = (id C ∗ τ × Mir) ∗ (cid:0) QDM( X Σ , E ) | C ∗ τ × B ∗ ε (cid:1) , FL loc C t (cid:0) H π + O S ×KM ◦ (cid:1) | C ∗ τ × B ∗ ε (cid:48) ∼ = (id C z × Mir (cid:48) ) ∗ (cid:16) QDM( E ∨ ) | C ∗ τ × B ∗ ε (cid:48) (cid:17) , FL loc C t (cid:0) H Π + M IC ( Z ◦ X ) (cid:1) | C ∗ τ × B ∗ ε ∼ = (id C ∗ τ × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) | C ∗ τ × B ∗ ε . Here B ∗ ε , B ∗ ε (cid:48) are some (pointed) convergency neighborhoods of the large volume limit point in KM ◦ , Mir is the mirror map (see, e.g., [CG07, MM11]) and
Mir (cid:48) is some other coordinate change (which alsoinvolves the mirror map
Mir ). From the pureness property of M IC ( Z ◦ X ) we can deduce the following corollary, which is (part of) theHodge theoretic aspect of our mirror correspondence. As mentioned earlier, it relies on the notion of non-commutative Hodge structures (see [Sab11] for an overview) which is adapted to the occurrence ofirregular singularities in the various quantum D -modules. Corollary 1.11 (see corollary 6.14) . Under the assumptions of the last theorem, the ambient quantum D -module QDM( X Σ , E ) (or at least its restriction to the convergency neighborhood B ∗ ε ) is part of avariation of non-commutative Hodge structures. We conjecture in 6.15 below that QDM( X Σ , E ) is itself a non-commutative Hodge structure, however, theproof of this conjecture would need some additional results on the Hodge filtration of H p M IC ( Z ◦ X )which are not yet available. The aim of the next two subsection is to give some ideas on the relation of our construction to othermirror models for Calabi-Yau resp. nef-complete intersections inside a smooth toric variety X Σ . Thereader should be warned that a complete comparison of the construction presented in this paper to othermodels is not yet available, and will be subject to some future work. Nevertheless, we hope that thefollowing remarks indicate that our mirror model can be considered as a unification and generalizationof other constructions.First we consider a construction that can be found (although in a very sketchy form) in [Giv98b, page10-11]. Let as above X Σ be smooth, projective and toric, and let L = O X Σ ( L ) , . . . , L c = O X Σ ( L c )be nef line bundles such that − K X Σ − (cid:80) cj =1 L j is nef. Let again b , . . . , b t be the primitive integral10enerators of the rays of Σ (cid:48) . Notice that if a , . . . , a m are the generators of the rays of Σ, then t = m + c (namely, we have m generators b i projecting to the a (cid:48) i s , and c generators b i that projects to zero underΣ (cid:48) (cid:16) Σ.) Consider the affine space C m + c with coordinates w , . . . , w m + c . Let l , . . . , l r be a basis ofthe module of relations between the vectors b , . . . , b m + n (so that r = m − n , since the b i ’s ly in Z n + c ).Actually, this basis should not be chosen in an arbitrary way, it is the basis dual to the basis p , . . . , p r one of H ( X Σ , Z ) chosen in section 6 (see the discussion after the exact sequence (59) below). Write l a = ( l a , . . . , l at ) and consider the affine variety E := ( w, q ) ∈ ( C ∗ ) m × C c × KM ◦ | (cid:32) m + c (cid:89) i =1 w l ai i = q a (cid:33) a =1 ,...,r . Actually, Givental has a slightly different definition as he consider equivariant quantum cohomology, butwe ignore this aspect here and concentrate on the case of the non-equivariant limit. Notice also that inGivental’s paper appears only the restriction E q := E | ( C ∗ ) m × C c ×{ q } .It can be shown that the closure of E inside C m + c × KM ◦ equals E (see the argument in Proposition 4.8below), and that the projection E → KM ◦ ; ( w, q ) (cid:55)→ q is precisely the mapping α ◦ β ◦ γ : Z ◦ X aff → KM ◦ as appearing in the diagram (60) in section 6.In [Giv98b, page 10-11], Givental very briefly mentions the following oscillating integral (cid:90) Γ ⊂ E e τ · ( (cid:80) mi =1 w i − (cid:80) cj =1 w m + j ) · d log( w ) ∧ . . . ∧ d log( w m ) ∧ dw m +1 ∧ . . . ∧ dw m + c d log( q ) ∧ . . . ∧ d log( q m − n ) (5)where Γ is some real non-compact n + c -dimensional cycle inside E , i.e., a Lefschetz thimble. Noticehowever that E is singular in general, so that in any case one would need to specify further how to definethis cycle. It is claimed in loc.cit (and easily verified) for any relation l = ( l , . . . , l t ) with (cid:80) m + ci =1 l i b i = 0satisfying l ≥ , . . . , l m ≥
0, this integral is annihilated by the differential operators∆ l := m (cid:89) i =1 l i − (cid:89) ν =0 ( r (cid:88) a =1 l ai q a τ ∂ q a − τ ν ) − r (cid:89) a =1 q (cid:104) p a ,l (cid:105) a · m + c (cid:89) i = m +1 l i (cid:89) ν =1 ( r (cid:88) a =1 l ai τ q a ∂ q a + τ ν )In order to connect this statement to our construction, one needs to discuss the relation between oscillat-ing integrals and Gauß-Manin systems in some detail. This is a rather classical subject, although theredoes not seem to exist a general reference covering the present situation. One can find in [Pha83, Pha85]a definition of oscillating integrals for certain polynomial mappings, and in [Sab08, section 1.b] a dis-cussion of the topological Fourier-Laplace transformation which yields a cohomological description ofLefschetz thimbles.Assuming that this relation between oscillating integrals and Gauß-Manin systems is properly established,one may conjecture that the integral (5) yields a solution of the moduleFL loc C t (cid:0) H π † O S ×KM ◦ (cid:1) that appeared in Theorem 1.10. However, even if this were proved, it is still unclear whether this integralsatisfy a stronger differential equation (this has been noticed by Givental himself in [Giv98b, page 10]),namely, one would like to show that it is even a solution of the system FL loc C t (cid:0) H Π + M IC ( Z ◦ X ) (cid:1) . However,we do not have any further evidence at this point for this conjecture. There is a special case of the construction described in the last subsection, for which a more completedescription of a mirror model is available in the literature. Namely, assume that the nef line bundles L , . . . , L c are obtained as (line bundles associated to) a sum of some of the torus invariant divisors of X Σ . Then already in [Giv98b] is a sketch of how the above mentioned general construction of oscillatingintegrals can be made more precise. We will use the more recent paper [Iri11] as a references, which alsoincorporates ideas from [BB96b]. Let us give a very brief reminder of the part of [Iri11] relevant in thepresent situation. 11 efinition 1.12 ([Giv98b, Iri11]) . Let X Σ be a n -dimensional, smooth toric variety given by a fan Σ with torus-invariant divisors D , . . . , D m . A nef-partition is a partition { , . . . , m } = I (cid:116) I . . . (cid:116) I c suchthat L j = O ( (cid:80) k ∈ I j D k ) is nef for j = 0 , . . . , c . Notice that Iritani’s paper covers a larger domain of applications than this definition since he considers(nef partitions of) toric orbifolds. However, in the main body of our paper we are only concerned withcomplete intersections in manifolds, so we restrict to this situation here.To a nef-partition one associates as above a vector bundle E = (cid:76) cj =1 L j (notice that the sum here isrunning only from 1 to c and does not include the bundle L ). Choosing a generic section s ∈ Γ( X Σ , E )gives a smooth nef complete intersection Y ⊂ X Σ . Hence we see that the data of a toric manifold with anef partition are a particular case of the setup considered in the main part of our paper. We will show inremark 1.17 below an example which falls in the scope of this paper but which does not come from a nefpartition. In that sense our construction is a true generalization of the Givental-Iritani model. Noticealso that in the case I = ∅ the complete intersection Y is a Calabi-Yau manifold, this is exactly thesituation considered by Batyrev and Borisov in [BB96b].In the approach of Iritani the mirror model of Y is given by a function on a family of complete intersectionsof Laurent polynomials inside an n -dimensional torus ˇ T = ( C ∗ ) n with coordiantes t , . . . , t n . In order toconstruct this, one associates to each I j the following Laurent polynomial W ( j ) α = (cid:88) i ∈ I j α i t b i for j = 0 , . . . , c where the b i are again the primitive integral generators of the one-dimensional cones of Σ. The familyof complete intersections ˇ Y α over α ∈ ( C ∗ ) m is then given byˇ Y α = { t ∈ ˇ T | W (1) α = . . . = W ( c ) α = 1 } . Assumption 1.13 ([Iri11, page 2936]) . In the above situation, we suppose that the affine variety ˇ Y α isa smooth complete intersection in ˇ T for generic α ∈ ( C ∗ ) m . To the best of our knowledge, there is up to now no result available which would show how restrictivethis assumption is. There are some speculations in [Iri11, remark 5.6] that the smoothness of ˇ Y α shouldbe related to the smoothness of Y ⊂ X Σ , at least in the case I = ∅ . Definition 1.14.
Let X Σ and a nef partition { , . . . , m } = I (cid:116) I . . . (cid:116) I c be given. Suppose thatassumption 1.13 holds true. If I (cid:54) = ∅ , we call the restriction W (0) α : ˇ Y α −→ C the nef-partition Landau-Ginzburg model of ( X Σ , ( I j ) j =1 ,...,m ) . If I = ∅ we consider the (affine) Calabi-Yau complete intersection ˇ Y α ⊂ ˇ T (smooth by the above assumption) itself as the nef-partition mirrormodel . In the case I = ∅ , Baytrev and Borisov considered a compactification Y α of ˇ Y α inside the projetice toricvariety P ∇ , given by the polytope ∇ = ∇ + . . . + ∇ c with ∇ j = Conv( { b i } i ∈ I j ). Since the Calabi-Yauvarieties Y α are usual singular and the ambient variety P ∇ does not always admit a crepant resolutionBatyrev and Borisov introduced so-called string-theoretic Hodge numbers h p,qst ( Y α ) and could show in[BB96a] that h p,qst ( Y α ) = h n − p,qst ( Y ) for 0 ≤ p, q ≤ n where n = n − c is the dimension of Y .In the case where I is non-empty Iritani defines (under assumption 1.13) an oscillating integral (cid:90) Γ R ( α ) e − W (0) α ( t ) · τ Ω α where Γ R ( α ) = ˇ Y α ∩ ˇ T R is a non-compact cycle in ˇ Y α ( ˇ T R := ( R > ) n ⊂ ˇ T being the real torus), i.e., areal Lefschetz thimble, and Ω α = dt t ∧ . . . ∧ dt n t n dW (1) α ∧ . . . ∧ dW ( c ) α
12s a holomorphic volume form on ˇ Y α . Notice that the definition of the volume form uses the smoothnessassumption 1.13 in an essential way. Notice also that in loc.cit., the variable z := τ is used.In order to get a mirror theorem, Iritani defines a so-called A -period of the complete intersection Y ⊂ X Σ Π(1 , O Y ) = ( J Y ( q, − z ) , z n − deg z ρ ˆΓ Y )here J Y is the J -function, which is a particular solution of the quantum D -module of Y , and ˆΓ Y is theGamma class of Y . We refer the reader to [Iri11] for details. The aforementioned mirror theorem ofIritani is the equality (see [Iri11, Theorem 5.7])Π Y (1 , O Y ) = 1 F ( α ) (cid:90) Γ R ( α ) e − W (0) α ( t ) · τ Ω α α ∈ KM ◦ (6)where F ( α ) is a certain coordinate change.From the remarks in the last subsection on the relation between oscillating integrals and Gauß-Maninsystems it seems plausible that the oscillating integral on the right hand side of equation (6) gives asolution of the following (Fourier-Laplace transformed) Gauß-Manin systemFL loc H ( W (0) α ) + O ˇ Y α ∈ Mod hol ( D C τ )As before, we need to consider families of such differential systems by letting the parameter α vary within KM ◦ , hence, we rather look at the D C τ ×KM ◦ -moduleFL loc KM ◦ H ( W (0) α , pr α ) + O Y α ×KM ◦ Finally, we have to eliminate the asymmetry between W (0) α and W ( j ) α for j = 1 , . . . , c . This can beeasily done in the following way: As Y α = (cid:84) cj =1 (cid:16) W ( j ) α (cid:17) − (1), instead of considering the direct image H ( W (0) α ) + O Y α , we can consider the direct image H (cid:16) W (0) α , W (1) α , . . . , W ( c ) α (cid:17) + O ˇ T and restrict it to thesubspace where the last c coordinates are set to 1 (this follows from the base change theorem for holonomic D -modules, see theorem 2.1 below). Finally, in order to be able to use the Radon transformation functoralluded to above, we need to allow α to vary within C m , and not just in the subspace KM ◦ . Thismotivates the following definition. Definition 1.15.
Let us be given a nef partition { , . . . , m } = I (cid:116) I . . . (cid:116) I c on X Σ . Then we call themorphism Θ : ˇ T × C m −→ C t +1 = C × C m × C c ( t, α ) (cid:55)→ ( λ , . . . , λ m + c ) = ( − W (0) α , α, − W (1) α , . . . , − W ( c ) α ) the nef partition Landau-Ginzburg model of ( X Σ , ( I j ) j =0 ,...,c )We proceed by comparing the nef partition Landau-Ginzburg model to our construction, as outlinedbefore in this introduction. First notice that if we consider the variety X Σ , then the collection of linebundles L = O ( (cid:80) k ∈ I D k ) , . . . , L c = O ( (cid:80) k ∈ I c D k ) satisfies (almost) the assumptions of our construc-tion: Namely we have that − K X Σ − (cid:80) cj =1 (cid:16)(cid:80) k ∈ I j D k (cid:17) is nef, since it is simply equal to (cid:80) k ∈ I D k andthen its nefness follows from the nef partition assumption. For our main result (Theorem 6.13, see alsoTheorem 1.10), we need the stronger assumption that L , . . . , L c are ample but this is unnecessary formany intermediate results.In any case, given a nef partion, we have nef line bundles L , . . . , L c on X Σ such that the bundle O X Σ ( − K X Σ ) ⊗ L − ⊗ . . . ⊗ L − c is nef, and we can consider the matrix B as constructed in the beginningof this introduction from the primitive integral generators of the fan Σ (cid:48) of the total space V ( E ∨ ) with E = ⊕ cj =1 L j . If we denote by B j the matrix with columns ( b i ) i ∈ I j for j = 0 , . . . , c , then B is an( n + c ) × ( m + c ) integer matrix given by B := B B B · · · B c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
13s before we get an associated family of Laurent polynomials ϕ : S × C t −→ C × C t ( y , . . . , y s , λ , . . . , λ t ) (cid:55)→ − W (0) λ − c (cid:88) i =1 ( W ( i ) λ + λ m + i ) · y n + i , remember that S = ( C ∗ ) s where s := n + c and t := m + c .With these definition, we can state the following conjectural relationship of the nef-partion Landau-Ginzburg to our non-affine Landau-Ginzburg model. Conjecture 1.16.
Consider a smooth projective toric variety X Σ with torus invariant divisors D , . . . , D m and a nef partition { , . . . , m } = I (cid:116) I . . . (cid:116) I c . Then1. There exists a morphism FL loc C t ( H Θ + O ˇ T × C m ) −→ FL loc C t ( H ϕ + O S × C t ) (cid:39) FL loc C t ( M (cid:101) B ) . of holonomic D (cid:98) C t +1 -modules.2. This morphism induces an epimorphism D (cid:98) C t +1 -modules FL loc C t ( W min H Θ + O ˇ T × C m ) (cid:16) FL loc C t ( W min H ϕ + O S × C t ) (cid:39) im (cid:16) FL loc C t ( M (cid:101) γ (cid:101) B ) D −→ FL loc C t ( M (cid:101) B ) (cid:17) where W min is the minimal step of the weight filtration on the Gauß-Manin system H Θ + O ˇ T × C m resp. H ϕ + O S × C t (which underlies a mixed Hodge modules, i.e., an element of the abelian categoryMHM ( D C t +1 ) ). We are actually able to show the first part of this conjecture, but the proof is far to technical to bereproduced here. The second part is still open. Some evidence for this part of the conjecture comesfrom the fact that the D C t +1 -module W min H ϕ + O S × C t (which underlies a pure Hodge module) is, aswe will see later, irreducible, and hence the above morphism must be surjective if it is not the zero map.Using the mirror symmetry statement of [Iri11] together with our main result, one may even speculatefurther that this map must be an isomorphism. We postpone a thorough discussion of these matters toa subsequent paper. Remark 1.17.
We give an example of a smooth toric variety X Σ with two nef line bundles L = O X Σ ( L ) , L = O X Σ ( L ) and − K X Σ − L − L = 0 , which is not representable as a nef partition.Consider the two-dimensional toric variety given by the fan b b b b b b b b The primitive generators of the rays give rise to a matrix A = (cid:18) − − − − − − (cid:19) The rows of the following matrix L = − − −
10 0 0 1 0 0 1 −
10 0 0 0 1 0 1 00 0 0 0 0 1 0 1 rovide a basis for the module of relations among the columns of A . The well-known sequence −→ M A t −→ (cid:77) i =1 Z D i L t −→ H ( X, Z ) −→ for torus-invariant Weil divisors endows the free Z -module H ( X, Z ) (which has rank ) with a basis.The coordinates of the image [ D i ] of D i with respect to this basis are given by the i -th column vector ofthe matrix L . The closure of the K¨ahler cone is generated by the vectors , − , , , , , , , − , − , , − One easily sees that the vector [ L ] := − and [ L ] := , which lie in the closure of the K¨ahler cone, are not of the type [ D i ] + . . . + [ D i s ] for { i , . . . , i s } ⊂{ , . . . , } . But [ L ] + [ L ] = [ D ] + . . . + [ D ] = [ − K X Σ ] Therefore the line bundles L := O (2 D + D + D ) and L := O (2 D + D + D ) as well as − K X Σ − L − L are nef.Notice that although most of the constructions of our article apply to this example, it does not satisfy theassumptions of our main theorem 6.13 simply because the bundles L and L are nef but not ample. Wecould also give an example which consists of ample line bundles on a toric variety that do not come froma nef partition, but which would even be more complicated. Actually, we need ampleness only to applyresults from quantum cohomology (like those presented in [MM11]), whereas for the constructions of thepresent paper, the nef assumption is sufficient. We thank Claude Sabbah for continuing support and interest in our work, Hiroshi Iritani, Etienne Mannand Thierry Mignon for useful discussions and for sending us an early version of their paper [IMM16].We are very grateful to Takuro Mochizuki for his careful reading of the paper and for several very usefulremarks. Special thanks go to J¨org Sch¨urmann for pointing us to the reference [KW06].We thank the anonymous referees for pointing out a number of inaccuracies and for suggesting severalimprovements of the paper.
In this section we use the comparison result between Gauß-Manin systems of Laurent polynomials andGKZ-systems from [Rei14] to describe the direct image of the intersection complex of a natural compact-ification of a generic family of Laurent polynomials. The input data is an integer matrix B of maximalrank and the GKZ-system in question will be defined by a certain homogenized matrix (cid:101) B . The maintool is the Radon transformation resp. the Fourier-Laplace transformation for monodromic D -modules([Bry86]).We start by a short remainder on some basic notions from the theory of algebraic D -modules. Then wediscuss Gauß-Manin systems, GKZ-systems and intersection cohomology D -modules associated to the15bove mentioned families. Finally, we show using some facts about quasi-equivariant D -modules thatmost of the objects considered here behave well with respect to a natural torus action on the parameterspace of the families of Laurent polynomials resp. of their compactification. We review very briefly some basic results from the theory of algebraic D -modules, which will be neededlater. Let X be a smooth algebraic variety (we only consider algebraic varieties defined over C in thepaper) of dimension n and D X be the sheaf of algebraic differential operators on X . We denote by M ( D X ) the abelian category of algebraic D X -modules on X and the abelian subcategory of (regular)holonomic D X -modules by M h ( D X ) (resp. ( M rh ( D X )). The full triangulated subcategory in D b ( D X )consisting of objects with (regular) holonomic cohomology is denoted by D bh ( D X ) (resp. D brh ( D X )).Let f : X → Y be a map between smooth algebraic varieties. Let
M ∈ D b ( D X ) and N ∈ D b ( D Y ) begiven, then we denote by f + M := Rf ∗ ( D Y←X L ⊗ M ) resp. f + N := D X →Y L ⊗ f − N the direct resp.inverse image for D -modules. Notice that the functors f + , f + preserve (regular) holonomicity (see e.g.,[HTT08, Theorem 3.2.3]). We denote by D : D bh ( D X ) → ( D bh ( D X )) opp the holonomic duality functor.Recall that for a single holonomic D X -module M , the holonomic dual is also a single holonomic D X -module ([HTT08, Proposition 3.2.1]) and that holonomic duality preserves regular holonomicity ([HTT08,Theorem 6.1.10]). For a morphism f : X → Y between smooth algebraic varieties we additionally definethe functors f † := D ◦ f + ◦ D and f † := D ◦ f + ◦ D .In [HTT08], the definition of the inverse image functors from above follows a different convention, whichis better adapted to the Riemann-Hilbert correspondence. Our functor f + corresponds to f † [dim( Y ) − dim( X )] from [HTT08, page 31], whereas our functor f † corresponds to f (cid:70) [dim( X ) − dim( Y )] fromloc.cit, Definition 3.2.13.Let i : Z → X be a closed embedding of a smooth subvariety of codimension d and j : U → X be theopen embedding of its complement. This gives rise to the following triangles for
M ∈ D brh ( D X ) i + i + M [ − d ] −→ M −→ j + j + M +1 −→ , (7) j † j † M −→ M −→ i † i † M [ d ] +1 −→ . (8)The first triangle is [HTT08, Proposition 1.7.1] and the second triangle follows by dualization. We willoften use the following base change theorem. Theorem 2.1 ([HTT08, Theorem 1.7.3]) . Consider the following cartesian diagram of algebraic varieties Z f (cid:48) (cid:47) (cid:47) g (cid:48) (cid:15) (cid:15) W g (cid:15) (cid:15) Y f (cid:47) (cid:47) X then we have the canonical isomorphism f + g + [ d ] (cid:39) g (cid:48) + f (cid:48) + [ d (cid:48) ] , where d := dim Y − dim X and d (cid:48) :=dim Z − dim W . Remark 2.2.
Notice that by symmetry we have also the canonical isomorphism g + f + [ ˜ d ] (cid:39) f (cid:48) + g (cid:48) + [ ˜ d (cid:48) ] with ˜ d := dim W − dim X and ˜ d (cid:48) := dim Z − dim Y . In the former case we say we are doing a basechange with respect to f , in the latter case with respect to g . Remark 2.3.
Using the duality functor we get isomorphisms: f † g † [ − d ] (cid:39) g (cid:48)† f (cid:48)† [ − d (cid:48) ] and g † f † [ − ˜ d ] (cid:39) f (cid:48)† g (cid:48)† [ − ˜ d (cid:48) ] . In the sequel, we will consider Fourier-Laplace transformations of various D -modules. We give a shortreminder on the definition and basic properties of the Fourier-Laplace transformation. Let X be a smoothalgebraic variety, U be a finite-dimensional complex vector space and U (cid:48) its dual vector space. Denoteby E (cid:48) the trivial vector bundle τ : U (cid:48) × X → X and by E its dual. Write can : U × U (cid:48) → C for thecanonical morphism defined by can( a, ϕ ) := ϕ ( a ). This extends to a function can : E × E (cid:48) → C .16 efinition 2.4. Define L := O E (cid:48) × X E e − can , this is by definition the free rank one module with differentialgiven by the product rule. Denote by p : E (cid:48) × X E → E (cid:48) , p : E (cid:48) × X E → E the canonical projections. TheFourier-Laplace transformation is then defined byFL X ( M ) := p ( p +1 M L ⊗ L ) M ∈ D bh ( D E (cid:48) ) . If the base X is a point we will simply write FL. In general, the Fourier-Laplace transformation does notpreserve regular holonomicity. However, it does preserve regular holonomicity for the derived categoryof complexes of D -modules the cohomology of which are so-called monodromic D -modules. We will givea short reminder on this notion. Let χ : C ∗ × E (cid:48) → E (cid:48) be the natural C ∗ action on the fiber U (cid:48) and let θ be a coordinate on C ∗ . We denote the push-forward χ ∗ ( θ∂ θ ) as the Euler vector field E . Definition 2.5. [Bry86] A regular holonomic D E (cid:48) -module M is called monodromic, if the Euler field E acts locally finite on τ ∗ ( M ) , i.e. for a local section v of τ ∗ ( M ) the set E n ( v ) , ( n ∈ N ) , generates afinite-dimensional vector space. We denote by D bmon ( D E (cid:48) ) the derived category of bounded complexes of D E (cid:48) -modules with regular holonomic and monodromic cohomology. Theorem 2.6. [Bry86]1. FL X preserves complexes with monodromic cohomology.2. In D bmon ( D E (cid:48) ) we have FL X ◦ FL X (cid:39) Id and D ◦ FL X (cid:39) FL X ◦ D . FL X is t -exact with respect to the natural t -structure on D bmon ( D E (cid:48) ) resp. D bmon ( D E ) .Proof. The above statements are stated in [Bry86] for constructible monodromic complexes. One hasto use the Riemann-Hilbert correspondence [Bry86, Proposition 7.12, Theorem 7.24] to translate thestatements. So the first statement is Corollaire 6.12, the second statement is Proposition 6.13 and thethird is Corollaire 7.23 in [Bry86].We will make occasionally use of the so-called R -modules. More precisely, let M be a smooth algebraicvariety and consider the product of M with the affine line C z where z is a fixed coordinate. Then bydefinition R C z × M is the O C z × M -subalgebra of D C z × M locally generated by z ∂ z and by z∂ x , . . . , z∂ x n where ( x , . . . , x n ) are local coordinates on M . Notice that j ∗ M R C z × M ∼ = D C ∗ z × M , where j M : C ∗ z × M (cid:44) → C z × M is the canonical open embedding.We will also consider the O C z × M -subalgebra R (cid:48) C z × M of R C z × M which is locally generated by z∂ x , . . . , z∂ x n only. Sometimes we omit the subscript which denotes the underlying space, so we write R resp. R (cid:48) in-stead of R C z × M resp. R (cid:48) C z × M . The inclusion R (cid:48) (cid:44) → R induces a functor from the category of R -modulesto the category of R (cid:48) -modules, which we denote by For z ∂ z (“forgetting the z ∂ z -structure”). D -modules and the Radon trans-formation In this subsection we adapt some results from [Rei14] to our situation. More precisely, for a givengeneric family of Laurent polynomials, we describe the canonical morphism between its Gauß-Manin-systems with compact support and its usual Gauß-Manin-systems. This mapping can be expressed asa morphism between the corresponding GKZ-systems. We will use this result in the next subsection todescribe certain intersection cohomology modules.We start by fixing our initial data and by introducing the GKZ-hypergeometric D -modules. Let B be a s × t -integer matrix such that the columns of B , which we denote by ( b , . . . , b t ), generate Z s . Considerthe torus S = ( C ∗ ) s and the t + 1-dimensional vector space V (with coordinates λ , λ , . . . , λ t ) as well asits dual V (cid:48) (with coordinates µ , µ , . . . , µ t ). Define the map g : S −→ P ( V (cid:48) ) , ( y , . . . , y s ) (cid:55)→ (1 : y b , . . . , y b t ) , (9)17here y b i := (cid:81) sk =1 y b ki k for i ∈ { , . . . , t } . The condition on the columns of the matrix B ensures thatthis is an embedding. If we denote the closure of the image of g in P ( V (cid:48) ) by X , then X is a (possiblynon-normal) toric variety in the sense of [GKZ08, Chapter 5]. So we have the following sequence of maps S j −→ X i −→ P ( V (cid:48) ) , (10)where j is an open embedding and i a closed embedding.We will denote the homogeneous coordinates on P ( V (cid:48) ) by ( µ : . . . : µ t ). Let Q be the convex hull of theelements { b = 0 , b , . . . , b t } in R s . Then by [GKZ08, Chapter 5, Prop 1.9] the projective variety X hasa natural stratification by torus orbits X (Γ), which are in one-to-one correspondence with faces Γ ofthe polytope Q . The orbit X (Γ) is isomorphic to ( C ∗ ) dim(Γ) and is specified inside X by the conditions µ i = 0 for all b i / ∈ Γ , µ i (cid:54) = 0 for all b i ∈ Γ . (11)In particular the torus S ⊂ X is given by the face Γ = Q , i.e. by the equations µ i (cid:54) = 0 for all i ∈ { , . . . , t } .To this setup we associate the following D -modules. Write W = C t with coordinates λ , . . . , λ t so that V = C λ × W . Definition 2.7 ([GKZ90], [Ado94]) . Consider a lattice Z s and vectors b , . . . , b t ∈ Z s . Moreover, let β = ( β , . . . , β s ) be an element in C s . Write L for the module of relations among the columns of B . Anyelement l ∈ L will be written as a vector l = ( l , . . . , l t ) in Z t . Define M βB := D W / (cid:0) ( (cid:3) l ) l ∈ L + ( E k − β k ) k =1 ,...s (cid:1) , where (cid:3) l := (cid:81) i : l i < ∂ − l i λ i − (cid:81) i : l i > ∂ l i λ i , l ∈ L E k := (cid:80) si =1 b ki λ i ∂ λ i , k ∈ { , . . . , s } where b ki is the k -th component of b i . The D W -module M βB is called a GKZ-system. As GKZ-systems are defined on the affine space W , we will often work with the D W -modules of global sec-tions M βB := Γ( W, M βB ) rather than with the sheaves themselves, where D W = C [ λ , . . . , λ t ] (cid:104) ∂ λ , . . . , ∂ λ t (cid:105) is the Weyl algebra.We will also consider a homogenization of the systems above. Let (cid:101) B be the ( s + 1) × ( t + 1) integermatrix with columns (cid:101) b := (1 , , (cid:101) b := (1 , b ) , . . . , (cid:101) b t := (1 , b t ). Definition 2.8.
Consider the hypergeometric system M (cid:101) β (cid:101) B on V = C t +1 associated to the vectors (cid:101) b , (cid:101) b , . . . , (cid:101) b t ∈ Z s +1 and (cid:101) β ∈ C s +1 . More explicitly, M (cid:101) β (cid:101) B := D V / I , where I is the sheaf of left ide-als in D V defined by I := D V ( (cid:3) l ) l ∈ L + D V ( E k − β k ) k =0 ,...,s , where (cid:3) l := ∂ lλ · (cid:81) i : l i < ∂ − l i λ i − (cid:81) i : l i > ∂ l i λ i if l ≥ , (cid:3) l := (cid:81) i : l i < ∂ − l i λ i − ∂ − lλ · (cid:81) i : l i > ∂ l i λ i if l < ,E k := (cid:80) ti =1 b ki λ i ∂ λ i ,E := (cid:80) ti =0 λ i ∂ λ i . The generic rank of the GKZ-systems M βB resp. M (cid:101) β (cid:101) B may be difficult to predict depending the pa-rameter (see, e.g., [MMW05]), but if we suppose that the matrix B resp. (cid:101) B satisfies the normalityassumption (see Proposition 5.1 and its proof below), then it is known that the rank of both modules18quals s ! · vol(Conv( b , . . . , b t )) (where vol denotes the normalized volume, i.e. such that the volume ofthe hypercube is [0 , d is one).Let h be the map given by h : T −→ V (cid:48) , (12)( y , . . . , y s ) (cid:55)→ ( y (cid:101) b , . . . , y (cid:101) b t ) = ( y , y y b , . . . , y y b t ) , where T = C ∗ × S = ( C ∗ ) s +1 . Notice that the restriction of h to { } × S is exactly the map g fromformula (9), when seen as a map to the affine chart { µ = 1 } ⊂ P ( V (cid:48) ). We will later also need theclosure of the image of h in V (cid:48) , which we denote by Y . Hence Y is the affine cone over X .As a piece of notation, for any matrix C = ( c , . . . , c k ), we write N C for the semi-group generated bythe columns c , . . . , c k , that is N C := (cid:80) ki N c i , where we adopt the convention that the set N of naturalnumbers contains the element 0. Then we can consider the semi-group ring C [ N (cid:101) B ], which is naturally Z -graded due to the first line of the matrix (cid:101) B . Hence we can consider the ordinary spectrum of thisring as well as its projective spectrum, and it is clear that we have Y = Spec C [ N (cid:101) B ] and X = Proj C [ N (cid:101) B ].We will now consider natural D V -linear maps between GKZ-systems, which will induce a shift of theparameter. Let (cid:101) B be as above and consider the map of monoids ρ : N t +1 −→ N (cid:101) B (13) e i (cid:55)→ (cid:101) b i where the e i are the standard generators of N t +1 . Let c ∈ N t +1 be given and put (cid:101) γ := ρ ( c ). Notice thatfor every (cid:101) β ∈ C s +1 the morphism M (cid:101) β (cid:101) B −→ M (cid:101) β + (cid:101) γ (cid:101) B P (cid:55)→ P · ∂ c is well-defined. Now let c , c ∈ ρ − ( (cid:101) γ ). Because c and c map to the same image, their difference c − c is a relation l among the columns of the matrix (cid:101) B , thus ∂ c − ∂ c ∈ ( (cid:3) l ). This shows that P · ∂ c = P · ∂ c in M (cid:101) β + (cid:101) γ (cid:101) B . Thus, we are lead to the following definition. Definition 2.9.
Let (cid:101) B and (cid:101) β be as above. For every (cid:101) γ ∈ N (cid:101) B define the morphism M (cid:101) β (cid:101) B · ∂ (cid:101) γ (cid:47) (cid:47) M (cid:101) β + (cid:101) γ (cid:101) B given by right multiplication with ∂ c for any c ∈ ρ − ( (cid:101) γ ) . In the next lemma, we establish a relation between a direct image under this morphism h and theGKZ-systems just introduced. Lemma 2.10.
There exists a δ (cid:101) B ∈ N (cid:101) B such that we have an isomorphism a : FL( h + O T ) (cid:39) −→ M (cid:101) β (cid:101) B (14) for every (cid:101) β ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) . Furthermore, we have a dual isomorphism a ∨ : FL( h † O T ) (cid:39) ←− M − (cid:101) β (cid:48) (cid:101) B (15) for every (cid:101) β (cid:48) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . For every (cid:101) β, (cid:101) β (cid:48) as above, the diagram below commutes up to a non-zeroconstant M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) (cid:47) (cid:47) (cid:39) a ∨ (cid:15) (cid:15) M (cid:101) β (cid:101) B FL( h † O T ) (cid:47) (cid:47) FL( h + O T ) , (cid:39) a (cid:79) (cid:79) where the lower horizontal morphism is induced by the natural morphism h † O T → h + O T . roof. By [SW09, Corollary 3.7] we have the isomorphism FL( h + ( O T · y (cid:101) β )) (cid:39) M (cid:101) β (cid:101) B for every (cid:101) β / ∈ sRes ( (cid:101) B )where sRes ( (cid:101) B ) is the set of so-called strongly resonant parameters ([SW09, Definition 3.4]). Here O T · y (cid:101) β is again the free rank one module with differential given by the product rule. Using [Rei14, Lemma 1.16],which says that there exists an δ (cid:101) B ∈ N (cid:101) B such that δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) ∩ sRes ( (cid:101) B ) = ∅ and the factthat O T (cid:39) O T · y (cid:101) γ for every (cid:101) γ ∈ Z s +1 , the first statement follows. The second statement follows fromtaking the holonomic dual of (14), namely, we put a ∨ := D a : D M (cid:101) β (cid:101) B (cid:39) −→ D FL( h + O T ) (cid:39) FL( D h + O T ) (cid:39) FL( h † O T )and then we conclude by applying [Rei14, Proposition 1.23].The last statement follows from the fact that the only non-zero morphism between M − (cid:101) β (cid:48) (cid:101) B and M (cid:101) β (cid:101) B isright multiplication ∂ (cid:101) β + (cid:101) β (cid:48) up to a non-zero constant (cf. [Rei14, Proposition 1.24]).We will denote by Z ⊂ P ( V (cid:48) ) × V the universal hyperplane given by Z := { (cid:80) ti =0 λ i µ i = 0 } and by U := ( P ( V (cid:48) ) × V ) \ Z its complement. Consider the following diagram U π U (cid:40) (cid:40) π U (cid:117) (cid:117) (cid:127) (cid:95) j U (cid:15) (cid:15) P ( V (cid:48) ) P ( V (cid:48) ) × V π (cid:111) (cid:111) π (cid:47) (cid:47) V ,Z π Z (cid:105) (cid:105) (cid:63)(cid:31) i Z (cid:79) (cid:79) π Z (cid:54) (cid:54) We will use in the sequel several variants of the so-called Radon transformation. These are functors from D brh ( D P ( V (cid:48) ) ) to D brh ( D V ) given by R ( M ) := π Z ( π Z ) + M (cid:39) π i Z + i + Z π +1 M , R ◦ ( M ) := π U ( π U ) + M (cid:39) π j U + j + U π +1 M , R ◦ c ( M ) := π U † ( π U ) + M (cid:39) π j U † j + U π +1 M , R cst ( M ) := π ( π ) + M ,
The adjunction triangle corresponding to the open embedding j U and the closed embedding i Z gives riseto the following triangles of Radon transformations. R [ − M ) −→ R cst ( M ) −→ R ◦ ( M ) +1 −→ , (16) R ◦ c ( M ) −→ R cst ( M ) −→ R [1]( M ) +1 −→ , (17)where the second triangle is dual to the first.We can now introduce the generic family of Laurent polynomials mentioned at the beginning of thissubsection. It is defined by the columns of the matrix B , more precisely, we put ϕ B : S × W −→ V = C λ × W , (18)( y , . . . , y s , λ , . . . , λ t ) (cid:55)→ ( − t (cid:88) i =1 λ i y b i , λ , . . . , λ t ) . The following theorem of [Rei14] constructs a morphism between the Gauß-Manin system H ( ϕ B, + O S × W )resp. the its proper version H ( ϕ B, † O S × W ) and certain GKZ-hypergeometric systems. For this we applythe triangle (16) to M = g † O S and the triangle (17) to M = g + O S , which gives us the result.20 heorem 2.11. [Rei14, Lemma 1.16, Theorem 2.7] There exists an δ (cid:101) B ∈ N (cid:101) B such that for every (cid:101) β ∈ δ (cid:101) B + R ≥ (cid:101) B ∩ Z s +1 and every (cid:101) β (cid:48) ∈ ( N (cid:101) B ) ◦ = N (cid:101) B ∩ ( R ≥ (cid:101) B ) ◦ , the following sequences of D V -modulesare exact and dual to each other: H s − ( S, C ) ⊗ O V H ( ϕ B, + O S × W ) M (cid:101) β (cid:101) B H s ( S, C ) ⊗ O V (cid:47) (cid:47) H − ( R cst ( g + O S )) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) H ( R ( g + O S )) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) H ( R ◦ c ( g + O S )) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) H ( R cst ( g + O S )) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) H ( R cst ( g † O S )) (cid:111) (cid:111) (cid:39) (cid:15) (cid:15) H ( R ( g † O S )) (cid:111) (cid:111) (cid:39) (cid:15) (cid:15) H ( R ◦ ( g † O S )) (cid:111) (cid:111) (cid:39) (cid:15) (cid:15) H ( R cst ( g † O S )) (cid:111) (cid:111) (cid:39) (cid:15) (cid:15) . (cid:111) (cid:111) H s +1 c ( S, C ) ⊗ O V H ( ϕ B, † O S × W ) M − (cid:101) β (cid:48) (cid:101) B H sc ( S, C ) ⊗ O V If moreover N (cid:101) B is saturated, then the vector δ (cid:101) B can be taken to be ∈ N (cid:101) B , in particular, the abovestatement holds for (cid:101) β = 0 ∈ Z s +1 . Thus we get the following exact 4-term sequences which can be connected vertically by the map η : H ( R ( g † O S )) → H ( R ( g + O S )) induced by the natural morphism g † O S → g + O S . Define θ to be thecomposition κ ◦ η ◦ κ . The next result gives a concrete description of this morphism:0 (cid:47) (cid:47) H s − ( S, C ) ⊗ O V (cid:47) (cid:47) H ( R ( g + O S )) κ (cid:47) (cid:47) M (cid:101) β (cid:101) B (cid:47) (cid:47) H s ( S, C ) ⊗ O V (cid:47) (cid:47) H s +1 c ( S, C ) ⊗ O V (cid:111) (cid:111) H ( R ( g † O S )) (cid:111) (cid:111) η (cid:79) (cid:79) M − (cid:101) β (cid:48) (cid:101) Bκ (cid:111) (cid:111) θ (cid:79) (cid:79) H sc ( S, C ) ⊗ O V (cid:111) (cid:111) . (cid:111) (cid:111) Lemma 2.12.
The morphism θ is induced by right multiplication with ∂ (cid:101) β + (cid:101) β (cid:48) up to a non-zero constant.Proof. Once we can prove that κ ◦ η ◦ κ is not equal to zero we apply a rigidity result of [Rei14,Proposition 1.24] which says that the only maps between M − (cid:101) β (cid:48) (cid:101) B and M (cid:101) β (cid:101) B is right-multiplication with c · ∂ (cid:101) β + (cid:101) β (cid:48) for c ∈ C . We only have to show that κ ◦ η ◦ κ becomes an isomorphism after micro-localizing with respect to ∂ · · · ∂ t . This is sufficient as the microlocalization of the GKZ-systems M (cid:101) β (cid:101) B resp. M − (cid:101) β (cid:48) (cid:101) B are not zero for otherwise the sheaves h + O T and h † O T would be supported on the divisor { µ · µ · . . . · µ t = 0 } , which is obviously wrong.It is clear that κ and κ become isomorphisms after (micro-)localization with respect to ∂ · · · ∂ t becausethese maps have O V -free kernel and cokernel. It remains to prove that η is an isomorphism afterthis micro-localization. To prove this we will use a theorem of [DE03] which compares the Radontransformation with the Fourier-Laplace transformation for D -modules. Consider the following diagram T h (cid:47) (cid:47) ˜ h (cid:35) (cid:35) π T (cid:15) (cid:15) V (cid:48) Bl ( V (cid:48) ) p (cid:111) (cid:111) q (cid:2) (cid:2) V (cid:48) \ { } j (cid:79) (cid:79) π (cid:15) (cid:15) S g (cid:47) (cid:47) P ( V (cid:48) )21here Bl ( V (cid:48) ) ⊂ P ( V (cid:48) ) × V (cid:48) is the blow-up of 0 in V (cid:48) and q is the restriction of the projection to thefirst component. Notice that the map h : T → V (cid:48) from formula (12) factors via V (cid:48) \{ } , that is, we have h = j ◦ (cid:101) h , where j : V (cid:48) \{ } (cid:44) → V (cid:48) is the canonical inclusion.It follows from [DE03, Proposition 1] that we have the following isomorphism R ( g + O S ) (cid:39) FL( p + q + g + O S ) (19)and its holonomic dual R ( g † O S ) (cid:39) FL( p + q + g † O S ) , (20)where we have used R ◦ D ∼ = D ◦ R , FL ◦ D ∼ = D ◦ FL, p + ◦ D ∼ = D ◦ p + ( p is proper) and q + ◦ D ∼ = D ◦ q + ( q is smooth). Recall that we want to show that the morphism H ( R ( g † O S )) η −→ H ( R ( g + O S )) , becomes an isomorphism after localization with respect to ∂ λ · · · ∂ λ t . Using the isomorphisms (19) and(20) and the fact that FL is an exact functor and that it exchanges the action of µ i and ∂ λ i we see thatit is enough to show that H ( p + q + g † O S ) −→ H ( p + q + g + O S ) (21)becomes an isomorphism after localization with respect to µ · · · µ t . In other words, we have to showthat the kernel and the cokernel of the morphism (21) are supported on { µ · · · µ t = 0 } ⊂ V (cid:48) . Obviously,we have { } ⊂ { µ · . . . · µ t = 0 } and hence V (cid:48) \{ µ · . . . · µ t = 0 } ⊂ V (cid:48) \{ } . It is thus sufficientto show that kernel and cokernel of the restriction of the morphism (21) to V (cid:48) \{ } are supported on { µ · . . . · µ t = 0 }\{ } . Notice that the restriction of H ( p + q + g † O S ) resp. H ( p + q + g + O S ) to V (cid:48) \ { } isisomorphic to H ( π + g † O S ) resp. H ( π + g + O S ). Thus the kernel and the cokernel of (21) are supportedon { µ · · · µ t = 0 } if and only if kernel and cokernel of H ( π + g † O S ) −→ H ( π + g + O S )are supported on { µ · · · µ t = 0 }\{ } . The map π is smooth and therefore π + is an exact functor. It istherefore enough to show that kernel and cokernel of H ( g † O S ) −→ H ( g + O S )are supported on { µ . . . µ t = 0 } ⊂ P ( V (cid:48) ). But this follows from the description of the map g , namely, bythe remark right after equation (11) the support of the cone of the morphism g † O S → g + O S is containedin { µ . . . µ t = 0 } . D -modules As mentioned in the beginning of this section, our aim is to describe a D V -module derived from theintersection complex of a natural compactification of the family of Laurent polynomials ϕ B as definedin formula (18). This module will actually appear as the Radon transformation of the ( D -modulecorresponding to the) intersection complex of the variety X ⊂ P ( V (cid:48) ).We start by fixing some notations concerning these D -modules. Let P be a smooth variety and U ⊂ P be a smooth subvariety, write X for the closure of U inside P , j U : U (cid:44) → X for the open embedding of U in X and i X : X → P for the closed embedding of the closure of X in P . Consider the abelian categoryPerv( P ) of perverse sheaves on P (with respect to middle perversity). For a reference about the definitionand basic properties of perverse sheaves, see [Dim04]. Recall that the simple objects in Perv( P ) are theobjects ( i X ) ! IC ( X , L ) where L is an irreducible local system on U and IC ( X , L ) is the intersectioncomplex of X with coefficient in L , that is the image of the morphism p H (( j U ) ! L ) → p H (( Rj U ) ∗ L ) in P erv ( X ). We will denote the corresponding D -module on P by M IC ( X , L ). If L is the constant sheaf C U we will simply write M IC ( X ). The p -th intersection cohomology group of X (see [GM83]) is denoted by IH p ( X ) and is obtained from the intersection complex by the formula IH p ( X ) = H p − dim( X ) ( IC ( X , C U )).We will apply this formalism to the special situation where U = g ( S ) (where g is the embedding defined byformula (9)), X = X and P = P ( V (cid:48) ). The module M IC ( X ) is the image of the morphism g † O S → g + O S .In the next result, we will compute the Radon transformation of this module.22 roposition 2.13. In the above situation, we have the following (non-canonical) isomorphism of D V -modules H R ( M IC ( X )) (cid:39) M IC ( X ◦ , L ) ⊕ ( IH s − ( X ) ⊗ O V ) , and H i R ( M IC ( X )) (cid:39) IH i + s +1 ( X ) ⊗ O V for i > , H i R ( M IC ( X )) (cid:39) IH i + s − ( X ) ⊗ O V for i < , where X ◦ is some subvariety of V and L some local system on some smooth open subset of X ◦ .Proof. Using the comparison isomorphism between the Radon transformation and the Fourier-Laplacetransformation (equation (19)) from above, we have H i R ( M IC ( X )) (cid:39) H i FL( p + q + M IC ( X )) (cid:39) FL H i ( p + q + M IC ( X )) (cid:39) FL H i ( p + M IC ( q − ( X ))) , where the second isomorphism follows from the exactness of FL and the last isomorphism follows fromthe smoothness of q . We now apply the decomposition theorem [Sai88, corollaire 3, equation 0.12] whichgives H i ( p + M IC ( q − ( X ))) (cid:39) (cid:77) k M IC ( Y ik , L ik ) (22)for some subvarieties Y ik ⊂ V (cid:48) and some local systems L ik on a Zariski open subset of Y ik . Notice that j +0 H i ( p + M IC ( q − ( X ))) (cid:39) j +0 H i ( p + q + M IC ( X )) (cid:39) H i ( j +0 p + q + M IC ( X )) (cid:39) H i ( π + M IC ( X )) (cid:39) H i ( M IC ( π − ( X ))) , which is equal to 0 for i (cid:54) = 0 and equal to M IC ( Y \ { } ) for i = 0 (recall from subsection 2.2, moreprecisely, from the discussion before Lemma 2.10, that Y is the cone of X in V (cid:48) ). Thus the decompositionfrom (22) becomes H ( p + M IC ( q − ( X ))) (cid:39) M IC ( Y ) ⊕ S , resp. H i ( p + M IC ( q − ( X ))) (cid:39) S i i (cid:54) = 0 , where the S i are D -modules with support at 0, i.e. S i (cid:39) i S i , where the S i are finite-dimensional vectorspaces and i : { } → V (cid:48) is the natural embedding. We now use the fact that FL is an equivalence ofcategories, which means that it transforms simple object to simple objects, so we set M IC ( X ◦ , L ) := FL( M IC ( Y )) . (23)It also transforms D -modules with support at 0 to free O -modules, i.e. FL( S i ) (cid:39) S i ⊗ O V . In order toshow the claim, we have to compute the S i . Recall that we have p + q + M IC ( X ) (cid:39) (cid:77) j H j ( p + q + M IC ( X ))[ − j ] (cid:39) (cid:77) j (cid:54) =0 S j [ − j ] ⊕ S ⊕ M IC ( Y ) , (24)where the first isomorphism is non-canonical. We compute H i ( a V (cid:48) ) + p + ( q + M IC ( X )) (cid:39) H i ( a P ) + q + ( q + M IC ( X )) (cid:39) H i ( a P ) + M IC ( X )[1] (cid:39) IH i + s +1 ( X )(here a V (cid:48) : V (cid:48) → { pt } resp. a P : P ( V (cid:48) ) → { pt } are the projections to a point), where the secondisomorphism follows from [KS94, Corollary 2.7.7 (iv)] and the Riemann-Hilbert correspondence. For the23ight hand side of Equation (24) we get H i ( a V (cid:48) ) + (cid:77) j (cid:54) =0 S j [ − j ] ⊕ S ⊕ M IC ( Y ) (cid:39) S i for i ≥ ,H i ( a V (cid:48) ) + (cid:77) j (cid:54) =0 S j [ − j ] ⊕ S ⊕ M IC ( Y ) (cid:39) S i ⊕ IH i + s +1 ( Y ) (cid:39) S i ⊕ IH i + s +1 p ( X ) for i < , where IH i + s +1 p ( X ) is the primitive part of IH i + s +1 ( X ) and where the last isomorphism follows from[KW06, Chapter 4.10]. Therefore we have S i (cid:39) IH i + s +1 ( X ) for i ≥ ,S i (cid:39) L ( IH i + s − ( X )) (cid:39) IH i + s − ( X ) for i < , where L : IH i + s − ( X ) → IH i + s +1 ( X ) is the Lefschetz operator which is injective for i ≤ λ ∈ V the Radon transformation R ( M IC ( X ))of M IC ( X ) measures the intersection cohomology of X ∩ H λ , where H λ is the hyperplane in P ( V (cid:48) )corresponding to λ . Proposition 2.14.
Let λ be a generic point of V and denote by i λ : { λ } −→ V its embedding. We havethe following isomorphism i + λ R ( M IC ( X )) (cid:39) R Γ( X ∩ H λ , IC X ∩ H λ ) , in particular H j ( i + λ R ( M IC ( X ))) (cid:39) IH j + s − ( X ∩ H λ ) . Proof.
Consider the following diagram where all squares are cartesian X i (cid:15) (cid:15) Z Xπ X (cid:111) (cid:111) η (cid:15) (cid:15) X ∩ H λi X (cid:111) (cid:111) η H (cid:15) (cid:15) P ( V (cid:48) ) Z π Z (cid:111) (cid:111) π Z (cid:15) (cid:15) H λπ H (cid:15) (cid:15) i H (cid:111) (cid:111) V { λ } i λ (cid:111) (cid:111) We have DR ( i + λ R ( M IC ( X ))) (cid:39) i ! λ Rπ Z ∗ ( π Z ) ! i ! IC ( X )[1] (cid:39) i ! λ Rπ Z ∗ Rη ∗ π X !1 IC ( X )[1] (cid:39) Rπ H ∗ i ! H Rη ∗ π X !1 IC ( X )[1] (cid:39) Rπ H ∗ Rη H ∗ i ! X π X !1 IC ( X )[1] (cid:39) R ( π H ◦ η H ) ∗ ( π X ◦ i X ) ! IC ( X )[1] (cid:39) R ( π H ◦ η H ) ∗ IC ( X ∩ H λ ) (cid:39) R Γ( X ∩ H λ , IC ( X ∩ H λ )) , where the first isomorphism follows from DR ◦ i + λ = i ! λ ◦ DR [ t + 1] and DR ◦ ( π Z ) + (cid:39) ( π Z ) ! ◦ DR [ − t ](see e.g. [HTT08, Theorem 7.1.1]), the second, third and fourth isomorphism follows from base change(see e.g. [Dim04, Theorem 3.2.13(ii)] and the sixth isomorphism follows from [GM83, Section 5.4.1](notice that their IC ( X ) is our IC ( X )[ n ] where n = dim C ( X )) and the fact that for a generic λ thehyperplane H λ is transversal to a given Whitney stratification of X . The first claim now follows fromthe fact that the de Rham functor DR is the identity on a point. The second claim follows from H j − s +1 ( X ∩ H λ , IC ( X ∩ H λ )) (cid:39) IH j ( X ∩ H λ ). 24 emark 2.15. Combining Proposition 2.13 and Proposition 2.14 we see that we have the followingdecomposition for generic λ ∈ V : IH s − ( X ∩ H λ ) (cid:39) H ( i + λ R ( M IC ( X ))) (cid:39) i + λ H ( R ( M IC ( X ))) (cid:39) i + λ M IC ( X ◦ , L ) ⊕ IH s − ( X ) . This is the intersection cohomology analogon of the decomposition of the cohomology of a smooth hyper-plane section of a smooth projective variety into its vanishing part and the ambient part.
We will now show that M IC ( X ◦ , L ) can expressed as an image of a morphism between GKZ-systems. Theorem 2.16.
Let (cid:101) β, (cid:101) β (cid:48) be as in Theorem 2.11, then M IC ( X ◦ , L ) (cid:39) im ( M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) −→ M (cid:101) β (cid:101) B ) .Proof. First recall that we have shown in the proof of Proposition 2.13. that M IC ( X ◦ , L ) (cid:39) FL( M IC ( Y )).On the other hand, as Y is the closure in V (cid:48) of the image of the morphism h , the module M IC ( Y ) isisomorphic to the image of h † O T → h + O T . As the Fourier-Laplace transformation is exact we canconclude that M IC ( X ◦ , L ) is isomorphic to the image of FL( h † O T ) → FL( h + O T ).By Lemma 2.10 we know that FL( h + O T ) is isomorphic to M (cid:101) β (cid:101) B for every (cid:101) β ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) andthat FL( h † O T ) is isomorphic to M − (cid:101) β (cid:48) (cid:101) B for every (cid:101) β (cid:48) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . It follows now from the laststatement of Lemma 2.10, that the induced morphism between M − (cid:101) β (cid:48) (cid:101) B and M (cid:101) β (cid:101) B is equal to · ∂ (cid:101) β + (cid:101) β (cid:48) up tosome non-zero constant.In general it is quite difficult to make any precise statement on the variety X ◦ and the local system L which define the module M IC ( X ◦ , L ). Nevertheless, if we restrict our attention to the situationwhere the matrix B defining the embedding g : S (cid:44) → P ( V (cid:48) ) is given by the primitive integral generatorsof a toric manifold which is given by a total bundle V ( E ∨ ) (cid:16) X Σ , where E is a split convex vectorbundle over another toric manifold X Σ such that the zero locus of a generic section is a nef completeintersection (i.e., the situation considered from section 4 on, see also the introduction in section 1), thenwe expect that X ◦ = V . In order to show this, one would need to prove that if we restrict the morphism M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) −→ M (cid:101) β (cid:101) B from the last theorem to a generic point of V , then it is not the zero map. It is wellknown (see, e.g., [Ado94]) that the restriction of M (cid:101) β (cid:101) B to a generic point is the quotient of C [ N (cid:101) B ] by theideal generated by the Euler vector fields (cid:80) ti =0 (cid:101) b ki λ i ∂ λ i ( k = 0 , . . . , s ), where now ( λ , . . . , λ t ) are thecomponents of the generic point we restrict to. Hence one needs to show that the monomial ∂ (cid:101) β + (cid:101) β (cid:48) doesnot lie in this ideal. Nevertheless, we do not have, at this moment, any further evidence for this to betrue.Even under the above restrictive assumptions on B and even if we suppose that X ◦ = V , it is not easyto predict the rank of L . What we expect is that the generic rank of the module (cid:99) M IC ( X ◦ , L ) fromtheorem 3.6 below can be identified with the dimension of the image of the map H ∗ ( X Σ , C ) ∪ c top ( E ) −→ H ∗ ( X Σ , C ) . However, the module M IC ( X ◦ , L ) resp. the local system L may contain constant subobjects that vanishafter localized Fourier-Laplace transformation (see section 3 below for more details). Hence its rank maybe different from that of (cid:99) M IC ( X ◦ , L ).For applications like those in the last section, we need a description of M IC ( X ◦ , L ) as a quotient of aGKZ-system, rather than submodule of it. For this purpose, denote by K the kernel of the morphism M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) −→ M (cid:101) β (cid:101) B , then M IC ( X ◦ , L ) is isomorphic to the quotient M − (cid:101) β (cid:48) (cid:101) B / K in the abelian category ofregular holonomic D -modules. The next result gives a concrete description of K as a submodule of M − (cid:101) β (cid:48) (cid:101) B .First, we define a sub- D V -module Γ ∂,c ( M − (cid:101) β (cid:48) (cid:101) B ) of M − (cid:101) β (cid:48) (cid:101) B , where c ∈ ρ − ( (cid:101) β + (cid:101) β (cid:48) ) (cf. Equation (13)) :Γ ∂,c ( M − (cid:101) β (cid:48) (cid:101) B ) := { m ∈ M − (cid:101) β (cid:48) (cid:101) B | ∃ n ∈ N with ( ∂ c ) n · m = 0 } Recall that two elements ∂ c and ∂ c with c , c ∈ ρ − ( (cid:101) β + (cid:101) β (cid:48) ) differ by some element P · (cid:3) l , where P ∈ C [ ∂ , . . . , ∂ s ] and l = c − c . Any element m ∈ M − (cid:101) β (cid:48) (cid:101) B is eliminated by left multiplication with25ome high enough power of P · (cid:3) l . This shows that Γ ∂,c ( M − (cid:101) β (cid:48) (cid:101) B ) is actually independent of the chosen c ∈ ρ − ( (cid:101) β + (cid:101) β (cid:48) ). Thus we denote it just by Γ ∂ ( M − (cid:101) β (cid:48) (cid:101) B ) and the corresponding sub- D V -module of M − (cid:101) β (cid:48) (cid:101) B by Γ ∂ ( M − (cid:101) β (cid:48) (cid:101) B ). Proposition 2.17.
Let (cid:101) β , (cid:101) β (cid:48) be as in Theorem 2.11 and let K be the kernel of M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) −→ M (cid:101) β (cid:101) B . Then K (cid:39) Γ ∂ ( M − (cid:101) β (cid:48) (cid:101) B ) , in particular M IC ( X ◦ , L ) (cid:39) M − (cid:101) β (cid:48) (cid:101) B / Γ ∂ ( M − (cid:101) β (cid:48) (cid:101) B ) . Proof.
Recall that the morphism M − (cid:101) β (cid:48) (cid:101) B · ∂ (cid:101) β + (cid:101) β (cid:48) −→ M (cid:101) β (cid:101) B is induced by the morphism FL( h † O T ) → FL( h + O T ),where we used the isomorphisms M − (cid:101) β (cid:48) (cid:101) B (cid:39) FL( h † O T ) and M (cid:101) β (cid:101) B (cid:39) FL( h + O T ). Applying the Fourier-Laplace transformation again and using FL ◦ FL = Id , we see that the morphism FL( M − (cid:101) β (cid:48) (cid:101) B ) · w (cid:101) β + (cid:101) β (cid:48) −→ FL( M (cid:101) β (cid:101) B ) is induced by the morphism h † O T −→ h + O T . We will calculate the kernel of FL( M − (cid:101) β (cid:48) (cid:101) B ) · w (cid:101) β + (cid:101) β (cid:48) −→ FL( M (cid:101) β (cid:101) B ). First notice that the map h can be factorized as h = k ◦ l , where k is the canonical inclusionof ( C ∗ ) t +1 → V (cid:48) and the map l is given by l : T −→ ( C ∗ ) t +1 , ( y , . . . , y r ) (cid:55)−→ ( y (cid:101) b , . . . , y (cid:101) b t ) = ( y , y y b , . . . , y y b t ) . This shows that FL( M (cid:101) β (cid:101) B ) (cid:39) k + l + O T is localized along V (cid:48) \ ( C ∗ ) t +1 , i.e. FL( M (cid:101) β (cid:101) B ) (cid:39) k + k + FL( M (cid:101) β (cid:101) B ).Let D = { w (cid:101) β + (cid:101) β (cid:48) = 0 } , set U := V (cid:48) \ D and denote by j : U → V (cid:48) the canonical inclusion. Because( C ∗ ) t +1 ⊂ U , the D -module FL( M (cid:101) β (cid:101) B ) is also localized along D , i.e, FL( M (cid:101) β (cid:101) B ) (cid:39) j j +1 FL( M (cid:101) β (cid:101) B ).Notice that the induced morphism j +1 FL( M − (cid:101) β (cid:48) (cid:101) B ) → j +1 FL( M (cid:101) β (cid:101) B ) is an isomorphism, because w (cid:101) β + (cid:101) β (cid:48) isinvertible on U . Therefore we can conclude that j j +1 FL( M − (cid:101) β (cid:48) (cid:101) B ) → j j +1 FL( M (cid:101) β (cid:101) B ) (cid:39) FL( M (cid:101) β (cid:101) B ) is anisomorphism. It is therefore enough to calculate the kernel of FL( M − (cid:101) β (cid:48) (cid:101) B ) → j j +1 FL( M − (cid:101) β (cid:48) (cid:101) B ). On thelevel of global sections this is H D (FL( M − (cid:101) β (cid:48) (cid:101) B )) (cf. [HTT08, Proposition 1.7.1]) which is given by H D (FL( M − (cid:101) β (cid:48) (cid:101) B )) = { m ∈ FL( M − (cid:101) β (cid:48) (cid:101) B ) | ∃ n ∈ N with ( w (cid:101) β + (cid:101) β (cid:48) ) n · m = 0 } . Applying the Fourier-Laplace transformation to this kernel shows the claim.
In this section we show that the D -modules discussed above are quasi-equivariant with respect to anatural torus action. We review the definition of an quasi-equivariant D -modules from [Kas08, Chapter3] and prove some simple statements for these.Let X be a smooth, complex, quasi-projective variety and G be a complex affine algebraic group, whichacts on X . Denote by ν : G × X → X the action of G on X and by p : G × X → X the secondprojection. A D X -module M is called quasi- G -equivariant if it satisfies ν + M (cid:39) p +2 M as O G (cid:2) D X -modules together with an associative law (cf. [Kas08, Definition 3.1.3]). We denote the abelian categoryof quasi- G -equivariant D X -modules by M ( D X , G ) and the subcategories of coherent, holonomic and reg-ular holonomic quasi- G -equivariant D X -modules by M coh ( D X , G ) resp. M h ( D X , G ) resp. M rh ( D X , G ).The corresponding bounded derived categories are denoted by D b ∗ ( D X , G ) for ∗ = ∅ , coh, h, rh .A O X -module F is called G -equivariant if ν ∗ F (cid:39) pr ∗ F as O G ×X -modules and if it satisfies an associativelaw (expressed as the commutativity of a certain diagram, see [Kas08, Definition 3.1.2]). We denote by M od ( O X , G ) the category of G -equivariant O X -modules and by M od coh ( O X , G ) the subcategory ofcoherent G -equivariant O X -modules. 26et f : X → Y be a G -equivariant map. Then the direct image resp. the inverse image functors preservequasi- G -equivariance (cf. [Kas08, Equation (3.4.1), Equation (3.5.2)].We will now show that the duality functor preserves quasi- G -equivariance. Proposition 2.18.
Let M ∈ D bcoh ( D X , G ) then D ( M ) ∈ D bcoh ( D X , G ) opp .Proof. By a d´evissage we may assume that M is a single degree complex, i.e. M ∈ M od coh ( D X , G ).By [Kas08, Lemma 3.3.2] for every N ∈ M od coh ( O X , G ) there exists a G -equivariant locally-free O X module L of finite rank and a surjective G -equivariant morphism L (cid:16) N . Notice that there exists a G -equivariant coherent O X -submodule K of M with D X ⊗ K = M . This enables us to construct alocally-free, G -equivariant resolution · · · → L → L → L → K → K in M od coh ( O X , G ), which gives rise to a resolution of M · · · → D X ⊗ L → D X ⊗ L → D X ⊗ L → M → M od coh ( D X , G ) by the exactness of D X ⊗ O X . We have D M = R H om D X ( M, D X ) ⊗ Ω ⊗− X [ dim X ] (cid:39) H om D X ( D X ⊗ L • , D X ) ⊗ Ω ⊗− X [ dim X ] (cid:39) ( H om O X ( L • , O X ) ⊗ D X ) ⊗ Ω ⊗− X [ dim X ] (cid:39) D X ⊗ H om O X ( L • , O X )[ dim X ] . But H om O X ( L • , O X ) is again a complex in M od coh ( O X , G ), which can be easily seen by the local-freenessof the L i . Thus we can conclude that D M ∈ D bcoh ( D X , G ) opp . Corollary 2.19.
Let f : X → Y be a G -equivariant map. Then the proper direct image and theexceptional inverse image functor preserves quasi- G -equivariance.Proof. This follows from f † = D ◦ f + ◦ D and f † = D ◦ f + ◦ D .In the next proposition we will show that the characteristic variety of a quasi- G -equivariant D -moduleis G -invariant. For that purpose, we will consider the action induced by ν on the cotangent bundle T ∗ X . More precisely, consider the differential dν of the action map, which is a map of vector bundles dν : ν ∗ T ∗ X → T ∗ ( G × X ) = T ∗ G (cid:2) T ∗ X over G × X , or, equivalently, a map dν : ( G × X ) × X T ∗ X → T ∗ G × T ∗ X of smooth complex varieties. Notice that t : G × T ∗ X −→ ( G × X ) × X T ∗ X = { (( g, x ) , v ) | π ( v ) = ν ( g, x ) ∈ X } , ( g, v ) (cid:55)−→ ( g, ν ( g − , π ( v )) , v )is an isomorphism, with inverse map sending (( g, x ) , v ) to ( g, v ). Now consider the composition ξ : (cid:101) p ◦ dν ◦ t : G × T ∗ X → T ∗ X , where (cid:101) p : T ∗ G × T ∗ X → T ∗ X is the second projection. One easily checksthat we have ξ ( g · g , x ) = ξ ( g , ξ ( g , x )), i.e., that we obtain an action of G on T ∗ X . Notice that forany g ∈ G , the map ξ ( g, − ) : T ∗ X → T ∗ X is nothing but the differential dν g of the map ν g : X → X where ν g ( x ) := ν ( g, x ). Notice that for M ∈ D b ( D X , G ) one has ν + g M (cid:39) M by the quasi- G -equivarianceof M . Proposition 2.20.
Let M ∈ D bcoh ( D X , G ) , then the characteristic variety char( M ) of M is invariantunder the G -action on T ∗ X given by ξ . Moreover, if G is irreducible then the irreducible components of char( M ) are also G -invariant.Proof. For both statements it is sufficient to show invariance under the morphism ν g for any g ∈ G .We are going to use the following fact (cf. [HTT08, Lemma 2.4.6(iii)]). Let f : X → Y be a morphismbetween smooth algebraic varieties. One has the natural morphisms T ∗ X X × Y T ∗ Y ρ f (cid:111) (cid:111) ω f (cid:47) (cid:47) T ∗ Y . M ∈ M od coh ( D Y ). If f is non-characteristic then char( f + M ) ⊂ ρ f ω − f (char( M )).We want to apply this to the case f = ν g . Notice that in this case the maps ρ ν g and ω ν g are isomorphismsand ρ ν g ◦ ω − ν g = dν g . Thus we havechar( M ) = char( ν + g M ) ⊂ dν g (char( M )) . Repeating the argument with ν g − gives char( M ) ⊂ dν g − (char( M )). Now applying dν g to both sides ofthe latter inclusion shows the first claim.Now assume that G is irreducible and let C i be an irreducible component of Ch ( M ). Notice that G × C i is irreducible. Consider the scheme-theoretic image I of G × C i under the induced action map ξ : G × char( M ) → char( M ). Then ξ : G × C i → I is a dominant morphism. We want to show that I isirreducible. Let U ⊂ I be an affine open set. The restriction ξ − ( U ) → U is still dominant and inducesan injective ring homomorphism O I ( U ) → O G × C i ( ξ − ( U )). As G × C i is irreducible and reduced thering O G × C i ( ξ − ( U )) is a domain. Thus O I ( U ) is also a domain and because U was chosen arbitrary weconclude that I is irreducible. Notice that we have C i ⊂ I ⊂ char( M ) and therefore C i = I , which showsthe claim.The proposition above enables us to prove that a section of a quotient map of a free action is non-characteristic with respect to quasi- G -equivariant D -modules. Lemma 2.21.
Let G × X → X be a free action and π G : X → X /G a geometric quotient. Let i G : X /G → X be a section of π G , then i G is non-characteristic with respect to every M ∈ D brh ( G, D X ) .Proof. We consider X /G as smooth subvariety of X . Notice that X /G is transversal to the orbits of the G -action on X given by ν . Let char( M ) = (cid:83) i ∈ I C i be the decomposition into irreducible componentsand put X i := π ( X i ) so that C i = T ∗X i X . From Proposition 2.20 we know that C i is invariant under theaction given by ξ , and hence a union of orbits of this G -action. On the other hand, the image underthe projection π : T ∗ X → X of such an orbit is necessarily an orbit of the original action given by ν . Hence X i is a union (cid:83) j X ( j ) i of G -orbits, more precisely, these orbits form a Whitney stratificationof X i (see, [Dim92, Proposition 1.14]). Whitney’s condition A then implies that T ∗X i X ⊂ (cid:83) j T ∗X ( j ) i X i .Transversality of X /G and the orbits X ( j ) i means that T ∗X /G X ∩ T ∗X ( j ) i X i ⊂ T ∗X X , from which we deducethat T ∗X /G X ∩ T ∗X i X i ⊂ T ∗X X and hence T ∗X /G X ∩ char( M ) ⊂ T ∗X X . Thus i G non-characteristic withrespect to M as required.Let V ∗ = C × ( C ∗ ) t and let j V ∗ : V ∗ → V be the canonical embedding. Consider the following diagram S j (cid:15) (cid:15) Γ π S (cid:111) (cid:111) θ (cid:15) (cid:15) Γ ∗ j Γ ∗ (cid:111) (cid:111) ζ (cid:15) (cid:15) X i (cid:15) (cid:15) Z Xη (cid:15) (cid:15) (cid:111) (cid:111) Z ∗ Xε (cid:15) (cid:15) j Z ∗ X (cid:111) (cid:111) P ( V (cid:48) ) Z π Z (cid:111) (cid:111) π Z (cid:15) (cid:15) Z ∗ j Z ∗ (cid:111) (cid:111) δ (cid:15) (cid:15) V V ∗ j V ∗ (cid:111) (cid:111) (25)where the varieties Z ∗ , Z ∗ X , Γ ∗ together with the maps j Z ∗ , j Z ∗ X , j Γ ∗ and δ, ε, ζ are induced by the basechange j V ∗ . Thus all squares in the diagram above are cartesian.28e now specify to the case G = ( C ∗ ) s . We let G act on S and V by G × S −→ S , (26)( g , . . . , g s , y , . . . , y s ) (cid:55)→ ( g y , . . . g s y s ) ,G × V −→ V , ( g , . . . , g s , λ , . . . , λ t ) (cid:55)→ ( λ , g − b λ , . . . , g − b t λ t ) . We also define the following G -action on P ( V (cid:48) ): G × P ( V (cid:48) ) −→ P ( V (cid:48) ) , (27)( g , . . . , g s , ( µ : . . . : µ t )) (cid:55)→ ( µ : g b µ : . . . : g b t µ t ) . This makes map g = i ◦ j : S → P ( V (cid:48) ) G -equivariant. There is a natural action of G on P ( V (cid:48) ) × V resp. S × V which leaves the subvarieties Z = { (cid:80) ti =0 λ i µ i = 0 } resp. Γ = { λ + (cid:80) ti =1 λ i y b i = 0 } invariant. Itis now easy to see, using the induced actions on Γ resp. Z , that the maps π Z , π Z , π S as well as η and θ are G -equivariant.Notice that G leaves V ∗ invariant and acts freely on it, but this shows that G acts also freely on Z ∗ , Z ∗ X and Γ ∗ . Therefore also the maps δ, ε, ζ are G -equivariant. Notice that the action of G on P ( V (cid:48) ) as definedin formula (27) is not free, there are orbits of dimension strictly smaller dimension than s = dim( G ).Because we have Z B = Z s , there exist matrices N ∈ Gl ( s × s, Z ) and N ∈ Gl ( t × t, Z ) such that B = N · ( I s | s × r ) · N , (28)where r := t − s . Define matrices L := N − · (cid:18) s × r I r (cid:19) , M := (0 r × s | I r ) · N , C := N − · (cid:18) I s r × s (cid:19) · N − , D := ( C · B ) t , whose entries we denote by l ij , m ji , c ik and d il , respectively. Then M · L = I r , B · C = I s , B · L = 0, M · C = 0 and C · B + L · M = I t . (29)Consider the following map, where F := ( C ∗ ) s and KM := ( C ∗ ) s : T P : P ( V (cid:48) ) × C × F × KM −→ P ( V (cid:48) ) × V ∗ , (( µ : . . . : µ t ) , λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ (( µ : f − b µ : . . . : f − b t µ t ) , λ , f b · q m , . . . , f b t · q m t )with f b i = (cid:81) sk =1 f b ki k , q m i = (cid:81) rj =1 q m ji j and inverse T − P : P ( V (cid:48) ) × V ∗ −→ P ( V (cid:48) ) × C × F × KM , (( µ : . . . : µ t ) , λ , . . . , λ t ) (cid:55)→ (( µ : λ d µ : . . . : λ d t µ t ) , λ , λ c , . . . , λ c s , λ l , . . . , λ l r )with λ c k := (cid:81) ti =1 λ c ik i , λ l j = (cid:81) ti =1 λ l ij i and λ d l := (cid:81) ti =1 λ d il i = (cid:81) ti =1 λ (cid:80) k c ik b kl i .Notice that the space KM will reappear in section 6 (see the explanations after the exact sequence (59)),where it similarly denotes the r -dimensional torus ( C ∗ ) r . There is however a difference: in the presentsection, our input data is the matrix B , and the map T P and its inverse T − P are defined by the choice ofthe matrices N and N which have to satisfy only equation 28. In section 6, we work with a toric variety(and the matrix B is given by the primitiv integral generators of its rays), and here these choices haveto satisfy much finer conditions. Nevertheless, we will use the same symbol in order to avoid overloadingthe notation too much.Recall the following G -action on P ( V (cid:48) ) × V ∗ G × ( P ( V (cid:48) ) × V ∗ ) −→ P ( V (cid:48) ) × V ∗ , ( g , . . . , g s , ( µ : . . . : µ t ) , λ , . . . , λ t ) (cid:55)→ (( µ : g b µ : . . . : g b t µ t ) , λ , g − b λ , . . . , g − b t λ t ) . G -action on P ( V (cid:48) ) × C × F × KM G × ( P ( V (cid:48) ) × C × F × KM ) −→ P ( V (cid:48) ) × C × F × KM , ( g , . . . , g s , ( µ : . . . : µ t ) , λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ (( µ : µ : . . . : µ t ) , λ , g − f , . . . , g − s f s , q , . . . , q r ) . It is easy to see that T P resp. T − P is G -equivariant with respect to the G -actions above.Consider the map T S : S × C × F × KM −→ S × V ∗ , ( y , . . . , y s , λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ ( f − y , . . . , f − s y s , λ , f b · q m , . . . , f b t · q m t )and its inverse T − S : S × V ∗ −→ S × C × F × KM , ( y , . . . , y s , λ , . . . , λ t ) (cid:55)→ ( λ c y , . . . , λ c s y s , λ , λ c , . . . , λ c s , λ l , . . . , λ l r ) , where one has to use (29).Recall the G -action on S × V ∗ G × ( S × V ∗ ) −→ S × V ∗ , ( g , . . . , g s , λ , . . . , λ t ) (cid:55)→ ( g y , . . . , g s y s , λ , g − b λ , . . . , g − b t λ t )and consider the following G -action on S × C × F × KM G × ( S × C × F × KM ) −→ S × C × F × KM , ( g , . . . , g s , y , . . . , y s , λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ ( y , . . . , y s , λ , g − f , . . . , g − s f s , q , . . . , q r ) . It is again easy to see that T S resp. T − S is G -equivariant with respect to the G -actions above.The subvarieties Z ∗ resp. Γ ∗ are then given by λ µ + (cid:80) ti =1 µ i · q m i = 0 resp. λ + (cid:80) ti =1 y b i · q m i = 0.Finally consider the maps T : C × F × KM −→ V ∗ , ( λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ ( λ , f b · q m , . . . f b t · q m t ) ,T − : V ∗ −→ C × F × KM , ( λ , . . . , λ t ) (cid:55)→ ( λ , λ c , . . . , λ c s , λ l , . . . , λ l r ) , (30)which are G -equivariant with respect to the G -action on V ∗ and the following G -action on C × F × KM G × ( C × F × KM ) −→ C × F × KM , ( g , . . . , g s , λ , f , . . . , f s , q , . . . , q r ) (cid:55)→ ( λ , g − f , . . . , g − s f s , q , . . . , q r ) . The G -equivariant isomorphisms above show that the geometric quotients of V ∗ , Z ∗ and Γ ∗ by G existand are given by C × KM , Z := { λ µ + t (cid:88) i =1 q m i µ i = 0 } ⊂ P ( V (cid:48) ) × C × KM and G := { λ + t (cid:88) i =1 q m i y b i = 0 } ⊂ S × C × KM , respectively. We denote the corresponding quotient maps by π V ∗ G , π Z ∗ G and π Γ ∗ G .30otice that we have a natural section i V ∗ G to π V ∗ G , which is induced by the inclusion C × KM −→ C × F × KM , ( λ , q , . . . , q r ) (cid:55)→ ( λ , , . . . , , q , . . . , q r )and the isomorphism above. This gives also rise to sections i Z ∗ G and i Γ ∗ G of π Z ∗ G resp. π Γ ∗ G . Consider thefollowing diagram S j (cid:15) (cid:15) Γ π S (cid:111) (cid:111) θ (cid:15) (cid:15) Γ ∗ j Γ ∗ (cid:111) (cid:111) ζ (cid:15) (cid:15) π Γ ∗ G (cid:55) (cid:55) G γ (cid:15) (cid:15) i Γ ∗ G (cid:111) (cid:111) X i (cid:15) (cid:15) Z Xη (cid:15) (cid:15) (cid:111) (cid:111) Z ∗ Xε (cid:15) (cid:15) j Z ∗ X (cid:111) (cid:111) Z Xβ (cid:15) (cid:15) i Z ∗ XG (cid:111) (cid:111) P ( V (cid:48) ) Z π Z (cid:111) (cid:111) π Z (cid:15) (cid:15) Z ∗ j Z ∗ (cid:111) (cid:111) δ (cid:15) (cid:15) π Z ∗ G (cid:55) (cid:55) Z α (cid:15) (cid:15) i Z ∗ G (cid:111) (cid:111) V V ∗ j V ∗ (cid:111) (cid:111) π V ∗ G (cid:53) (cid:53) C × KM i V ∗ G (cid:111) (cid:111) (31)Notice also that all squares are cartesian. Proposition 2.22.
Let i Z ∗ G : Z → Z ∗ resp. i V ∗ G : C × KM → V ∗ be the sections constructed above.1. The D Z ∗ -modules ( ε ◦ ζ ) † O Γ ∗ , ( ε ◦ ζ ) + O Γ ∗ and M IC ( Z ∗ X ) are quasi- G -equivariant and non-characteristic with respect to i Z ∗ G .2. The D V ∗ -modules H ( ϕ B, † O S × W ∗ ) and H ( ϕ B, + O S × W ∗ ) are quasi- G -equivariant and non-characteristic with respect to i V ∗ G .3. We have ( i Z ∗ G ) + M IC ( Z ∗ X ) (cid:39) M IC ( Z X ) . In particular we have α + M IC ( Z X ) (cid:39) i + KM R (cid:0) M IC ( X ) (cid:1) , (32) where i KM := j V ∗ ◦ i V ∗ G is non-characteristic with respect to R ( M IC ( X )) .Proof.
1. First notice that because the map ( i ◦ j ) : S → P ( V (cid:48) ) is affine and this property is preservedby base change, the map ( ε ◦ ζ ) is also affine. Thus the direct image as well as the proper directimage of O Γ ∗ is a single D Z ∗ -module. The closure of Γ ∗ in Z ∗ is Z ∗ X , therefore we have M IC ( Z ∗ X ) = im (( ε ◦ ζ ) † O Γ ∗ → ( ε ◦ ζ ) + O Γ ∗ ) ∈ M od rh ( D Z ∗ ) . (33)To show the first claim, it is enough by Lemma 2.21 to show that the corresponding D -modulesare quasi- G -equivariant. First recall that Γ ∗ ⊂ S × V ∗ and denote by ι : Γ ∗ → S the restrictionof the projection to the first factor. Notice that ι is G -equivariant and O Γ ∗ (cid:39) ι + O S . Therefore O Γ ∗ is a quasi- G -equivariant D -module. Because ε, ζ is G -equivariant we see that ( ε ◦ ζ ) † O Γ ∗ and( ε ◦ ζ ) + O Γ ∗ are quasi- G -equivariant. Furthermore, because of Equation (33) and the fact that M od ( G, D Z ∗ ) is an abelian category the D -module M IC ( Z ∗ X ) is quasi- G -equivariant.31. For the second point, consider the action of G on W ∗ = ( C ∗ ) t which is given by G × W ∗ −→ W ∗ , ( g , . . . , g s , λ , . . . , λ t ) (cid:55)→ ( g − b λ , . . . , g − b t λ t ) . This action together with the action (26) induces a G -action on S × W ∗ . It is easy to see that ϕ B | S × W ∗ is G -equivariant. Thus the D V ∗ -modules H ( ϕ B, † O S × W ∗ ) and H ( ϕ B, + O S × W ∗ ) arequasi- G -equivariant.The fact that i V ∗ G is non-characteristic with respect to these D V ∗ -modules fol-lows now again from Lemma 2.21.3. To show the third claim, consider the following isomorphisms M IC ( Z X ) (cid:39) im (( β ◦ γ ) † O G → ( β ◦ γ ) + O G ) (cid:39) im (cid:16) ( β ◦ γ ) † ( i Γ ∗ G ) † O Γ ∗ → ( β ◦ γ ) + ( i Γ ∗ G ) + O Γ ∗ (cid:17) (cid:39) im (cid:16) ( i Z ∗ G ) † ( ε ◦ ζ ) † O Γ ∗ → ( i Z ∗ G ) + ( ε ◦ ζ ) + O Γ ∗ (cid:17) (cid:39) ( i Z ∗ G ) + im (( ε ◦ ζ ) † O Γ ∗ → ( ε ◦ ζ ) + O Γ ∗ ) (cid:39) ( i Z ∗ G ) + M IC ( Z ∗ X ) , where the second isomorphism follows from ( i Γ ∗ G ) + O Γ ∗ (cid:39) O G , the fact that O Γ ∗ is non-characteristicfor i Γ ∗ G and [HTT08, Theorem 2.7.1(ii)]. The third isomorphism follows by base change and thefourth isomorphism follows from the fact that i Z ∗ G is non-characteristic with respect to ( ε ◦ ζ ) † O Γ ∗ and ( ε ◦ ζ ) + O Γ ∗ .For the last claim consider the following diagram Z π Z (cid:15) (cid:15) Z ∗ j Z ∗ (cid:111) (cid:111) δ (cid:15) (cid:15) Z i Z ∗ G (cid:111) (cid:111) α (cid:15) (cid:15) V V ∗ j V ∗ (cid:111) (cid:111) C × KM i V ∗ G (cid:111) (cid:111) We have the following isomorphisms α + M IC ( Z X ) (cid:39) α + ( i Z ∗ G ) + M IC ( Z ∗ X ) (cid:39) α + ( i Z ∗ G ) + j + Z ∗ M IC ( Z X ) (cid:39) ( i V ∗ G ) + j + V ∗ π Z M IC ( Z X ) (cid:39) i + KM π Z M IC ( Z X ) (cid:39) i + KM π Z ( π Z ) + M IC ( X ) (cid:39) i + KM R ( M IC ( X )) . The non-characteristic property of i KM = j V ∗ ◦ i V ∗ G follows from Lemma 2.21 and the fact that j + V ∗ R ( M IC ( X )) is quasi- G -equivariant. In this section we apply the Fourier-Laplace transformation functor FL W to the various D -modulesconsidered in section 2. For the families of Laurent polynomials resp. compactifications thereof thatappear in mirror symmetry, we obtain D -modules that can eventually be matched with the differentialsystems defined by quantum cohomology. They have in general irregular singularities, and this is reflectedin the fact that although the modules considered in section 2 were monodromic on V , they do not have32ecessarily that property with respect to the vector bundle V = C λ × W → W . Hence the functor FL W will in general not preserve regularity.In the second part of this section, we study a lattice in the Fourier-Laplace transformation of the Gauß-Manin system of the family of Laurent polynomials ϕ B . It is given by a so-called twisted de Rhamcomplex, however, in order to obtain a good hypergeometric description of it, we have to introduce acertain intermediate compactification of ϕ B and replace this de Rham complex by a logarithmic version.Moreover, the parameters of the family ϕ B have to be restricted to a Zariski open set excluding certain(but not all) singularities at infinity. Then we can show the necessary finiteness and freeness of thelattice. It will later correspond to the twisted quantum D -module (see section 4), seen as a family ofalgebraic vector bundles over C z (not only over C ∗ z ) with connection operator which is meromorphicalong { z = 0 } . We discuss here a partial localized Fourier-Laplace transform of the Gauß-Manin systems of ϕ B and ofthe D -module M IC ( X ◦ , L ).Consider the product decomposition V = C λ × W , where W is the hyperplane given by λ = 0. Weinterpret V as a rank one bundle with base W and consider the Fourier-Laplace transformation withrespect to the base W as in Definition 2.4, where we denote the coordinate on the dual fiber by τ . Set z = 1 /τ and denote by j τ : C ∗ τ × W (cid:44) → C τ × W and j z : C ∗ τ × W (cid:44) → ˆ V := C z × W = P τ \ { τ = 0 } × W the canonical embeddings. Let N be a D V -module, the partial, localized Fourier-Laplace transformationis defined by FL locW ( N ) := j z + j + τ FL W ( N ) . The localized Fourier-Laplace transformations of the Gauß-Manin systems are denoted by G + := FL locW ( H ( ϕ B, + O S × W )) , (34) G † := FL locW ( H ( ϕ B, † O S × W )) . (35)We also consider the partial, localized Fourier-Laplace transform of the D -modules M (cid:101) β (cid:101) B . The followingnotation will be useful. Definition 3.1.
Let (cid:99) M ( β ,β ) B be the D (cid:98) V -module D (cid:98) V [ z − ] /I , where I is the left ideal generated by theoperators (cid:98) (cid:3) l , (cid:98) E k − β k z and (cid:98) E − β z , which are defined by (cid:98) (cid:3) l := (cid:81) i : l i < ( z · ∂ λ i ) − l i − (cid:81) i : l i > ( z · ∂ λ i ) l i , l ∈ L B (cid:98) E := z ∂ z + (cid:80) ti =1 zλ i ∂ λ i , (cid:98) E k := (cid:80) ti =1 b ki zλ i ∂ λ i , k = 1 , . . . , s. We denote the corresponding D (cid:98) V -module by (cid:99) M ( β ,β ) B . Lemma 3.2.
We have the following isomorphism FL locW ( M (cid:101) β (cid:101) B ) (cid:39) (cid:99) M ( β +1 ,β ) B for every (cid:101) β = ( β , β ) ∈ Z s +1 .Proof. This is an easy calculation, using the substitution λ → − ∂ τ = z ∂ z and ∂ λ → τ = 1 /z and the fact that (cid:99) M ( β ,β ) B is localized along z = 0. 33otice that in the lemma above we used the subscript (cid:101) B for the GKZ-system on the left hand side andthe subscript B for its localized Fourier-Laplace transform on the right hand side. This notation takesinto account the fact that the properties of the system M (cid:101) β (cid:101) B are governed by the geometry of the semi-group N (cid:101) B , whereas the properties of its localized Fourier-Laplace transform (cid:99) M ( β +1 ,β ) depend on thegeometry of N B . This explains the different sets of allowed parameters in Proposition 3.3 resp. Theorem3.6 in contrast to Theorem 2.11 resp. Theorem 2.16 and Proposition 2.17.Notice that under the normality assumption on the semi-group N (cid:101) B , the rank of (cid:99) M ( β ,β ) B is also equal to d ! · vol(Conv( b , . . . , b t )) (this can be shown by an argument similar to [RS15, Proposition 2.7]).The following proposition gives an isomorphism between the localized partial Fourier-Laplace transformof the Gauß-Manin systems G + and G † and the hypergeometric systems (cid:99) M ( β ,β ) B introduced above. Proposition 3.3.
There exists a δ B ∈ N B such that we have an isomorphism G + (cid:39) (cid:99) M ( β ,β ) B for every β ∈ Z and β ∈ δ B + ( R ≥ B ∩ Z s ) . If N B is saturated, then δ B can be taken to be ∈ N B (inparticular, the statement holds for ( β , β ) = ( β , ∈ Z s ).Furthermore, we have an isomorphism G † (cid:39) (cid:99) M ( β (cid:48) , − β (cid:48) ) B for every β (cid:48) ∈ Z and β (cid:48) ∈ ( R ≥ B ) ◦ ∩ Z s .Proof. We construct the isomorphisms by applying the Fourier-Laplace transform FL W to the exactsequences in Theorem 2.11. First notice that the first and last term in the exact sequences are free O V -modules, thus their Fourier-Laplace transform has support on τ = 0, i.e. their localized Fourier-Laplacetransform is 0. Thus there is some δ (cid:101) B ∈ N (cid:101) B such that we have the following isomorphisms G + = FL locW ( H ( ϕ B, + O S × W )) (cid:39) FL locW ( M (cid:101) β (cid:101) B )and G † = FL locW ( H ( ϕ B, † O S × W )) (cid:39) FL locW ( M − (cid:101) β (cid:48) (cid:101) B )for any (cid:101) β ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) and any (cid:101) β (cid:48) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . Write δ (cid:101) B = ( δ , δ B ) with δ B ∈ Z s . Nowgiven any ( β , β ) ∈ Z × ( δ B + ( R ≥ B ∩ Z s )) resp. ( β (cid:48) , β (cid:48) ) ∈ Z × (( R ≥ B ) ◦ ∩ Z s ) we can find a γ , γ (cid:48) ∈ Z such that ( γ , β ) ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) resp. ( γ (cid:48) , β (cid:48) ) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . It remains to show that thereare isomorphism (cid:99) M ( β ,β ) B (cid:39) (cid:99) M ( γ ,β ) B (36)for ( β , β ) ∈ Z × ( δ B + ( R ≥ B ∩ Z s )) and ( γ , β ) ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 )) resp. (cid:99) M ( β (cid:48) , − β (cid:48) ) B (cid:39) (cid:99) M ( − γ (cid:48) , − β ) B (37)for ( β (cid:48) , β (cid:48) ) ∈ Z × (( R ≥ B ) ◦ ∩ Z s ) and ( − γ (cid:48) , − β (cid:48) ) ∈ (( R ≥ (cid:101) B ) ◦ ∩ Z s +1 ) . Notice that (cid:99) M ( β ,β ) B is localizedalong z = 0 for all ( β , β ) ∈ Z s +1 by Lemma (3.2). Therefore the morphism given by right multiplicationwith z (cid:99) M ( β ,β ) B · z −→ (cid:99) M ( β − ,β ) B (38)is an isomorphism, which shows (36) and (37).Concerning the last statement, suppose that N B is saturated. Let β ∈ N B = ( R ≥ B ∩ Z s ) and let β ∈ Z be arbitrary. By [Rei14, Lemma 1.17] we have β / ∈ sRes ( B ), where sRes ( B ) ⊂ C s is theset of strongly resonant values (cf. [SW09, Definition 3.4]). Using [Rei14, Lemma 1.19] there exists a γ ∈ Z such that ( γ , β ) / ∈ sRes ( (cid:101) B ). Now we argue as above, i.e. by [Rei14, Theorem 2.7] we have G + = FL locW ( H ( ϕ B, + O S × W )) (cid:39) FL locW ( M ( γ ,β ) (cid:101) B ) which in turn is isomorphic to (cid:99) M ( β ,β ) B .If the semigroup N B is saturated, we will compute the isomorphism above explicitly for ( β , β ) = (0 , G + . 34 emma 3.4. Write ϕ B = ( F, pr ) , where F : S × W → C , ( y, λ ) (cid:55)→ − (cid:80) ti =1 λ i y b i and pr : S × W → W is the projection. Recall from formula (34) that we denote by G + the localized Fourier-Laplacetransformation of the Gauß-Manin system of the morphism ϕ B . Write G + := H ( (cid:98) V , G + ) for its moduleof global sections. Then there is an isomorphism of D (cid:98) V -modules G + ∼ = H (cid:16) Ω • + sS × W/W [ z ± ] , d − z − · d y F ∧ (cid:17) , where d is the differential in the relative de Rham complex Ω • S × W/W . The structure of a D (cid:98) V -module onthe right hand side is defined as follows ∂ z ( ω · z i ) = i · ω · z i − + F · ω · z i − ,∂ λ i ( ω · z i ) := ∂ λ i ( ω ) · z i − ∂ λ i F · ω · z i − = ∂ λ i ( ω ) · z i + y b i · ω · z i − , where ω ∈ Ω sS × W/W .Proof.
The expression for the module G + as well as the formulas for the D (cid:98) V -structure are an immediateconsequence of the definition of the direct image functor. See, e.g. [Rei14, equations 2.0.18, 2.0.19], fromwhich the desired formulas can be easily obtained.Using the description of G + via relative differential forms, we find a distinguished element, which is (theclass of) the volume form on S , that is ω := dy y ∧ . . . ∧ dy s y s . In the next lemma we compute the image of ω under the isomorphisms in Proposition 3.3 under theassumption of normality of N B . Lemma 3.5.
Let N B be a saturated semigroup, then the isomorphism from Proposition 3.3 Φ : G + (cid:39) −→ (cid:99) M (0 , B maps ω to .Proof. Recall from the proof of Proposition 3.3, that there exists a γ ∈ Z such that ( γ , / ∈ sRes ( (cid:101) B )(notice that here we only assume that N B is saturated which does not imply that N (cid:101) B is saturated).Denote by ψ ( γ , : Γ( V, H ( ϕ B, + O S × W )) → M ( γ , (cid:101) B the morphism from Theorem 2.11. We first compute the image of ω under the morphism ψ ( γ , usingthe description of H ( ϕ B, + O S × W ) by relative differential forms (see e.g. [Rei14, Equation 2.0.17]).We will use the following two facts of loc. cit. Proposition 2.8 whose proofs extend directly to ourslightly more general situation (there it was assumed that N (cid:101) B is saturated). Namely first, that thereexists a non-zero morphism M ( − , (cid:101) B → Γ( V, H ( ϕ B, + O S × W )) which sends 1 to ω and second that ψ ( γ , ( ω ) (cid:54) = 0. Concatenating this morphism with ψ ( γ , gives a non-zero morphism M ( − , (cid:101) B → M ( γ , (cid:101) B ,where 1 ∈ M ( − , (cid:101) B is sent to the image of ω under ψ ( γ , . By [Rei14, Proposition 1.24] this morphism isuniquely given by right multiplication with ∂ γ +1 λ (up to a non-zero constant). Applying now the partiallocalized Fourier-Laplace transform to the morphism ψ ( γ , , we see that ψ ( γ , ( ω ) = z − γ − . Usingthe isomorphism (cid:99) M ( γ (cid:48) , B · z −→ (cid:99) M ( γ (cid:48) − , B , which holds for any γ (cid:48) ∈ Z , shows the claim.By Proposition 2.16, we can now give a concrete description of the partial, localized Fourier-Laplacetransform (cid:100) M IC ( X ◦ , L ):= FL locW ( M IC ( X ◦ , L )) of the intersection cohomology D -module M IC ( X ◦ , L ).35 heorem 3.6. Let β ∈ δ B +( R ≥ B ∩ Z s ) , β (cid:48) ∈ ( R ≥ B ) ◦ ∩ Z s and β , β (cid:48) ∈ Z , then we have the followingisomorphisms (cid:91) M IC ( X ◦ , L ) (cid:39) im (cid:32) (cid:99) M ( β (cid:48) , − β (cid:48) ) B · z β (cid:48) − β ∂ β + β (cid:48) (cid:47) (cid:47) (cid:99) M ( β ,β ) B (cid:33) , resp. (cid:91) M IC ( X ◦ , L ) (cid:39) (cid:99) M ( β (cid:48) , − β (cid:48) ) B / (cid:98) Γ ∂ (cid:16) (cid:99) M ( β (cid:48) , − β (cid:48) ) B (cid:17) , where (cid:98) Γ ∂ (cid:16) (cid:99) M ( β (cid:48) , − β (cid:48) ) B (cid:17) is the sub- D -module corresponding to the sub- D -module (cid:98) Γ ∂ (cid:16) (cid:99) M ( β (cid:48) , − β (cid:48) ) B (cid:17) := { m ∈ (cid:99) M ( β (cid:48) , − β (cid:48) ) B | ∃ n ∈ N with (cid:16) ∂ β + β (cid:48) (cid:17) n · m = 0 } . Furthermore, if N B is saturated, then δ B can be taken to be ∈ N B (so that, similarly to Proposition3.3, the statement holds true for ( β , β ) = ( β , ∈ Z s ).Proof. Using the isomorphism (cid:99) M ( β ,β ) B · z −→ (cid:99) M ( β − ,β ) B , (39)which holds for every ( β , β ) ∈ Z s +1 , we can assume that ( β + 1 , β ) ∈ δ (cid:101) B + ( R ≥ (cid:101) B ∩ Z s +1 ) resp.( β (cid:48) + 1 , β (cid:48) ) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . Then the first isomorphism follows by applying the functor FL locW to theisomorphism in Theorem 2.16 and Lemma 3.2.For the second isomorphism we can assume again that ( β (cid:48) + 1 , β (cid:48) ) ∈ ( R ≥ (cid:101) B ) ◦ ∩ Z s +1 . Now the desiredstatement is obtained by applying FL locW to the second isomorphism in Proposition 2.17 and the fact that (cid:98) Γ ∂ ( (cid:99) M ( β (cid:48) , − β (cid:48) ) B ) is stable under left multiplication with z .Now assume that N B is saturated and let β ∈ N B . Arguing as in the last part of the proof of Proposition3.3 we can find a γ ∈ Z such that ( γ , β ) / ∈ sRes ( (cid:101) β ). By [SW09, Corollary 3.7] we have an isomorphismFL( h + O T ) (cid:39) M ( γ ,β ) (cid:101) B . Now the proof of Theorem 2.16 shows that M IC ( X ◦ , L ) (cid:39) im ( M − (cid:101) β (cid:48) (cid:101) B · ∂ ( γ ,β )+ (cid:101) β (cid:48) −→ M ( γ ,β ) (cid:101) B ) . Now applying the functor F locW and using the isomorphism (39) shows the claim in the saturated case. In this section we define a natural lattice in the Fourier-Laplace transformed Gauß-Manin system G + outside some bad locus where the Laurent polynomial acquires singularities at infinity. For this we needto study the characteristic variety of the Gauß-Manin system of ϕ B and the corresponding GKZ system M (cid:101) β (cid:101) B . Throughout this section we assume that N B is a saturated semigroup. Recall the embedding ofthe torus S in the projective space from formula (10) S j −→ X i −→ P ( V (cid:48) ) . The projective variety X serves as a convenient ambient space to compactify fibers of the family ofLaurent polynomials ϕ B . However, we will also need an intermediate partial compactification of S ,which is still an affine variety. Definition 3.7.
The restriction of X to the affine chart of P ( V (cid:48) ) given by µ = 1 is called X aff , inother words, X aff is the closure of the map g B : S −→ C t , ( y , . . . , y s ) (cid:55)→ ( y b , . . . , y b t ) , and thererfore isomorphic to Spec ( C [ N B ]) . θ (cid:47) (cid:47) (cid:15) (cid:15) Z X aff (cid:15) (cid:15) θ (cid:47) (cid:47) Z X η (cid:47) (cid:47) (cid:15) (cid:15) Z π Z (cid:47) (cid:47) π Z (cid:15) (cid:15) VS j (cid:47) (cid:47) X aff j (cid:47) (cid:47) X i (cid:47) (cid:47) P ( V (cid:48) ) (40)where j and j are the canonical inclusions and the three squares are cartesian. Recall that Z ⊂ P ( V (cid:48) ) × V was given by the incidence relation (cid:80) ti =0 λ i µ i = 0 and the composed map g = i ◦ j = i ◦ j ◦ j was defined by formula (9). Thus Γ resp. Z X aff is the subvariety of S × V = S × C λ × W resp. X aff × V given by the equation λ + (cid:80) ri =1 λ i y b i = 0. It follows from the definition that Γ is the graph of ϕ B .Therefore the maps π Z X := π Z ◦ η : Z X −→ V resp. π Z X aff := π Z ◦ η ◦ θ : Z X aff −→ V provide natural (partial) compactifications of the family of Laurent polynomials ϕ B . Putting H (cid:101) λ := { (cid:80) ti =0 λ i µ i = 0 } ⊂ P ( V (cid:48) ) for any (cid:101) λ ∈ V , we see that the fiber π − Z X ( (cid:101) λ ) resp. π − Z X aff ( (cid:101) λ ) is given by X ∩ H (cid:101) λ resp. { λ + (cid:80) ti =1 λ i y b i = 0 } ⊂ X aff .Recall that the toric variety X has a natural stratification by torus orbits X (Γ), which are in one-to-one correspondence with the faces Γ of the polytope Q , which is the convex hull of the elements { b := 0 , b , . . . , b t } . Notice that the stratification S := { X (Γ) } is a Whitney stratification of X (seee.g. [Dim92, Proposition 1.14].By [GKZ08, Chapter 5, Prop 1.9] the orbit X (Γ) (cid:39) ( C ∗ ) dim(Γ) is the image of the map g Γ : S −→ P ( V (cid:48) ) , ( y , . . . , y s ) (cid:55)→ ( ε ε y b : . . . : ε t y b t ) , where ε i = 0 if b i / ∈ Γ and ε i = 1 if b i ∈ Γ. It is easy to see that X aff = (cid:91) Γ | ∈ Γ X (Γ)and this induces a Whitney stratification of X aff .The preimage of X (Γ) ∩ H (cid:101) λ under g Γ is given by { ( y , . . . , y s ) ∈ S | (cid:88) b i ∈ Γ λ i y b i = 0 } . It follows from [GKZ08, Chapter 5.D] that the morphism g Γ : S −→ X (Γ) (cid:39) ( C ∗ ) dim(Γ) is a trivialfibration with fiber being isomorphic to ( C ∗ ) d − dim(Γ) .Denote by S crit, (cid:101) λ Γ the set (cid:26) ( y , . . . , y s ) ∈ S | (cid:88) b i ∈ Γ (cid:101) λ i y b i = 0 ; y k ∂ y k ( (cid:88) b i ∈ Γ (cid:101) λ i y b i ) = 0 for all k ∈ { , . . . , s } (cid:27) . (41)Then its image under g Γ is exactly the singular set sing ( X (Γ) ∩ H (cid:101) λ ) of X (Γ) ∩ H (cid:101) λ . This motivatesthe following definition. Definition 3.8.
Let (cid:101) λ ∈ V . The fiber π − Z X ( (cid:101) λ ) has stratified singularities in X (Γ) if X (Γ) ∩ H (cid:101) λ is singular, i.e. S crit, (cid:101) λ Γ (cid:54) = 0 .2. The set ∆ B := { (cid:101) λ ∈ V | S crit, (cid:101) λQ (cid:54) = ∅} = { (cid:101) λ ∈ V | ϕ − B ( (cid:101) λ ) is singular } is called the discriminant of ϕ B .3. The fiber ϕ − B ( (cid:101) λ ) has singularities at infinity if there exists a proper face Γ of the Newtonpolyhedron Q so that S crit, (cid:101) λ Γ (cid:54) = ∅ . The set ∆ ∞ B := { (cid:101) λ ∈ V | ∃ Γ (cid:54) = Q so that S crit, (cid:101) λ Γ (cid:54) = ∅} is called the non-tame locus of ϕ B .4. The fiber ϕ − B ( (cid:101) λ ) has bad singularities at infinity if there exists a proper face Γ of the Newtonpolyhedron Q not containing the origin such that S crit, (cid:101) λ Γ (cid:54) = ∅ . The set ∆ badB := { (cid:101) λ ∈ V | ∃ Γ (cid:54) = Q, / ∈ Γ so that S crit, (cid:101) λ Γ (cid:54) = ∅} ⊂ ∆ ∞ B is called the bad locus of ϕ B . Remark 3.9.
Notice that ∆ badB is independent of λ . We denote its projection to W by W bad . Let W ∗ = W \ { λ . . . λ t = 0 } and define W ◦ := W ∗ \ W bad , which we call the set of good parameters for ϕ B . Recall that X aff is isomorphic to Spec ( R B ) with R B := C [ N B ]. Let λ ∈ W and set f λ ( • ) := ϕ B ( • , λ ).Notice that the Laurent polynomials f λ and y k ∂f λ /∂y k for k = 1 , . . . , s , which were defined on S beforeare actually elements of R B and can thus naturally be considered as functions on X aff . Lemma 3.10.
Let λ ∈ W ◦ be a good parameter, then dim C (cid:0) R B / ( y k ∂f λ /∂y k ) k =1 ,...,s (cid:1) = vol( Q ) , where the volume of a hypercube [0 , s ⊂ R s is normalized to s ! . Moreover, we have supp ( R B / ( y k ∂f λ /∂y k ) k =1 ,...,s ) = (cid:91) λ ∈ C sing S ( π − Z X ( λ , λ )) , where we see π − Z X ( λ , λ ) as a subset of X ⊂ P ( V (cid:48) ) and where sing S ( π − Z X ( λ , λ )) denotes the stratifiedsingular locus with respect to the stratification S of X by torus orbits defined above.Proof. For the first claim consider the following increasing filtration on R B . Let as above Q be the convexhull of b , . . . , b t and 0 in R s . Let u ∈ N B then the weight of y u is defined by inf { λ ∈ R ≥ | u ∈ λ · Q } .It is easy to see that there is an integer e so that all weights lie in e − N . Denote by R ke B the elementsin R B with weight ≤ k/e . Let grR B be the graduated ring with respect to this filtration. By [Ado94,Equation 5.12] we have dim C gr ( R B ) / ( y k ∂f λ /∂y k ) k =1 ,...,s = vol( Q ) , where y k ∂f λ /∂y k is the image of y k ∂f λ /∂y k in gr ( R B ). It remains to show thatdim C gr ( R B ) / ( y k ∂f λ /∂y k ) k =1 ,...,s = dim C R B / ( y k ∂f λ /∂y k ) k =1 ,...,s . The proof of this equality is an easy adaptation of the proof of [Ado94, Theorem 5.4].38or the proof of the second statement we notice first that sing S ( π − Z X ( λ , λ )) = (cid:91) Γ | ∈ Γ sing ( X (Γ) ∩ H ( λ ,λ ) )because the fiber over ( λ , λ ) has no bad singularities at infinity.Define the following r hyperplanes H kλ for k ∈ { , . . . , s } and λ ∈ W ◦ : H kλ := { ( µ : . . . : µ t ) ∈ P ( V (cid:48) ) | t (cid:88) i =1 b ki λ i µ i = 0 } . We have sing ( X (Γ) ∩ H ( λ ,λ ) ) = X (Γ) ∩ H ( λ ,λ ) ∩ ( (cid:84) sk =1 H kλ ) by equation (41) and therefore sing S ( π − Z X ( λ , λ )) = X aff ∩ H ( λ ,λ ) ∩ ( s (cid:92) k =1 H kλ ) . Notice that (cid:91) λ ∈ C ( X aff ∩ H ( λ ,λ ) ∩ ( s (cid:92) k =1 H kλ )) = (cid:91) λ ∈ C supp ( R B /R B ( f λ − λ ) + R B ( ∂f λ /∂y k ) k =1 ,...,s )= supp ( R B /R B ( ∂f λ /∂y k ) k =1 ,...,s ) , which shows the claim.Let (cid:101) B be the ( s + 1) × ( t + 1)-matrix as introduced before Definition 2.8. Let (cid:101) Q be the convex hullof (cid:101) b , . . . , (cid:101) b t in R s +1 . Notice that (cid:101) Q ⊂ { } × R s and therefore no face (cid:101) Γ of (cid:101) Q contains the origin.Adolphson characterized the characteristic variety char( M (cid:101) β (cid:101) B ) of the GKZ system M (cid:101) β (cid:101) B as follows. Let T ∗ V (cid:39) V × V (cid:48) be the holomorphic cotangent bundle with coordinates ( λ , . . . , λ t , µ , . . . , µ t ). Define thefollowing Laurent polynomials on ( C ∗ ) s +1 (cid:101) f (cid:101) λ ( y ) := (cid:101) f (cid:101) λ, (cid:101) Q ( y ) := t (cid:88) i =0 λ i y (cid:101) b i , (cid:101) f (cid:101) λ, (cid:101) Γ ( y ) := (cid:88) (cid:101) b i ∈ (cid:101) Γ λ i y (cid:101) b i , where we define y (cid:101) b i := (cid:81) rk =0 y (cid:101) b ki k . Lemma 3.11 ([Ado94] Lemma 3.2, Lemma 3.3) .
1. For each ( (cid:101) λ (0) , (cid:101) µ (0) ) ∈ char( M (cid:101) β (cid:101) A ) there exists a (possibly empty) face (cid:101) Γ such that (cid:101) µ (0) j (cid:54) = 0 if andonly if (cid:101) b j ∈ (cid:101) Γ .2. If (cid:101) λ (0) is a singular point of M (cid:101) β (cid:101) B and (cid:101) Γ the corresponding (non-empty) face, then the Laurentpolynomials ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Γ /∂y , . . . , ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Γ /∂y s have a common zero in ( C ∗ ) s +1 . We can use this result in the next lemma to compute the singular locus of the D -modules we are interestedin. Lemma 3.12.
The singular locus of M (cid:101) β (cid:101) B as well as the singular locus of the modules H ( ϕ B + O S × W ) resp. H ( ϕ B † O S × W ) is given by ∆ S := ∆ B ∪ ∆ ∞ B . roof. Notice that the polytope (cid:101) Q ⊂ { } × R s is just the shifted polytope Q ⊂ R s defined above.One easily sees that the Laurent polynomials ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Q /∂y , . . . , ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Q /∂y s have a common zero in( C ∗ ) s +1 if and only if ϕ − B ( (cid:101) λ (0) ) is singular, i.e. the set of (cid:101) λ (0) ’s which satisfy this condition is exactlythe discriminant ∆ B of ϕ B . If there exists a proper face (cid:101) Γ of (cid:101) Q such that the Laurent polynomials ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Γ /∂y , . . . , ∂ (cid:101) f (cid:101) λ (0) , (cid:101) Γ /∂y s have a common zero in ( C ∗ ) s +1 , then then fiber ϕ − B ( (cid:101) λ (0) ) has a singularityat infinity, i.e. its compactification has a singularity in X (Γ), where Γ is the corresponding face of Q . Lemma 3.13.
The restriction of the discriminant ∆ S to C × W ◦ ⊂ V is finite over W ◦ ⊂ W .Proof. We will first show quasi-finiteness of the map p : ∆ S | C × W ◦ → W ◦ . First notice that we have∆ S | C × W ◦ = (∆ S \ ∆ badB ) | C × W ◦ . Fix some λ ∈ W ◦ . We have to show that ∆ S | C ×{ λ } is a finite set. Bythe definition of ∆ S it is enough to show that sing S ( π − Z X ( λ , λ )) is a finite set, but this is Lemma 3.10.To prove finiteness of the map p : ∆ S | C × W ◦ → W ◦ it remains to show that it is proper. Let K be anycompact subset of W ◦ . Suppose that p − ( K ) is not compact, then it must be unbounded in V (cid:39) C t +1 for the standard metric. Hence there is a sequence ( λ ( i )0 , λ ( i ) ) ∈ p − ( K ) with lim i →∞ | λ ( i )0 | = ∞ , as K isclosed and bounded in W ◦ ⊂ W = C t .In order to construct a contradiction, we use the partial compactification of the family ϕ B from above.Recall the spaces Z := { (cid:80) ti =0 λ i · µ i = 0 } ⊂ P ( V (cid:48) ) × V and Z X := ( X × V ) ∩ Z . Introduce thespaces Z k := { (cid:80) ti =1 b ki λ i µ i = 0 } for k ∈ { , . . . , t } . Then Z X ∩ ( (cid:84) dk =1 Z k ) is the stratified critical locus crit S ( π Z X ) of the family π Z X , where we denote by abuse of notation by S also the stratification on Z X induced from the torus stratification on X used above.Because the projection from the stratified critical locus crit S ( π Z X ) of π Z X to ∆ S is onto, there is asequence (( µ ( i )0 : µ ( i ) ) , ( λ ( i )0 , λ ( i ) )) ∈ X aff × p − ( K ) projecting under π Z X | X aff × p − ( K ) to ( λ ( i )0 , λ ( i ) ) (Noticethat we consider here X aff as a subset of P ( V (cid:48) ) under the embedding i ◦ j ). Consider the first componentof the sequence (( µ ( i )0 : µ ( i ) ) , ( λ ( i )0 , λ ( i ) )), then this is a sequence ( µ ( i )0 : µ ( i ) ) in X which converges (afterpossibly passing to a subsequence) to a limit (0 : µ lim1 : . . . : µ lim t ) (this is forced by the incidence relation (cid:80) ti =0 λ i µ i ). In other words this limit lies in X \ X aff by the definition of X aff (see Definition 3.7). Butbecause ( X × V ) ∩ Z ∩ (cid:84) dk =1 Z k = Z X ∩ (cid:84) dk =1 Z k is closed, the point ((0 : µ lim1 : . . . : µ lim t ) , ( λ lim0 , λ lim ))lies in (( X \ X aff ) × p − ( K )) ∩ Z ∩ (cid:84) dk =1 Z k , i.e. π − Z X (lim i →∞ ( λ ( i )0 , λ ( i ) )) has a bad singularity at infinity,which is a contradiction by the definition of W ◦ .We can now prove the following regularity property of (cid:99) M ( β ,β ) B , which is essentially the same proof as in[RS15, Lemma 4.4]. Lemma 3.14.
Consider (cid:99) M ( β ,β ) B as a D P × W -module, where W is a smooth projective compactificationof W . Then (cid:99) M ( β ,β ) B is regular outside ( { z = 0 } × W ) ∪ ( P z × ( W \ W ◦ )) and smooth on C ∗ z × W ◦ .Proof. It suffices to show that any λ = ( λ , . . . , λ t ) ∈ W ◦ has a small analytic neighborhood W ◦ λ ⊂ W ◦ an such that the partial analytization O anW ◦ λ [ τ, τ − ] ⊗ O C ∗ τ × W ◦ (cid:99) M ( β ,β ) B is regular on C τ × W ◦ λ (but not at τ = ∞ ). This is precisely the statement of [DS03, Theorem 1.11 (1)], taking into account the regularityof M (cid:101) β (cid:101) B (c.f. [Hot98, section 6]), the fact that the singular locus of M (cid:101) β (cid:101) B coincides with ∆ S (see Lemma3.12) as well as the last lemma (notice that the non-characteristic assumption in [DS03, Theorem 1.11(1)] is satisfied, see, e.g., [Pha79, page 281]).The next step is to study several natural lattices in (cid:99) M ( β ,β ) B . They are defined in terms of R -modules,see the end of subsection 2.1. Definition 3.15.
1. Consider the left ideal I := D C z × W ∗ ( (cid:98) (cid:3) l ) l ∈ L + D C z × W ∗ ( (cid:98) E k − z · β k ) k =1 ,...,r + D C z × W ∗ ( (cid:98) E − z · β ) in D C z × W ∗ and write ∗ (cid:99) M ( β ,β ) B for the cyclic D -module D C z × W ∗ / I . Here the operators (cid:98) (cid:3) l , (cid:98) E k and (cid:98) E are those from Definition 3.1. . Consider the left ideal I := R C z × W ∗ ( (cid:98) (cid:3) l ) l ∈ L + R C z × W ∗ ( (cid:98) E k − z · β k ) k =1 ,...,r + R C z × W ∗ ( (cid:98) E − z · β ) in R C z × W ∗ and write ∗ (cid:99) M ( β ,β ) B for the cyclic R -module R C z × W ∗ / I .3. Consider the open inclusions W ◦ ⊂ W ∗ ⊂ W and define ◦ R := R | C z × W ◦ with ring of global sections ◦ R . Define the D C z × W ◦ -module ◦ (cid:99) M ( β ,β ) B := (cid:16) (cid:99) M ( β ,β ) B (cid:17) | C z × W ◦ and the R C z × W ◦ -module ◦ (cid:99) M ( β ,β ) B := (cid:16) ∗ (cid:99) M ( β ,β ) B (cid:17) | C z × W ◦ . Remark 3.16.
1. We have D C z × W ∗ ⊗ R C z × W ∗ ∗ (cid:99) M ( β ,β ) B = (cid:99) M ( β ,β ) B | C z × W ∗ .2. The restriction of ∗ (cid:99) M ( β ,β ) B to C ∗ z × W ∗ equals the restriction of (cid:99) M ( β ,β ) B to C ∗ z × W ∗ .3. For z ∂ z ( ∗ (cid:99) M ( β ,β ) B ) = R (cid:48) / I (cid:48) , where I (cid:48) is given by I (cid:48) := R (cid:48) ( (cid:98) (cid:3) l ) l ∈ L + R (cid:48) ( (cid:98) E k − z · β k ) k =1 ,...,r . Lemma 3.17.
The quotient ∗ (cid:99) M ( β ,β ) B /z · ∗ (cid:99) M ( β ,β ) B is the sheaf of commutative O W ∗ -algebras associatedto C [ λ ± , . . . , λ ± t , κ , . . . , κ t ]( (cid:81) l i < κ − l i i − (cid:81) l i > κ l i i ) l ∈ L + ( (cid:80) ti =1 b ki λ i κ i ) k =1 ,...,s (cid:39) C [ N B ][ λ ± , . . . , λ ± t ] y k ∂f λ /∂y k , (42) where y k ∂f λ /∂y k = (cid:80) ti =1 b ki λ i y b i .Proof. Let κ i be the class of z∂λ i . Because the commutator [ κ i , λ i ] is zero we see that ∗ (cid:99) M ( β ,β ) B /z · ∗ (cid:99) M ( β ,β ) B is a commutative algebra and isomorphic to the module on the left hand side of equation (42).To show the isomorphism (42), consider the C [ λ ± , . . . , λ ± t ]-linear morphism ψ : C [ λ ± , . . . , λ ± t , κ , . . . , κ t ] −→ C [ N B ][ λ ± , . . . , λ ± t ] ,κ i (cid:55)→ y b i which is surjective by the definition of C [ N B ]. The kernel of this map is equal to ( (cid:81) l i < κ − l i i − (cid:81) l i > κ l i i ) l ∈ L by [MS05, Theorem 7.3]. Finally notice that ψ ( (cid:80) ti =1 b ki λ i κ i ) = y k ∂f λ /∂y k , which showsthe claim.We need the following result saying that the GKZ-system M βB is isomorphic to the restriction of theFourier-Laplace transformed GKZ system (cid:99) M ( β ,β ) B . Lemma 3.18.
Let i : { } × W −→ ˆ V = C z × W be the canonical inclusion. Then H (cid:16) i +1 (cid:99) M ( β ,β ) B (cid:17) (cid:39) M βB . Proof.
During the proof we will work with modules of global sections rather with the D -modules itself.Recall that the left ideal defining the quotient (cid:99) M ( β ,β ) B is generated by the operators (cid:98) (cid:3) l , (cid:98) E k − β k z and (cid:98) E − β z , where (cid:98) (cid:3) l := (cid:81) i : l i < ( z · ∂ λ i ) − l i − (cid:81) i : l i > ( z · ∂ λ i ) l i , (cid:98) E := z ∂ z + (cid:80) ti =1 zλ i ∂ λ i , (cid:98) E k := (cid:80) ti =1 b ki zλ i ∂ λ i . z − ( (cid:98) E − β z ) in this ideal show that have the an isomorphism of C [ z ± , λ , . . . , λ n ] (cid:104) ∂ λ , . . . ∂ λ n (cid:105) -modules (cid:99) M (cid:39) C [ z ± , λ , . . . , λ n ] (cid:104) ∂ λ , . . . ∂ λ n (cid:105) / C [ z ± , λ , . . . , λ n ] (cid:104) ∂ λ , . . . ∂ λ n (cid:105) (cid:98) I (43)where the left C [ z ± , λ , . . . , λ n ] (cid:104) ∂ λ , . . . ∂ λ n (cid:105) -ideal (cid:98) I is generated by (cid:98) (cid:3) l ∈ L and (cid:98) E k − β k for k ∈ { , . . . , d } .The D W -module corresponding to H (cid:16) i +1 (cid:99) M (cid:17) is given by (cid:99) M / ( z − (cid:99) M . Using the isomorphism (43) oneeasily sees that (cid:99) M / ( z − (cid:99) M (cid:39) M βB , which shows the claim. Proposition 3.19.
The O C z × W ◦ -module ◦ (cid:99) M ( β ,β ) B is locally-free of rank vol( Q ) .Proof. Notice that it is sufficient to show that ◦ (cid:99) M ( β ,β ) B is O C × W ◦ -coherent. Namely, ◦ (cid:99) M ( β ,β ) B /z · ◦ (cid:99) M ( β ,β ) B is O W ◦ -locally free of rank vol( Q ) by Lemma 3.10. Moreover, the restriction of ◦ (cid:99) M ( β ,β ) B to C ∗ z × W ◦ is a locally-free O C ∗ z × W ◦ -module by Lemma 3.14. Its restriction to { } × W ◦ is isomorphic tothe restriction of M βB to W ◦ by Lemma 3.18 which is locally free of rank vol( Q ). Now we use the factthat a coherent O -module which has everywhere the same rank is locally-free.It is actually sufficient to show the coherence of N := For z ∂ z ( ◦ (cid:99) M ( β ,β ) B ), as this is the same as ◦ (cid:99) M ( β ,β ) B when considered as an O C z × W ◦ -module. Let us denote by F • the natural filtration on R (cid:48) C z × W ◦ definedby F k R (cid:48) C z × W ◦ := P ∈ R (cid:48) C z × W ◦ | P = (cid:88) | α |≤ k g α ( z, λ )( z∂ λ ) α · . . . · ( z∂ λ t ) α t . This filtration induces a filtration F • on N which satisfies F k R (cid:48) C z × W ◦ · F l N = F k + l N . Obviously, forany k , F k N is O C z × W ◦ -coherent, so that it suffices to show that the filtration F • becomes eventuallystationary. Let P = (cid:80) | α |≤ k g α ( z, λ )( z∂ λ ) α · . . . · ( z∂ λ t ) α t then its symbol is defined as σ k ( P ) := (cid:88) | α | = k g α ( z, λ )( κ ) α · . . . · ( κ t ) α t ∈ O C z × W ◦ [ κ , . . . , κ t ] , which is a function on C z × T ∗ W ◦ with fiber variables κ , . . . , κ t . Let I be the radical ideal of the idealgenerated by the symbols of (cid:98) (cid:3) l ∈ L and (cid:98) E k − z · β k for k = 1 , . . . , t . Then the vanishing locus of I isthe R (cid:48) C z × W ◦ -characteristic variety of N . Notice that N is O C z × W ◦ -coherent if and only if its R (cid:48) C z × W ◦ -characteristic variety is a subset of C z × T ∗ W ◦ W ◦ . The proof of this fact is completely parallel to the D -module case (see e.g. [Pha79, Proposition 10.3]).To compute the R (cid:48) C z × W ◦ -characteristic variety, notice that the symbols of (cid:98) (cid:3) l ∈ L and (cid:98) E k − z · β k areindependent of z . Thus it is enough to compute its restriction to { } × W ◦ . Now notice that thegenerators of the ideal corresponding to the GKZ-system M βB have exactly the same symbols as theoperators above. Thus it is enough to show that the restriction of the GKZ-system M βB to W ◦ is O W ◦ -coherent. But this follows from [Ado94, Lemma 3.2 and 3.3] and the definition of W ◦ (see Definition 3.8and Lemma 3.12). Corollary 3.20.
The natural map ◦ (cid:99) M ( β ,β ) B → ◦ (cid:99) M ( β ,β ) B which is induced by the inclusion R C z × W ∗ →D C z × W ∗ is injective.Proof. Recall that D C z × W ∗ ⊗ R (cid:99) M ( β ,β ) B (cid:39) (cid:99) M ( β ,β ) B | C z × W ∗ and D C z × W ∗ (cid:39) R [ z ± ]. Thus the kernel of (cid:99) M ( β ,β ) B → (cid:99) M ( β ,β ) B | C z × W ∗ has z -torsion. On the open set C z × W ◦ ⊂ C z × W ∗ the module ◦ (cid:99) M ( β ,β ) B = (cid:99) M ( β ,β ) B | C z × W ◦ is O C z × W ◦ -locally free. In particular it has no z -torsion, but this shows the claim.In order to do this, consider once again the affine toric variety X aff = Spec ( C [ N B ]) from Definition3.7, which contains the torus g B ( S ) ∼ = S as an open subset. Denote by D the complement of S in X aff . We will consider X aff as a log scheme in the sense of logarithmic geometry (see, e.g., [Gro11]).More precisely, we endow X aff with divisorial log structure induced by D and W ∗ with the trivial logstructure. We consider the relative log de Rham complex Ω • X aff × W ∗ /W ∗ (log D ) ([Gro11, section 3.3]).We have isomorphisms Ω kX aff × W ∗ /W ∗ (log D ) ∼ = O X aff × W ∗ ⊗ Z (cid:86) k Z r .42 roposition 3.21. Let N B be a saturated semigroup. There exists the following R C z × W ◦ -linear iso-morphism H (cid:16) Ω • + sX aff × W ◦ /W ◦ (log D )[ z ] , zd − d y F ∧ (cid:17) ∼ = ◦ (cid:99) M (0 , B , which maps ω to .Proof. We first define the R C z × W -linear morphism ψ : (cid:99) M (0 , B −→ H (cid:16) Ω • + sX aff × W ∗ /W ∗ (log D )[ z ] , zd − d y F ∧ (cid:17) , (cid:55)→ ω , which is well-defined by 3.4. Let ω = (cid:88) α,γ,δ c αγδ λ γ . . . λ γ t t z δ y α · b . . . y α t · b t ω be a general element in Ω sX aff × W ∗ /W ∗ (log D )[ z ] with α ∈ N t , γ ∈ Z t and δ ∈ N . Then (cid:88) α,γ,δ c αγδ λ γ . . . λ γ t t z δ ( z∂ λ ) α . . . ( z∂ λ t ) α t is a preimage, which shows that the map ψ is surjective. Notice that the restricted map ◦ ψ : ◦ (cid:99) M (0 , B −→ H (Ω • + sX aff × W ◦ /W ◦ (log D )[ z ] , zd − d y F ∧ )is also surjective. Consider the following commutative diagram ◦ (cid:99) M (0 , B (cid:39) (cid:47) (cid:47) H (Ω • + sS × W ◦ /W ◦ [ z ± ] , zd − d y F ∧ ) ◦ (cid:99) M (0 , B ◦ ψ (cid:47) (cid:47) (cid:63)(cid:31) (cid:79) (cid:79) H (Ω • + sX aff × W ◦ /W ◦ (log D )[ z ] , zd − d y F ∧ ) (cid:79) (cid:79) where the upper horizontal map is an isomorphism by Proposition 3.3 and Lemma 3.4 , the left verticalmap is injective by Corollary 3.20 and the right vertical map is induced by the morphismΩ sX aff × W ◦ /W ◦ (log D )[ z ] −→ Ω sX aff × W ◦ /W ◦ ( ∗ D )[ z ± ] = Ω sS × W ◦ /W ◦ [ z ± ] . But this shows that ◦ ψ is also injective, which shows the claim. Notice that as a by-product, we alsoobtain that the morphism H (Ω • + sX aff × W ◦ /W ◦ (log D )[ z ] , zd − d y F ∧ ) −→ H (Ω • + sS × W ◦ /W ◦ [ z ± ] , zd − d y F ∧ )is injective. We recall in this section some rather well known notations and results concerning twisted Gromov-Witten invariants on the one hand, and basic constructions from toric geometry for smooth completeintersections in toric varieties on the other hand. Any of the statements of this section can be found ineither the original articles like [Kon95], [Giv98b, Giv98a], [CG07] (for twisted Gromov-Witten invariants),the references [Ful93], [CLS11] and [CK99], (for facts on toric geometry of complete intersections) butalso in the more recent paper [MM11], from which we borrow some of the notation. By collecting thematerial we need later here we hope to make this paper more self-contained.43 .1 Twisted and reduced quantum D -modules A smooth complete intersection inside a smooth projective variety can be described as the zero locus ofa generic section of a split vector bundle on that variety. Associated to such a bundle are the twistedGromov-Witten invariants , which we describe first. They give rise to the twisted quantum product,and to the twisted quantum- D -module. From this one can derive (basically by dividing by the kernel ofthe multiplication by the first Chern classes of the factors of the vector bundle) the reduced quantum D -module , which corresponds to the ambient part of the quantum cohomology of the subvariety. Wealso discuss this reduced module here, and we define pairings (coming from the Poincar´e pairing on theambient variety) on both the twisted and the reduced quantum D -module.Let X be a smooth projective n -dimensional variety. Let L , . . . , L c be line bundles on X which areglobally generated and define E := (cid:76) ci =1 L i . We are going to recall the construction of the so-calledtwisted quantum D -module QDM( X , E ) and the reduced quantum D -module QDM( X , E ). Our notationfollows the exposition in [MM11, Chapter 2.5].For l ∈ N and d ∈ H ( X , Z ) we denote by M ,l,d ( X ) the moduli space of stable maps of degree d fromcurves of genus 0 with l marked points to X . Denote by e i : M ,l,d ( X ) −→ X the evaluation at the i marked point for i ∈ { , . . . , l } and denote by π : M ,l +1 ,d ( X ) −→ M ,l,d ( X ) the map which forgets thelast marked point. Let E ,l,d be the locally free sheaf R π ∗ e ∗ l +1 E and for any j ∈ { , . . . , l } , let E ,l,d ( j )be the kernel of the surjective morphism E ,l,d −→ e ∗ j E which evaluates a section at the j -marked point.For i ∈ { , . . . , l } denote by N i the line bundle on M ,l,d ( X ) whose fiber at a point ( C, x , . . . , x l , f : C → X ) is the cotangent space T ∗ x i C . Put ψ i := c ( N i ) ∈ H ( M ,l,d ( X )). Definition 4.1.
Let l ∈ N , ( m , . . . , m l ∈ N l ) , γ , . . . , γ l ∈ H ∗ ( X ) and d ∈ H ( X , Z ) . The j -th twistedGromov-Witten invariant with descendants is denoted by (cid:104) τ m ( γ ) , . . . , (cid:94) τ m j ( γ j ) , . . . , τ m l ( γ l ) (cid:105) ,l,d := (cid:90) [ M ,l,d ( X )] vir c top ( E ,l,d ( j )) l (cid:89) i =1 ψ m i i e ∗ i γ i , where [ M ,l,d ( X )] vir is the virtual fundamental class of M ,l,d ( X ) . An invariant (cid:104) . . . , γ k , . . . (cid:105) ,l,d has tobe understood as (cid:104) . . . , τ ( γ k ) , . . . (cid:105) ,l,d . Below we will actually use only such non-descendant (i.e., with all m k = 0 ) invariants. Let ( T , T , . . . , T h ) be a homogeneous basis of H ∗ ( X ) such that T = 1 and T , . . . , T r is a basis of H ( X , Z ) modulo torsion which lies in the K¨ahler cone of X . Let T be the torus H ( X , C ) / πiH ( X , Z ).Then the basis T , . . . , T r of H ( X , Z ) gives rise to coordinates q = ( q , . . . , q r ) on T . Definition 4.2.
Let γ , . . . , γ ∈ H ∗ ( X , C ) and q ∈ T . The twisted small quantum product is definedby γ • twq γ := h (cid:88) a =1 (cid:88) d ∈ H ( X , Z ) q d (cid:104) γ , γ , (cid:101) T a (cid:105) , ,d T a . where q d is shorthand notation for q (cid:104) T ,d (cid:105) , . . . , q (cid:104) T r ,d (cid:105) r . Notice that (cid:104) γ , γ , (cid:101) T a (cid:105) , ,d (cid:54) = 0 only if d liesin the semigroup of effective classes, i.e. d ∈ Eff X ⊂ H ( X , Z ). Hence, by our assumption on thebasis T , . . . , T r , only positive powers of the q i appear in the formula above. Let ¯ T = C r be a partialcompactification of T with respect to the coordinates q , . . . q r . In the following we assume that thereexists an open neighborhood ¯ U of 0 ∈ ¯ T such that the twisted quantum product is convergent on ¯ U asa power series in q , . . . , q r . The twisted quantum product is associative, commutative and has T as aunit (see, e.g., [MM11, Proposition 2.14]).Put U := ¯ U ∩ T . In analogy to the untwisted case one defines a trivial vector bundle F on H ( X ) × C z × U with fiber H ∗ ( X ) together with a flat meromorphic connection ∇ ∂ t := ∂ t + 1 z T • twq , ∇ q a ∂ qa := q a ∂ q a + 1 z T a • twq , ∇ z∂ z := z∂ z − z E • twq + µ , where µ is the diagonal morphism defined by µ ( T a ) := ( deg ( T a ) − (dim C X − rk E )) T a and E := t T + c ( T X ) − c ( E ) is the so-called Euler field. 44efine a twisted pairing on H ∗ ( X ) by:( γ , γ ) tw := (cid:90) X γ ∪ γ ∪ c top ( E ) for γ , γ ∈ H ∗ ( X ) . This pairing is degenerate with kernel equal to ker m c top , where m c top is defined by m c top : H ∗ ( X ) −→ H ∗ ( X ) ,α (cid:55)→ c top ( E ) ∪ α and satisfies the Frobenius relation:( γ • twq γ , γ ) tw = ( γ , γ • twq γ ) tw for γ , γ , γ ∈ H ∗ ( X ) . Denote by F the sheaf sections of F and define an involution ι by ι : H ( X ) × C z × U −→ H ( X ) × C z × U , ( t , z, q ) (cid:55)→ ( t , − z, q ) . We define a ∇ -flat sesquilinear pairing S : ι ∗ ( F ) × F −→ O , ( s , s ) (cid:55)→ S ( s , s )( t , z, q ) = ( s ( t , − z, q ) , s ( t , z, q )) tw . We call H ∗ ( X ) := H ∗ ( X ) / ker m c top the reduced cohomology ring of ( X , E ). For γ ∈ H ∗ ( X ) denoteby γ its class in H ∗ ( X ). The pairing ( · , · ) tw gives rise to a pairing ( · , · ) red on H ∗ ( X ) by( γ , γ ) red := ( γ , γ ) tw for γ , γ ∈ H ∗ ( X ) . Because the kernel of ( · , · ) tw is ker m c top this pairing is well-defined and non-degenerate. Denote by F the trivial bundle on H ( X ) × C z × U with fiber H ∗ ( X ). The pairing S induces a pairing S on F by S ( s , s ) := S ( s , s ) , which is non-degenerate.Notice that H ∗ ( X ) is naturally graded because m c top is a graded morphism. Let ( φ , . . . , φ s (cid:48) ) be ahomogeneous basis of H ∗ ( X ) and denote by ( φ , . . . , φ s (cid:48) ) its dual basis w.r.t. ( · , · ) red . The reducedGromov-Witten invariants are defined by (cid:104) γ , . . . , γ l (cid:105) red ,l,d := (cid:104) γ , . . . , (cid:94) c top ( E ) γ l (cid:105) ,l,d and the reduced quantum product is γ • redq γ := s (cid:48) (cid:88) a =0 (cid:88) d ∈ H ( X , Z ) q d (cid:104) γ , γ , φ a (cid:105) red , ,d φ a , where the restriction is compatible with the multiplication , i.e. γ • twq γ = γ • redq γ . The bundle F carries the following connection: ∇ ∂ t := ∂ t + 1 z T • redq , ∇ q a ∂ qa + 1 z T a • redq , ∇ z∂ z := z∂ z − z E • redq + µ , where µ is the diagonal morphism defined by µ ( φ A ) := ( deg ( φ a ) − ( dim C X − rk E )) φ a and E := t T + c ( T X ) − c ( E ). One can show that ∇ is flat and S is ∇ -flat. Definition 4.3.
Consider the above situation of a smooth projective variety X and globally generatedline bundles L , . . . , L c .1. The triple ( F, ∇ , S ) is called the twisted quantum D -module QDM( X , E ) .2. The triple ( F , ∇ , S ) is called the reduced quantum D -module QDM( X , E ) . .2 Toric geometry of complete intersection subvarieties In this subsection we consider the case where the variety X from above is toric. It will be denoted by X Σ , where Σ is the defining fan (see below). We recall some well-known results on the toric descriptionof the total space of the bundle E resp. its dual, on Picard groups, K¨ahler cones etc. All this is neededin section 6 below.Let, as usual, N be a free abelian group of rank n for which we choose once and for all a basis whichidentifies it with Z n . Let Σ be a complete smooth fan in N R := N ⊗ R and X Σ the associated toricvariety, which is compact and smooth. We recall the toric description of the K¨ahler resp. the nef coneof Σ. Let Σ(1) = { R ≥ a , . . . , R ≥ a m } be the rays of Σ, where a i ∈ N ∼ = Z n are the primitive integralgenerators of the rays of Σ. Then we have an exact sequence0 −→ L A −→ Z Σ(1) = Z m −→ N −→ , (44)where the morphism Z m (cid:16) N is given by the matrix (henceforth called A ) having the vectors a , . . . , a m as columns. L A is the module of relations between these vectors. We also consider the dual sequence0 −→ M −→ ( Z Σ(1) ) ∨ = Z m −→ L ∨ A −→ , where M := N ∨ is the dual lattice. It is well known that as X Σ is smooth and compact, we have H ( X Σ , Z ) (cid:39) Pic( X Σ ) ∼ = L ∨ A , moreover, the group ( Z Σ(1) ) ∨ is the free abelian group generated by the torus invariant divisors on X Σ .We denote these generators by D , . . . , D m . Its images in L ∨ A (called D i ) are thus the cohomology classeswhich are Poincar´e dual to these divisors, and they generate the Picard group.Any element in (cid:0) Z Σ(1) (cid:1) ∨ ⊗ R can be considered as a function on N R (actually on the support of Σ,but this equals N R by completeness), which is linear on each cone of Σ, these are called piecewise linearfunctions with respect to Σ. For a given divisor D ∈ Div( X Σ ) ∼ = ( Z Σ(1) ) ∨ , we denote the piecewise linearfunction it corresponds to by ψ Σ D . Inside (cid:0) Z Σ(1) (cid:1) ∨ ⊗ R we have the cone of convex functions, which arethose piecewise linear functions ψ having the property that for any cone σ ∈ Σ and for any n ∈ N R , wehave ψ ( n ) ≤ ψ σ ( n ), where ψ σ is the extension to a linear function on all of N R of the restriction ψ | σ .The interior of the cone of convex functions are those which are strictly convex, that is, those such thatthe above inequality is strict on N R \ σ . Notice that any linear function on N is piecewise linear andthis inclusion is precisely given by M R (cid:44) → (cid:0) Z Σ(1) (cid:1) ∨ ⊗ R . We define the nef cone K X Σ of X Σ to be theimage of the cone of convex functions in (cid:0) Z Σ(1) (cid:1) ∨ ⊗ R under the projection (cid:0) Z Σ(1) (cid:1) ∨ ⊗ R (cid:16) L ∨ A ⊗ R .Its interior is the K¨ahler cone K ◦ X Σ of Σ. We assume that K ◦ X Σ is non-empty, which amounts to say that X Σ is projective. Let us recall the following description of the cone K X Σ , the proof of this fact can befound, e.g., in [CK99, section 3.4.2]. Lemma 4.4.
For any cone σ ∈ Σ , put J σ := { i ∈ { , . . . , m } | R ≥ a i / ∈ σ } and define ˇ σ := (cid:88) i ∈ J σ R ≥ D i ⊂ ( L ∨ A ) R . We call ˇ σ the anticone associated to σ . Then we have K X Σ = (cid:84) σ ∈ Σ ˇ σ ⊂ ( L ∨ A ) R . We proceed by considering the toric analogue of the situation from subsection 4.1. More precisely, let L = O X Σ ( L ) , . . . , L c = O X Σ ( L c ) be line bundles on X Σ with L , . . . , L c ∈ Div ( X Σ ). We suppose thatthe following two properties hold Assumption 4.5.
1. For all j = 1 , . . . , c , the line bundle L j is nef. Notice that according to [Ful93, Section 3.4], on atoric variety, L j is nef iff it is globally generated.2. Let − K X Σ be the anti-canonical divisor of X Σ . Then we assume that − K X Σ − (cid:80) cj =1 L j is nef. E := ⊕ cj =1 L j and consider the dual bundle E ∨ := H om O X Σ ( E , O X Σ ). We have the following fact. Definition-Lemma 4.6.
The total space V ( E ∨ ) := Spec O X Σ (cid:16) Sym O X Σ ( E ) (cid:17) of E ∨ , is a smooth toricvariety defined by a fan Σ (cid:48) which is described in the following way. First we define the set of rays Σ (cid:48) (1) :For this, we choose divisors D m + j = (cid:80) mi =1 d ji D i with d ji ≥ and O ( D m + j ) = L j . This choice ispossible due to Lemma 4.4 as all L j are nef. Write d i := ( d i , . . . , d ci ) ∈ Z c and put a (cid:48) i := ( a i , d i ) ∈ N (cid:48) := N × Z c ∼ = Z n + c . Moreover, letting e n +1 , . . . , e n + c be the last c standard generators of Z n + c , weput a (cid:48) m + j := e n + j . Then we let Σ (cid:48) (1) := { R ≥ a (cid:48) , . . . , R ≥ a (cid:48) m + c } and we group, as before, the columnvectors a (cid:48) , . . . , a (cid:48) m + c in a matrix A (cid:48) ∈ Mat (( n + c ) × ( m + c ) , Z ) . This means that A (cid:48) = (cid:18) A n,c ( d ji ) Id c (cid:19) . (45) The fan Σ (cid:48) is now defined as follows: For any set of vectors b , . . . , b r ∈ R k define (cid:104) b , . . . , b r (cid:105) := (cid:80) rj =1 R ≥ b j . Then we put Σ (cid:48) := (cid:8) (cid:104) a (cid:48) i , . . . , a (cid:48) i k , e j , . . . , e j t (cid:105) ⊂ N (cid:48) R (cid:12)(cid:12) (cid:104) a i , . . . , a i k (cid:105) ∈ Σ( k ) , { j , . . . , j t } ⊂ { n + 1 , . . . , n + c } (cid:9) . In other words, considering the canonical projection π : N (cid:48) R → N R which forgets the last c components,we have that σ (cid:48) ∈ Σ (cid:48) iff π ( σ (cid:48) ) ∈ Σ . In the following proposition, we list some rather obvious properties of the cohomology (resp. its toricdescription) of the space V ( E ∨ ). Proposition 4.7.
Let X Σ , L , . . . , L c and the sum E resp. its dual E ∨ be as above.1. The projection map p : V ( E ∨ ) (cid:16) X Σ induces an isomorphism p ∗ : H ∗ ( X Σ , Z ) ∼ = H ∗ ( V ( E ∨ ) , Z ) .2. Consider the analogue of sequence (44) for the matrix A (cid:48) , that is, the sequence −→ L A (cid:48) −→ Z Σ (cid:48) (1) = Z m + c −→ N (cid:48) −→ , (46) then we have an isomorphism L A −→ L A (cid:48) l = ( l , . . . , l m ) (cid:55)−→ l (cid:48) := ( l , . . . , l m , l m +1 , . . . , l m + c ) , (47) where l m + j := − (cid:80) mi =1 l i d ji = −(cid:104) c ( L j ) , l (cid:105) for all j = 1 , . . . , c , and where (cid:104)− , −(cid:105) is the non-degenerate intersection product between L ∼ = H ( X Σ , Z ) and Pic( X Σ ) . Notice that in the definitionof this isomorphism we consider L A resp. L A (cid:48) as embedded into Z m resp. Z m + c .3. The scalar extension H ( X Σ , R ) ∼ = → H ( V ( E ∨ ) , R ) of the isomorphism p ∗ from above identifies theK¨ahler cones (resp. the nef cones) K ◦ X Σ and K ◦ V ( E ∨ ) (resp. K X Σ and K V ( E ∨ ) ).4. The manifold V ( E ∨ ) is nef. Moreover, if s ∈ Γ( X Σ , E ) is generic, and Y := s − (0) is the zero locusof this section, then also Y is smooth and also nef.Proof. The first point follows from the fact that V ( E ∨ ) and X Σ are homotopy equivalent. The secondpoint follows from a direct calculation. For the third point notice that the isomorphism p ∗ restricted to H ( X Σ ) is given by p ∗ : H ( X Σ ) (cid:39) m (cid:77) i =1 Z D i / ( m (cid:88) i =1 a ki D i ) k =1 ,...,n −→ m + c (cid:77) i =1 Z D (cid:48) i / ( m + c (cid:88) i =1 a (cid:48) ki D (cid:48) i ) k =1 ,...,n + c (cid:39) H ( V ( E ∨ )) , m (cid:88) i =1 d i D i (cid:55)→ m (cid:88) i =1 d i D (cid:48) i . We first prove p ∗ ( K X Σ ) ⊂ K V ( E ∨ ) . Let D = (cid:80) mi =1 d i D i be a divisor in X Σ with D ∈ K X Σ . Then ψ Σ D isgiven on a maximal cone σ ∈ Σ( n ) by u Σ σ ∈ M (cid:39) Z n which is defined by (cid:104) u Σ σ , a i (cid:105) = − d i for a i ∈ σ . The47L-function ψ Σ D is convex if and only if for all σ ∈ Σ( n ) the following inequalities hold (cid:104) u Σ σ , a i (cid:105) ≥ − d i for all i ∈ { , . . . , m } . Now consider the corresponding PL-function ψ Σ (cid:48) p ∗ ( D ) for p ∗ ( D ). Let σ (cid:48) ∈ Σ (cid:48) ( n + c )be a maximal cone in Σ (cid:48) with σ (cid:48) = (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) , where { i , . . . , i n } ⊂ { , . . . , m } . Then u Σ (cid:48) σ (cid:48) ∈ M (cid:48) (cid:39) Z n + c is defined by (cid:104) u Σ (cid:48) σ (cid:48) , a (cid:48) i (cid:105) = − d i for i ∈ { i , . . . , i n } and (cid:104) u Σ (cid:48) σ (cid:48) , e i (cid:105) = 0 for i ∈ { n + 1 , . . . , n + c } . (48)But because of equation (48) we have (cid:104) u Σ (cid:48) σ (cid:48) , a (cid:48) i (cid:105) = (cid:104) u Σ σ , a i (cid:105) ≥ − d i for i ∈ { , . . . , m } , which shows that ψ Σ (cid:48) p ∗ ( D ) is convex, i.e. p ∗ ( D ) ∈ K V ( E ∨ ) . Now assume D (cid:48) ∈ K V ( E ∨ ) . Because p ∗ is anisomorphism, we can assume that D (cid:48) has a presentation (cid:80) m + ci =1 d (cid:48) i D (cid:48) i in which d (cid:48) m + j = 0 for j ∈ { , . . . , c } ,i.e. D (cid:48) = p ∗ ( D ) with D = (cid:80) mi =1 d (cid:48) i D i . Let σ ∈ Σ( n ) and σ (cid:48) ∈ Σ( n + c ) be maximal cones with π ( σ (cid:48) ) = σ .Because of the presentation of D (cid:48) we have (cid:104) u Σ (cid:48) σ (cid:48) , e i (cid:105) = 0 for i ∈ { n + 1 , . . . , n + c } . Therefore we have (cid:104) u Σ σ , a i (cid:105) = (cid:104) u Σ (cid:48) σ (cid:48) , a (cid:48) i (cid:105) ≥ − d i , which shows that ψ Σ D is convex, i.e. D ∈ K X Σ . The statement for the open parts follows from the factthat p ∗ is a homeomorphism.For the fourth point recall that V ( E ∨ ) is nef, i.e. has a nef anticanonical divisor, if the class of the divisor − K V ( E ∨ ) = m (cid:88) i =1 D (cid:48) i + c (cid:88) j =1 D (cid:48) m + j lies in K V ( E ∨ ) . Because of 3. it is enough to show that ( p ∗ ) − ( − K V ( E ∨ ) ) lies in K X Σ . But we have( p ∗ ) − ( − K V ( E ∨ ) ) = m (cid:88) i =1 D i − c (cid:88) j =1 m (cid:88) i =1 d ji D i = − K X Σ − c (cid:88) j =1 c ( L j )and the term on the right hand side lies in K X Σ by Assumption 4.5 2. Let s ∈ Γ( X Σ , E ) be a genericsection, then one can show that Y = s − (0) is smooth by repeatedly applying Bertini’s theorem. Thenefness of Y is obtained by repeatedly applying the adjunction formula and Assumption 4.5 2. .We finish this section by the following remark, which will not be explicitly used in the sequel, butwhich helps to understand the geometry of the torus embedding considered in the beginning of section2. More precisely, let S := Spec C [ Z n + c ] and denote again by g : S → P m + c the map defined by( y , . . . , y m + c ) (cid:55)−→ (1 : y a (cid:48) : . . . : y a (cid:48) m + c ). In section 2 we considered the factorization g : S j (cid:44) → X i (cid:44) → P m + c (with X := Im ( g )) where j is an open embedding and i is a closed embedding. However, we will alsoneed to consider some other factorization, namely, we write g = g (2) ◦ g (1) , where g (1) : S −→ C m × ( C ∗ ) c sends y to ( y a (cid:48) i ) i =1 ,...,m + c and g (2) is the composition of the two open embeddings C m × ( C ∗ ) c (cid:44) → C m + c and C m + c (cid:44) → P m + c . Now we have the following fact. Proposition 4.8.
The morphism g (1) is a closed embedding. Hence, we have X \ Im ( g ) ⊂ { µ · µ m +1 · . . . · µ m + c = 0 } , where we use ( µ : µ : . . . : µ m + c ) as homogeneous coordinates on P m + c and µ , . . . , µ m as coordinateson C m + c (resp. on ( C ∗ ) m + c , C m × ( C ∗ ) c etc).Proof. It suffices obviously to show the first statement. We will use a method similar to the proof of[RS15, Proposition 2.1]. First notice that the embedding α : S (cid:44) → ( C ∗ ) m + c sending y to ( y a (cid:48) i ) i =1 ,...,m + c
48s obviously closed, so that it suffices to show that im( g (1) ) ∩ ( C m \ ( C ∗ ) m ) × ( C ∗ ) c = ∅ . Recall thatim( g (1) ) is the closed subvariety of C m × ( C ∗ ) c defined by the binomial equations (cid:89) i : l (cid:48) i > µ l (cid:48) i i − (cid:89) i : l i < µ − l (cid:48) i i for any l (cid:48) ∈ L A (cid:48) (these equations form the toric ideal of A (cid:48) ). It was shown in loc.cit. that due to thecompactness of X Σ , there is some l lying in L A ∩ Z m> . Hence, the image l (cid:48) of l under the isomorphism(47) lies in Z m> × Z c< , as the coefficients d ji appearing in formula (47) are non-negative (see Definition4.6) and moreover, for fixed j , not all d ji can be zero. It follows that the toric ideal of A (cid:48) contains anequation m (cid:89) i =1 µ l (cid:48) i i − m + c (cid:89) i = m +1 µ − l (cid:48) i i , (49)where none of the exponents is zero. Now suppose that there is a point x = ( x , . . . , x m , x m +1 , . . . , x m + c ) ∈ im( g (1) ) ∩ ( C m \ ( C ∗ ) m ) × ( C ∗ ) c , that is, we have x i = 0 for some i ∈ { , . . . , m } , then as equation (49)vanishes on x , we must have some j ∈ { , . . . , c } with x m + j = 0, which contradicts the assumption that x ∈ ( C m \ ( C ∗ ) m ) × ( C ∗ ) c . Hence the intersection im( g (1) ) ∩ ( C m \ ( C ∗ ) m ) × ( C ∗ ) c is indeed empty fromwhich it follows that g (1) : S (cid:44) → C m × ( C ∗ ) c is a closed embedding. Remark:
The GKZ-systems (see Definition 2.8) associated to the matrix A (cid:48) is not necessary regular,as the vectors a (cid:48) , . . . , a (cid:48) m + c do not necessarily lie on an affine hyperplane in Z m + c (see [Hot98] for thisregularity criterion). The situation is similar to that considered in our earlier paper [RS15], and forthe same reasons as in loc.cit., we will work with the extended matrix A (cid:48)(cid:48) ∈ Mat((1 + n + c ) × (1 + m + c ) , Z ) with columns a (cid:48)(cid:48) , a (cid:48)(cid:48) , . . . , a (cid:48)(cid:48) m + c , where a (cid:48)(cid:48) i := (1 , a (cid:48) i ) and a (cid:48)(cid:48) := (1 , , a (cid:48)(cid:48) m + j = (1 , e n + j ) ∈ Z n + c +1 for j = 1 , . . . , c where e n + j is the n + j -th standard vector in C n + c . Wewrite L A (cid:48)(cid:48) for the module of relations between the columns of A (cid:48)(cid:48) , obviously we have an isomorphism L A (cid:48) → L A (cid:48)(cid:48) sending l = ( l , . . . , l m + c ) to ( − (cid:80) m + ci =1 l i , l ). As a matter of notation, we will often write theparameter of the GKZ-systems defined by the matrix A (cid:48)(cid:48) , which are vectors in C m + c by definition, as( α, β, γ ) ∈ C m + c , where α ∈ C , β ∈ C m and γ ∈ C c . In this section, we show a duality result for the GKZ-systems associated to the toric situation justdescribed. We will explain how to calculate the holonomic dual of the system M βA (cid:48)(cid:48) for some specific β , this is used to get a more precise description of the various D -module considered in sections 2 and3. The methods used here somehow similar [RS15, section 2.3], but we have to take into account thenon-compactness of the toric varieties involved. Proposition 5.1.
Let X Σ be smooth, toric and projective and suppose that L = O X Σ ( L ) , . . . , L c = O X Σ ( L c ) are nef line bundles on X Σ . However, we do not make any assumption on the positivity of − K X Σ − (cid:80) cj =1 L j . Let A (cid:48) be the matrix from in Definition 4.6 (i.e. with columns the primitive integralgenerator of the fan of V ( E ∨ ) ) . Then the semi-group ring C [ N A (cid:48) ] is normal and Cohen-Macaulay. Themap Ψ : N A (cid:48) −→ ( N A (cid:48) ) ◦ ,m (cid:55)−→ m + a (cid:48) m +1 + . . . + a (cid:48) m + c is a bijection. Hence, C [ N A (cid:48) ] is a Gorenstein ring where the generator of the canonical module ω C [ N A (cid:48) ] is given by the monomial y a (cid:48) m +1 + ... + a (cid:48) m + c . We can deduce the following immediate corollary.
Corollary 5.2.
In the situation of the last proposition, suppose moreover that − K X Σ − (cid:80) cj L j is nef.Let A (cid:48)(cid:48) be the extension considered at the end of section 4. Then also the semi-group N A (cid:48)(cid:48) is normal andwe have ( N A (cid:48)(cid:48) ) ◦ = a (cid:48)(cid:48) + a (cid:48)(cid:48) m +1 + . . . + a (cid:48)(cid:48) m + c + N A (cid:48)(cid:48) . ence C [ N A (cid:48)(cid:48) ] is a normal, Cohen-Macaulay and Gorenstein ring, with ω C [ N A (cid:48)(cid:48) ] ∼ = C [ N A (cid:48)(cid:48) ] · y a (cid:48)(cid:48) + a (cid:48)(cid:48) m +1 + ... + a (cid:48)(cid:48) m + c . Proof.
This follows directly by applying proposition 5.1 to the toric variety X Σ and the collection of nefline bundles L , . . . , L c , L c +1 := O X Σ ( − K X Σ − (cid:80) xj =1 L j ).The following lemma is a rather obvious consequence of the nefness condition of the bundles L , . . . , L c . Lemma 5.3.
Let as before X Σ be toric and let L , . . . , L c be nef line bundles. Consider the fan Σ (cid:48) ofthe space V ( E ∨ ) , where E = ⊕ cj =1 L j . Then the support supp(Σ (cid:48) ) is convex. As a consequence, we havethe following equality supp(Σ (cid:48) ) = R ≥ A (cid:48) (50) where R ≥ A (cid:48) := (cid:80) m + ci =1 R ≥ a (cid:48) i .Proof. This is obvious from the construction of Σ (cid:48) as presented in definition 4.6. Namely, for any j ∈ { , . . . , c } , the functions ψ Σ D m + j = (cid:80) mi =1 d ji ψ Σ D i are convex due to the nefness of L j (remember that O ( D m + j ) = L j ), and one can describe the set supp(Σ (cid:48) ) assupp(Σ (cid:48) ) = (cid:26) ( x , . . . , x n , x n +1 , . . . , x n + c | ( x , . . . , x n ) ∈ supp(Σ) = R n ,x n + j ≥ − ψ Σ D m + j ( x , . . . , x n ) ∀ j = 1 , . . . , c (cid:27) . Then the convexity of the set supp(Σ (cid:48) ) is precisely the convexity condition on the functions ψ Σ D m + j .For the second statement, notice that the inclusion supp(Σ (cid:48) ) ⊂ R ≥ A (cid:48) is trivial (and does not dependon the convexity of supp(Σ (cid:48) )). On the other hand, if supp(Σ (cid:48) ) is convex, then we have the inclusionsupp(Σ (cid:48) ) ⊃ Conv( a (cid:48) , . . . , a (cid:48) m + c ) , (51)where Conv( a (cid:48) , . . . , a (cid:48) m + c ) denotes the convex hull of the vectors a (cid:48) , . . . , a (cid:48) m + c , since the left hand sidemust contain the convex hull of any of its subsets. On the other hand, we obviously have that R ≥ A = (cid:8) λ · x | x ∈ Conv( a (cid:48) , . . . , a (cid:48) m + c ) , λ ∈ R ≥ (cid:9) , so that the desired inclusion supp(Σ (cid:48) ) ⊃ R ≥ A (cid:48) follows from equation (51) and the fact that the setSupp(Σ (cid:48) ) is conical, i.e., for all x ∈ Supp(Σ (cid:48) ) and all λ ∈ R ≥ we have that λ · x ∈ Supp(Σ (cid:48) ) .
Proof of the proposition.
We first show the normality of N A (cid:48) : Given any vector x (cid:48) ∈ R ≥ A (cid:48) ∩ N (cid:48) , then byequation (50) there is some maximal cone (cid:104) a i , . . . , a i n (cid:105) ∈ Σ such that x (cid:48) ∈ (cid:104) a (cid:48) i , . . . , a (cid:48) i n , a (cid:48) i n +1 , . . . , a (cid:48) i n +1 (cid:105) ∈ Σ (cid:48) (recall that a (cid:48) i n + j = a (cid:48) m + j = e n + j ). Hence we have an equation x (cid:48) = n + c (cid:88) k =1 λ k a (cid:48) i k (52)with λ k ∈ R ≥ . We know that ( a (cid:48) i , . . . , a (cid:48) i n + c ) = ( a (cid:48) i , . . . , a (cid:48) i n , e m +1 , . . . , e m + c ) is a Z -basis of N (cid:48) as (cid:104) a (cid:48) i , . . . , a (cid:48) i n + c (cid:105) is a smooth n + c -dimensional cone in Σ (cid:48) . Hence λ k ∈ N for k = 1 , . . . , n + c , and x (cid:48) ∈ N A (cid:48) , which is the defining property of normality of N A (cid:48) . It follows that C [ N A (cid:48) ] is Cohen-Macaulayby Hochster’s theorem ([Hoc72, Theorem 1]).It remains to show the second statement concerning the characterization of the interior points of N A (cid:48) .We will actually show the following Claim:
Let x (cid:48) ∈ N A (cid:48) . Consider the representation (52) of x (cid:48) as an element of (cid:80) n + cj =1 R ≥ a (cid:48) i j , that is, anequation x (cid:48) = (cid:80) m + ci =1 λ i a (cid:48) i ∈ N A (cid:48) , where λ k = 0 if k ∈ { , . . . , m }\{ i , . . . , i n } . Then x (cid:48) lies in ( N A (cid:48) ) ◦ iff λ i > i ∈ { m + 1 , . . . , m + c } = { i n +1 , . . . , i n + c } .Notice that a representation as in the claim is unique, if there are two maximal cones of Σ( n ) such that x (cid:48) is contained in both of the cones generated by the corresponding column vectors of A (cid:48) , then it lies ona common boundary, and the two expressions (52) are equal.50he claim implies that the map Ψ from the proposition is well-defined and surjective, and it is obviouslyinjective. In order to show the claim, notice that( N A (cid:48) ) ◦ = ( R ≥ A (cid:48) \ ∂ ( R ≥ A (cid:48) )) ∩ N (cid:48) = ( R ≥ A (cid:48) ∩ N (cid:48) ) \ ( ∂ ( R ≥ A (cid:48) ) ∩ N (cid:48) ) = N A (cid:48) \ ( ∂ ( R ≥ A (cid:48) ) ∩ N (cid:48) ) , so that we have to show that the points in ∂ ( R ≥ A (cid:48) ) ∩ N (cid:48) are precisely those from N A (cid:48) where in theabove representation (52) there is at least one index i ∈ { m + 1 , . . . , m + c } with λ i = 0. From Formula(50) we deduce that ∂ ( R ≥ A (cid:48) ) ⊂ (cid:91) (cid:104) a i ,...,a in (cid:105)∈ Σ A ( n ) ∂ (cid:104) a (cid:48) i , . . . , a (cid:48) i n , a (cid:48) m +1 , . . . , a (cid:48) m + c (cid:105) . More precisely, for each (cid:104) a i , . . . , a i n (cid:105) ∈ Σ( n ) the cone (cid:104) a (cid:48) i , . . . , a (cid:48) i n , a (cid:48) m +1 , . . . , a (cid:48) m + c (cid:105) has two types offacets: those that are facets of ∂ ( R ≥ A (cid:48) ) (call them “outer boundary”) and those which are not (“innerboundary”). The union (over all n -dimensional cones of Σ) of the outer boundaries is the set ∂ ( R ≥ A (cid:48) )we are interested in.The fan Σ (cid:48) is smooth, in particular simplicial, this implies that for any cone (cid:104) a (cid:48) i , . . . , a (cid:48) i n , a (cid:48) m +1 , . . . , a (cid:48) m + c (cid:105) ∈ Σ (cid:48) we have the following description of its boundary. ∂ (cid:104) a (cid:48) i , . . . , a (cid:48) i n + c (cid:105) = ∂ (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) ! = n (cid:83) k =1 (cid:104) a (cid:48) i , . . . , (cid:98) a (cid:48) i k , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) ∪ c (cid:83) l =1 (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e m +1 , . . . , (cid:98) e m + l , . . . , e m + c (cid:105) . The facet (cid:104) a (cid:48) i , . . . , (cid:98) a (cid:48) i k , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) is an inner boundary, i.e., it is not contained in ∂ ( R ≥ A (cid:48) ).This is a consequence of the completeness of Σ, namely, there is some other cone (cid:104) a j , . . . , a j n (cid:105) ∈ Σ having (cid:104) a i , . . . , (cid:98) a i k , . . . , a i n (cid:105) as a facet, and then similarly the cone (cid:104) a (cid:48) i , . . . , (cid:98) a (cid:48) i k , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) is afacet of both (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e n +1 , . . . , e n + c (cid:105) and (cid:104) a (cid:48) j , . . . , a (cid:48) j n , e n +1 , . . . , e n + c (cid:105) , hence it is not containedin ∂ ( R ≥ A (cid:48) ). However, the facet (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e n +1 , . . . , (cid:98) e n + l , . . . , e n + c (cid:105) (for l = 1 , . . . , c ) is an outerboundary, i.e., a facets of R ≥ A (cid:48) . We conclude that ∂ ( R ≥ A (cid:48) ) = (cid:83) (cid:104) a i ,...,a in (cid:105)∈ Σ( n ) (cid:20) c (cid:83) l =1 (cid:104) a (cid:48) i , . . . , a (cid:48) i n , e n +1 , . . . , (cid:98) e n + l , . . . , e n + c (cid:105) (cid:21) . We see that for any point ∂ ( R ≥ A (cid:48) ) ∩ N (cid:48) , there must be some l ∈ { , . . . , c } such that in the representation(52) the coefficient λ m + l is zero. This shows the claim, and proves that the map Ψ is an isomorphism.Finally, it follows from standard arguments about semigroup rings (see, e.g. [BH93, corollary 6.3.8]) that C [ N A (cid:48) ] is Gorenstein, and that the generator of the canonical module ω C [ N A (cid:48) ] is as claimed.As a consequence, we obtain the following duality result for those GKZ systems that we will be interestedin the sequel. Theorem 5.4. et A (cid:48)(cid:48) be as above, that is, suppose that its columns ( a (cid:48)(cid:48) , a (cid:48)(cid:48) , . . . , a (cid:48)(cid:48) m + c ) are of the form a (cid:48)(cid:48) i = (1 , a (cid:48) i ) where a (cid:48)(cid:48) = (1 , and where a (cid:48) i ( i = 1 , . . . , m + c ) are the integral primitive generator of thefan of V ( E ∨ ) . For β ∈ Z m + c , consider the GKZ-system M βA (cid:48)(cid:48) as in Definition 2.7.1. There is an isomorphism D ( M (0 , , A (cid:48)(cid:48) ) ∼ = M − ( c +1 , , A (cid:48)(cid:48) = M − a (cid:48)(cid:48) − (cid:80) cl =1 a (cid:48)(cid:48) m + l A (cid:48)(cid:48) .
2. Consider the natural good filtration F • M βA (cid:48)(cid:48) induced by the order filtration on D . Let D ( M βA (cid:48)(cid:48) , F • ) be the dual filtered module in the sense of [Sai88, section 2.4], i.e., D ( M βA (cid:48)(cid:48) , F • ) = ( D M βA (cid:48)(cid:48) , F D • ) where F D • ( D M βA (cid:48)(cid:48) ) is the filtration dual to F • M βA (cid:48)(cid:48) . Then we have D (cid:16) M − a (cid:48)(cid:48) − (cid:80) cl =1 a (cid:48)(cid:48) m + l A (cid:48)(cid:48) , F • (cid:17) ∼ = ( M (0 , , A (cid:48)(cid:48) , F • + n − ( m + c +1) ) . roof.
1. The proof is parallel to [Wal07, Proposition 4.1] or [RS15, Theorem 2.15 and Proposition2.18], so that we only sketch it here, referring to loc.cit. for details. First one has to define theso-called Euler-Koszul complex resp. co-complex (see [MMW05]). Its global sections complex K • ( T, E − β ) is a complex of free D V ⊗ R T -modules where R = C [ ∂ , ∂ , . . . , ∂ m + c ] and where T is a so-called toric R -module. A particular case is T = C [ N A (cid:48)(cid:48) ]. Notice that the terms of K • ( T, E − β ) are not free over D V . However, for T = C [ N A (cid:48)(cid:48) ], this complex is a resolution by left D V -modules of the modules M βA (cid:48)(cid:48) .The differentials of K • ( T, E − β ) are defined by the operators E and Z k entering in the definition of M βA (cid:48)(cid:48) . From a resolution of the toric ring C [ N A (cid:48)(cid:48) ] by free C [ ∂ , ∂ , . . . , ∂ m + c ]-modules one can also construct a resolution of M (0 , , A (cid:48)(cid:48) by free D V -modules.Applying Hom D V ( − , D V ) yields basically the same complex, but where the parameters in thedifferentials are changed, and where the toric module is now the canonical module of the ring C [ N A (cid:48)(cid:48) ]. Now from the Gorenstein property of C [ N A (cid:48)(cid:48) ] with the precise description of the interiorideal from Corollary 5.2 we obtain the desired result by taking the cohomology of the two complexes,that is, we can show the identification of the holonomic dual of M (0 , , A (cid:48)(cid:48) with M − ( c +1 , , A (cid:48)(cid:48) .2. The proof is literally the same as in [RS15, Proposition 2.19, 2.] with the indices shifted appropri-ately.As a consequence, we can make more specific statements on the parameter vectors of the various GKZ-systems occurring in the results of the previous sections. Corollary 5.5.
Consider the situation in section 2 where the matrix B is A (cid:48) , i.e., given by the primitiveintegral generators of the fan of V ( E ∨ ) , in particular, both N B = N A (cid:48) and N (cid:101) B = N A (cid:48)(cid:48) are normalsemigroups. Then1. The statements of Theorem 2.11, Theorem 2.16 and of Proposition 2.17 hold true for the parametervalues (cid:101) β = (0 , , , (cid:101) β (cid:48) = ( c + 1 , , ∈ Z n + c .2. The statements of Proposition 3.3 and of Theorem 3.6 hold true for the parameter values β =(0 , , β (cid:48) = (0 , ∈ Z n + c and for any β , β (cid:48) ∈ Z . For later use, we introduce the following piece of notation.
Definition 5.6.
In the situation of Theorem 5.4, we call the map φ : M − ( c +1 , , A (cid:48)(cid:48) −→ M (0 , , A (cid:48)(cid:48) induced by right multiplication by ∂ · ∂ m +1 · . . . · ∂ m + c the duality morphism. For any β ∈ Z , we obtainan induced morphism (cid:98) φ : (cid:99) M ( β , , − A (cid:48) −→ (cid:99) M ( β + c, , A (cid:48) given by right multiplication with ∂ m +1 · . . . · ∂ m + c (see 3.1 for the definition of the modules (cid:99) M β ). Thecase β = − c will be particularly important, and we will also call the map (cid:98) φ : (cid:99) M − (2 c, , A (cid:48) −→ (cid:99) M ( − c, , A (cid:48) the duality morphism. Remark:
In our previous paper [RS15], we obtained from a similar construction a non-degeneratepairing on the Fourier-Laplace transformed GKZ-system (see [RS15, corollary 2.20], where this systemwas called (cid:99) M (cid:101) A ). It was given by an isomorphism of (cid:99) M (cid:101) A to its holonomic dual (which is isomorphic toits meromorphic dual, see also the proof of Lemma 6.6 below). The fact that in the current situation,we only have a morphism (cid:98) φ : (cid:99) M − (2 c, , A (cid:48) −→ (cid:99) M ( − c, , A (cid:48) which is not an isomorphism unless c = 0 (inwhich case we are exactly in the situation of [RS15], see the remark at the end of section 6 of thispaper) corresponds to the fact that the pairing S on the twisted quantum D -module as introduced inDefinition 4.3 is degenerate. As we have seen in the definition of the reduced quantum- D -module, itbecomes non-degenerate when we divide out the kernel of the cup product with the first Chern classes ofthe line bundles L j . We will show below in corollary 6.14 that the reduced quantum D -module is part ofa non-commutative Hodge structure, which implies in particular that it carries a non-degenerate pairinglike the one from [RS15]. 52
Mirror correspondences
In this section we combine the results obtained so far with the GKZ-type description of the ambientresp. reduced quantum D -modules from [MM11] for the toric case. We obtain a mirror statementwhich identifies them with D -modules constructed from our Landau-Ginzburg models. The results fromsection 2 will be applied for the case where the matrix B (used for the construction of GKZ-systemsand of families of Laurent polynomials) is given by A (cid:48) (see Definition 4.6) the columns of which are theprimitive integral generators of the fan of the total bundle V ( E ∨ ). Recall also (remark at the end ofsection 4) that we denote by A (cid:48)(cid:48) the matrix constructed from A (cid:48) by adding 1 as an extra component toall columns and by adding (1 ,
0) as extra column. Hence, if B is equal to A (cid:48) , then the matrix (cid:101) B used insection 2 is exactly the matrix A (cid:48)(cid:48) . Recall also that the parameter of the GKZ-systems of the matrix A (cid:48)(cid:48) is written as ( α, γ, δ ) ∈ C m + c with α ∈ C , γ ∈ C m and δ ∈ C c .The starting point for our discussion here is the duality morphism from the last section. We need toconsider a slight variation of it, which is defined only outside the boundary λ i = 0 and only outside thebad parameter locus as defined in subsection 3.2. Recall that V = C λ × W , and that this bad parameterlocus of the family ϕ A (cid:48) was called W bad ⊂ W . The complement of this locus outside the boundary λ i = 0was called W ◦ , that is, W ◦ := W ∗ \ W bad . Definition-Lemma 6.1.
For any β = ( β , β , . . . , β m , β m +1 , . . . , β n + c ) ∈ Z n + c , consider the re-stricted, Fourier-Laplace transformed GKZ-system ∗ (cid:99) M ( β ,β ) A (cid:48) we have ∗ (cid:99) M ( β ,β ) A (cid:48) = D C z × W ∗ [ z − ] D C z × W ∗ [ z − ]( (cid:98) (cid:3) ∗ l ) l ∈ L A (cid:48) + D C z × W ∗ [ z − ]( (cid:98) E k − zβ k ) k =0 ,...,n + c , where (cid:98) (cid:3) ∗ l := (cid:81) i ∈{ ,...,m + c } : l i > λ l i i ( z · ∂ i ) l i − m + c (cid:81) i =1 λ l i i · (cid:81) i ∈{ ,...,m + c } : l i < λ − l i i ( z · ∂ i ) − l i , l ∈ L A (cid:48) (cid:98) E := z ∂ z + (cid:80) m + ci =1 λ i · z∂ i , (cid:98) E k := (cid:80) m + ci =1 a (cid:48) ki λ i · z∂ i k = 1 , . . . , n + c and moreover, ∗ (cid:99) M βA (cid:48) is the R C z × W ∗ -subalgebra generated by [1] , and we have ∗ (cid:99) M βA (cid:48) = R C z × W ∗ R C z × W ∗ ( (cid:98) (cid:3) ∗ l ) + R C z × W ∗ ( (cid:98) E k − zβ k ) k =0 ,...,n + c . Moreover, we define the modules ∗ (cid:98) N ( β ,β ) A (cid:48) as the cyclic quotients of D C z × W ∗ [ z − ] by the left ideal generatedby (cid:101) (cid:3) l for l ∈ L A (cid:48) and (cid:98) E k − zβ k for k = 0 , . . . , n + c , where (cid:101) (cid:3) l := (cid:81) i ∈{ ,...,m } : l i > λ l i i ( z · ∂ i ) l i (cid:81) i ∈{ m +1 ,...,m + c } : l i > l i (cid:81) ν =1 ( λ i ( z · ∂ i ) − z · ν ) − m + c (cid:81) i =1 λ l i i · (cid:81) i ∈{ ,...,m } : l i < λ − l i i ( z · ∂ i ) − l i (cid:81) i ∈{ m +1 ,...,m + c } : l i < − l i (cid:81) ν =1 ( λ i ( z · ∂ i ) − z · ν ) . Consider the morphism
Ψ : ∗ (cid:98) N (0 , , A (cid:48) −→ ∗ (cid:99) M − (2 c, , A (cid:48) (53) given by right multiplication with z c · (cid:81) m + ci = m +1 λ i . As it is obviously invertible, the two modules ∗ (cid:98) N (0 , , A (cid:48) and ∗ (cid:99) M − (2 c, , A (cid:48) are isomorphic. We define (cid:101) φ to be the composition (cid:101) φ := (cid:98) φ ◦ Ψ , where (cid:98) φ is the dualitymorphism introduced in Definition 5.6. In concrete terms, we have: (cid:101) φ : ∗ (cid:98) N (0 , , A (cid:48) −→ ∗ (cid:99) M ( − c, , A (cid:48) ,m (cid:55)−→ (cid:98) φ ( m · z c · λ m +1 · . . . · λ m + c ) = m · ( zλ m +1 ∂ m +1 ) · . . . · ( zλ m + c ∂ m + c ) . n view of corollary 5.5, 2. (see also Theorem 3.6) we obtain im( (cid:101) φ ) ∼ = im( (cid:98) φ ) ∼ = (id C z × j ) + (cid:91) M IC ( X ◦ , L ) . (54) For any β ∈ Z n + c , consider the R C z × W ∗ -subalgebra of D C z × W ∗ [ z − ] (cid:46) D C z × W ∗ [ z − ] (cid:16) ( (cid:101) (cid:3) l ) l ∈ L A (cid:48) (cid:17) + D C z × W ∗ [ z − ] (cid:16) (cid:98) E k − zβ k ) k =0 ,...,n + c (cid:17) generated by the element [1] and denote its restriction to C z × W ◦ by ◦ (cid:98) N ( β ,β ) A (cid:48) . Similarly to corollary3.20, we have ◦ (cid:98) N ( β ,β ) A (cid:48) = C [ z, λ ± , . . . , λ ± m + c ] (cid:104) z ∂ z , z∂ λ , . . . z∂ λ m + c (cid:105) (cid:16) ( (cid:101) (cid:3) l ) l ∈ L A (cid:48) , ( (cid:98) E k − z · β k ) k =0 ,...,n + c (cid:17) | C z × W ◦ . In the next lemma we want to describe the restriction of the D -module (cid:91) M IC ( X ◦ , L ) to C z × W ∗ . Lemma 6.2.
Consider the morphism (cid:98) φ : (cid:99) M − (2 c, , A (cid:48) −→ (cid:99) M ( − c, , A (cid:48) from Definition 5.6 and the isomor-phisms (cid:91) M IC ( X ◦ , L ) (cid:39) im( (cid:98) φ ) (cid:39) (cid:99) M − (2 c, , A (cid:48) /ker ( (cid:98) φ ) from Corollary 5.5 (see also Theorem 3.6). We havethe following isomorphism ( id C z × j ) + (cid:91) M IC ( X ◦ , L ) (cid:39) ∗ (cid:99) M − (2 c, , A (cid:48) / (cid:98) K M (cid:39) ∗ (cid:98) N (0 , , A (cid:48) / (cid:98) K N , where (cid:98) K M resp. (cid:98) K N are the sub- D -modules associated to the sub- D -modules { m ∈ ∗ (cid:99) M − (2 c, , | ∃ p ∈ Z , k ∈ N such that ( λ∂ + p ) . . . ( λ∂ + p + k ) m = 0 } resp. { n ∈ ∗ (cid:98) N (0 , , | ∃ p ∈ Z , k ∈ N such that ( λ∂ + p ) . . . ( λ∂ + p + k ) n = 0 } with ( λ∂ + i ) := (cid:81) m + cj = m +1 ( λ j ∂ j + i ) for i ∈ Z .Proof. We will first compute the restriction of M IC ( X ◦ , L ) to V ∗ = C λ × W ∗ . Recall the morphism φ : M − ( c +1 , , A (cid:48)(cid:48) −→ M (0 , , A (cid:48)(cid:48) from Definition 5.6. We know from Theorem 2.16 and from Propostion 2.17that M IC ( X ◦ , L ) (cid:39) M − ( c +1 , , A (cid:48)(cid:48) / ker( φ ) where ker( φ ) is given by { m ∈ M − ( c +1 , , A (cid:48)(cid:48) | ∃ n ∈ N such that ( ∂ · ∂ m +1 · · · ∂ m + c ) n m = 0 } . Notice that C [ λ ± ] ⊗ C [ λ ] M IC ( X ◦ , L ) (cid:39) ∗ M − ( c +1 , , A (cid:48)(cid:48) / ( C [ λ ± ] ⊗ C [ λ ] ker ( φ )), where ∗ M − ( c +1 , , A (cid:48)(cid:48) is the mod-ule of global sections of ∗ M − ( c +1 , , A (cid:48)(cid:48) . The notation C [ λ ] is shorthand for C [ λ , . . . , λ m , λ m +1 , . . . , λ m + c ],and the notation C [ λ ± ] is shorthand for C [ λ , . . . , λ m , λ ± m +1 , . . . , λ ± m + c ] (and not, as it is usual, shorthandfor C [ λ ± , . . . , λ ± m , λ ± m +1 , . . . , λ ± m + c ]).We want to characterize C [ λ ± ] ⊗ C [ λ ] ker ( φ ) inside ∗ M − ( c +1 , , A (cid:48)(cid:48) = C [ λ ± ] ⊗ C [ λ ] M − ( c +1 , , A (cid:48)(cid:48) . For this wedefine the following submodule in ∗ M − ( c +1 , , A (cid:48)(cid:48) : K := { m ∈ ∗ M − ( c +1 , , A (cid:48)(cid:48) | ∃ p ∈ Z , k ∈ N such that ∂ k +10 ( λ∂ + p ) . . . ( λ∂ + p + k ) m = 0 } . Consider the following element of C [ λ ± ] ⊗ C [ λ ] ker ( φ ): λ − p m +1 . . . λ − p c m + c ⊗ m with p , . . . , p c ∈ N , (55)i.e. there exists an n ∈ N such that ( ∂ · ∂ m +1 . . . ∂ m + c ) n +1 m = 0. Therefore we have0 = λ − p m +1 . . . λ − p c m + c ⊗ ( ∂ · ∂ m +1 . . . ∂ m + c ) n +1 m = λ − p m +1 . . . λ − p c m + c ⊗ ( λ m +1 . . . λ m + c ) n +1 ( ∂ · ∂ m +1 . . . ∂ m + c ) n +1 m = ∂ n +10 · ( λ − p m +1 . . . λ − p c m + c ⊗ ( λ∂ ) . . . ( λ∂ − n ) m )= ∂ n +10 ( λ∂ + p max ) . . . ( λ∂ + p min − n ) · ( λ − p m +1 . . . λ − p c m + c ⊗ m )= ∂ k +10 ( λ∂ + p ) . . . ( λ∂ + p + k ) · ( λ − p m +1 . . . λ − p c m + c ⊗ m ) , p max := max { p i } , p min := min { p i } , p := p min − n and k := p max − p min + n . Because C [ λ ± ] ⊗ C [ λ ] ker ( φ ) is generated by elements of the form (55), we see that C [ λ ± ] ⊗ C [ λ ] ker ( φ ) ⊂ K . Therefore wehave a surjective morphism C [ λ ± ] ⊗ C [ λ ] M IC ( X ◦ , L ) (cid:39) ∗ M − ( c +1 , , A (cid:48)(cid:48) / ( C [ λ ± ] ⊗ C [ λ ] ker ( φ )) (cid:16) ∗ M − ( c +1 , , A (cid:48)(cid:48) /K . Because C [ λ ± ] ⊗ C [ λ ] M IC ( X ◦ , L ) corresponds to the restriction of the simple D -module M IC ( X ◦ , L ) tothe open subset V ∗ , it is itself simple. Thus, ∗ M − ( c +1 , , A (cid:48)(cid:48) /K is either equal to 0 or is isomorphic to C [ λ ± ] ⊗ C [ λ ] M IC ( X ◦ , L ).In order to prove the lemma, we are going to show that K (cid:32) ∗ M − ( c +1 , , A (cid:48)(cid:48) . Denote by F •∗ M − ( c +1 , , A (cid:48)(cid:48) the good filtration on ∗ M − ( c +1 , , A (cid:48)(cid:48) which is induced by the order filtration on D V ∗ . Notice that we have K (cid:32) ∗ M − ( c +1 , , A (cid:48)(cid:48) ⇐⇒ gr F K (cid:32) gr F ∗ M − ( c +1 , , A (cid:48)(cid:48) (56)In order to show that gr F K (cid:32) gr F ∗ M − ( c +1 , , A (cid:48)(cid:48) , we first remark that gr F K ⊂ { m ∈ gr F ∗ M − ( c +1 , , A (cid:48)(cid:48) | ∃ k ∈ N such that µ k +10 λ k +1 µ k +1 m = 0 } , where λ = ( λ m +1 · · · λ m + c ), µ = ( µ m +1 · · · µ m + c ) and µ i is the symbol σ ( ∂ λ i ).Thus, in order to show the right hand side of (56), it is enough to show that char( ∗ M − ( c +1 , , A (cid:48)(cid:48) ) = supp ( gr F ∗ M − ( c +1 , , A (cid:48)(cid:48) ) ⊂ T ∗ ( V ∗ ) is not contained in { µ · µ · λ = 0 } .Therefore it is enough to find a vector ( µ (cid:48) , λ (cid:48) ) ∈ char( ∗ M − ( c +1 , , A (cid:48)(cid:48) ) ⊂ T ∗ ( V ∗ ) with µ (cid:48) · µ (cid:48) · λ (cid:48) (cid:54) = 0, resp.a vector ( µ (cid:48) , λ (cid:48) ) ∈ char( M − ( c +1 , , A (cid:48)(cid:48) ) ⊂ T ∗ ( V ) with µ (cid:48) · µ (cid:48) (cid:54) = 0 and λ (cid:48) i (cid:54) = 0 for i = 1 , . . . , m + c .Notice that we havechar( M − ( c +1 , , A (cid:48)(cid:48) ) = char( M (0 , , A (cid:48)(cid:48) ) = char(FL( M (0 , , A (cid:48)(cid:48) )) = char( h + O T ) , where the first equality follows from [GKZ89, Theorem 4], the second equality follows e.g. from [Bry86,Corollaire 7.25] and the third equality follows from [SW09, Corollary 3.7]. Recall that the coordinateson V (cid:48) are denoted by µ i for i = 0 , . . . , m + c and the symbols of ∂ µ i are denoted by λ i . We now computethe fiber of char( h + O T ) → V (cid:48) over the point µ = (1 , . . . , h : T −→ V (cid:48) , ( y , . . . , y n + c ) (cid:55)→ ( y a (cid:48)(cid:48) , . . . , y a (cid:48)(cid:48) m + c )can be factored into a closed embedding h (cid:48) : T → ( C ∗ ) m + c +1 and an open embedding ( C ∗ ) m + c +1 → V (cid:48) .Therefore the fiber of the characteristic variety over (1 , . . . ,
1) is just the fiber of the conormal bundle of h (cid:48) ( T ) in ( C ∗ ) m + c +1 . The tangent space of h (cid:48) ( T ) at (1 , . . . ,
1) is generated by m + c (cid:88) i =0 a (cid:48)(cid:48) ki ∂ µ i for k = 0 , . . . , n + c . Therefore (1 , λ (cid:48) ) lies in char( h + O T ) if and only if (cid:80) m + ci =0 a (cid:48)(cid:48) ki λ (cid:48) i = 0 for all k = 0 , . . . , n + c . So it remainsto prove that there exists such a λ (cid:48) with λ (cid:48) i (cid:54) = 0 for i = 1 , . . . , m + c . First notice that it is enough toconstruct a ( λ ◦ , . . . , λ ◦ m + c ) with m + c (cid:88) i =1 a (cid:48) ki λ ◦ i = 0 (57)for all k = 1 , . . . , n + c and λ ◦ i (cid:54) = 0 for all i = 1 , . . . , m + c . Recall the structure of the matrix A (cid:48) : A (cid:48) = (cid:18) A n,c ( d ji ) Id c (cid:19) , (58)where d ji ≥ a i of the matrix A are the primitive integral generators of the raysof the fan Σ corresponding to a complete, smooth toric variety X Σ . This ensures the existence of55 λ ◦ , . . . , λ ◦ m ) ∈ Z m> with (cid:80) mi =1 λ ◦ i a i = 0. Setting λ ◦ m + j := − (cid:80) mi =1 d ji λ ◦ i , we have constructed anelement ( λ ◦ , . . . , λ ◦ m + c ) with λ ◦ j (cid:54) = 0 and satisfying (cid:80) m + ci =1 a (cid:48) ki λ ◦ i = 0. Summarizing, this shows that K (cid:32) ∗ M − ( c +1 , , A (cid:48)(cid:48) , i.e. C [ λ ± ] ⊗ C [ λ ] M IC ( X ◦ , L ) (cid:39) ∗ M − ( c +1 , , A (cid:48)(cid:48) /K . Applying the localized Fourier-Laplace transformation to this isomorphism, we obtain the first isomor-phism in the statement of the lemma. The second isomorphism follows from the D -linearity of theisomorphism ∗ (cid:99) M − (2 c, , A (cid:48) (cid:39) ∗ (cid:98) N (0 , , A (cid:48) .As in [RS15, section 3], we proceed by studying the restriction of the modules ∗ M βA (cid:48)(cid:48) , ∗ (cid:99) M βA (cid:48)(cid:48) and ∗ (cid:98) N βA (cid:48)(cid:48) to the K¨ahler moduli space of V ( E ∨ ) as described in the second part of section 4 (see Lemma 4.4 andProposition 4.7). The following construction has some overlap with the considerations in subsection 2.4on which we comment later.We apply Hom Z ( − , C ∗ ) to the exact sequence (46) to obtain the following exact sequence1 −→ ( C ∗ ) n + c −→ ( C ∗ ) m + c −→ L ∨ A (cid:48) ⊗ C ∗ −→ . (59)We will identify the middle torus with Spec C [ λ ± , . . . , λ ± m + c ], this space was called W ∗ in section 2.Choose a basis ( p , . . . , p r ) of L ∨ A (cid:48) with the following properties1. p a ∈ K V ( E ∨ ) = K X Σ for all a = 1 , . . . , r ,2. (cid:80) m + ci =1 D i ∈ (cid:80) ra =1 R ≥ p a .Using the basis ( p a ) a =1 ,...,r , we identify L ∨ A (cid:48) ⊗ C ∗ with ( C ∗ ) r and obtain coordinates q , . . . , q r on thisspace. We will write KM for this space and call it complexified K¨ahler moduli space. Notice that thechoice of coordinates is considered as part of the data of KM , that is, we really have KM = ( C ∗ ) r andnot only KM = L ∨ A (cid:48) ⊗ C ∗ . Notice that this space already occurred in section 2.4 in a slightly more generalcontext (which is consistent with the situation considered here, see the explanations after formula (30).Consider the embedding L A (cid:48) (cid:44) → Z m + c , which is given by a matrix L ∈ Mat(( m + c ) × r, Z ) with respectto the basis p ∨ a of L A (cid:48) and the natural basis of Z m + c . Choose a section Z m + c → L A (cid:48) of this inclusion,which is given by a matrix M ∈ Mat( r × ( m + c ) , Z ). This defines a section on the dual lattices, i.e. asection L ∨ A (cid:48) → Z m + c of the projection Z m + c → L ∨ A (cid:48) and a closed embedding (cid:37) (cid:48) : KM = ( C ∗ ) r (cid:44) → W ∗ .We will need to consider a slight twist of this morphism. Let ι : W ∗ → W ∗ be the involution given by ι ( λ i ) := ( − ε ( i ) λ i with ε ( i ) = 0 for i = 1 , . . . , m and ε ( i ) = 1 for i = m + 1 , . . . , m + c .We will further restrict our objects of study to that part of the complexified K¨ahler moduli spacewhich maps to the set of good parameters in W = C m + c as discussed in subsection 3.2. Hence we put KM ◦ := ( ι ◦ (cid:37) (cid:48) ) − ( W ◦ ) ⊂ KM , and write (cid:37) := ι ◦ (cid:37) (cid:48) : KM ◦ (cid:44) → W ∗ . We can now define the main object of study of this paper. We are going to use the constructions ofthe subsections 2.4 and 3.2, in particular, the diagrams (25), (31) and (40). We consider the composedmorphism α ◦ β : Z X → C λ × KM as defined by diagram (31). Let Z ◦ X := ( α ◦ β ) − ( C λ × KM ◦ ) ⊂ Z X be the subspace which is parameterized by the good parameter locus KM ◦ inside KM .For future reference, let us collect the relevant morphisms once again in a diagram, in which the spaces Z ◦ , Z ◦ X aff and Z ◦ X are defined by the requirement that all squares are cartesian. For simplicity of the56otation, we denote by α, β , γ and γ also the corresponding restrictions above C λ × KM ◦ . S j (cid:15) (cid:15) Γ ∼ = S × W π S (cid:111) (cid:111) θ (cid:15) (cid:15) Γ ∗ ∼ = S × W ∗ ζ (cid:15) (cid:15) (cid:111) (cid:111) S × KM γ (cid:15) (cid:15) (cid:111) (cid:111) S × KM ◦ (cid:111) (cid:111) γ (cid:15) (cid:15) X aff j (cid:15) (cid:15) Z X aff ∼ = X aff × W θ (cid:15) (cid:15) (cid:111) (cid:111) Z ∗ X aff ∼ = X aff × W ∗ ζ (cid:15) (cid:15) (cid:111) (cid:111) Z X aff ∼ = X aff × KM γ (cid:15) (cid:15) (cid:111) (cid:111) Z ◦ X aff ∼ = X aff × KM ◦ (cid:111) (cid:111) γ (cid:15) (cid:15) X i (cid:15) (cid:15) Z Xη (cid:15) (cid:15) (cid:111) (cid:111) Z ∗ Xε (cid:15) (cid:15) (cid:111) (cid:111) Z Xβ (cid:15) (cid:15) (cid:111) (cid:111) Z ◦ X (cid:111) (cid:111) β (cid:15) (cid:15) P ( V (cid:48) ) Z π Z (cid:111) (cid:111) π Z (cid:15) (cid:15) Z ∗ δ (cid:15) (cid:15) (cid:111) (cid:111) Z α (cid:15) (cid:15) (cid:111) (cid:111) Z ◦ (cid:111) (cid:111) α (cid:15) (cid:15) V V ∗ (cid:111) (cid:111) C λ × KM (cid:111) (cid:111) C λ × KM ◦ (cid:111) (cid:111) id C λ × (cid:37) (cid:104) (cid:104) (60) Definition 6.3.
The non-affine Landau-Ginzburg model associated to ( X Σ , L , . . . , L c ) is the mor-phism Π : Z ◦ X −→ C λ × KM ◦ , which is by definition the restriction of the universal family of hyperplane sections of X , i.e, of themorphism π Z ◦ η : Z X → V to the parameter space KM ◦ . We recall once again that X is defined as theclosure of the embedding g : S → P ( V (cid:48) ) sending ( y , . . . , y n + c ) to (1 : y a (cid:48) : . . . : y a (cid:48) m + c ) where a (cid:48) i are thecolumns of the matrix A (cid:48) from Definition 4.6.We also consider the restrictions π = α ◦ β ◦ γ : Z ◦ X aff ∼ = X aff × KM ◦ → C λ × KM ◦ resp. π = α ◦ β ◦ γ ◦ γ : S × KM ◦ : → C λ × KM ◦ . These are nothing but the family of Laurent polynomials ( y, q ) (cid:55)−→ (cid:32) − m (cid:88) i =1 q m i · y a (cid:48) i + m + c (cid:88) i = m +1 q m i · y a (cid:48) i , q (cid:33) , where the monomial y a (cid:48) i is seen as an element of O X aff in the first case and as an element of O S in thesecond case. Here m i is the i ’th column of the matrix M ∈ Mat( r × ( m + c ) , Z ) from above. Noticethat the first component of π has been split in two sums with opposite signs of each summand due to theaction of the involution ι entering in the definition of the morphism (cid:37) : KM ◦ (cid:44) → W ∗ . Both morphisms π and π are called the affine Landau-Ginzburg model of ( X Σ , L , . . . , L c ) . As we will see later, the affine Landau-Ginzburg model is related to the twisted quantum D -moduleQDM( X Σ , E ) whereas the reduced quantum D -module QDM( X Σ , E ) can be obtained from the non-affineLandau-Ginzburg model Π : Z ◦ X → C λ × KM ◦ . The next results are parallel to [RS15, corollary3.3. and corollary 3.4]. They show that the calculation of the Gauß-Manin system resp. the intersectioncohomology D -module from section 2 can be used to describe the corresponding objects for the morphismΠ.We consider, as in subsection 3.1, the localized partial Fourier-Laplace transformation, this time withbase KM ◦ , that is, let j τ : C ∗ τ × KM ◦ (cid:44) → C τ × KM ◦ , j z : C ∗ τ × KM ◦ (cid:44) → C z × KM ◦ then we putFL loc KM ◦ := j z, + j + τ FL KM ◦ . Lemma 6.4.
We have FL loc KM ◦ (cid:0) H π + O S ×KM ◦ (cid:1) ∼ = (id C z × (cid:37) ) + ∗ (cid:99) M ( − c, , A (cid:48) . Similarly, the isomorphism FL loc KM ◦ (cid:0) H π † O S ×KM ◦ (cid:1) ∼ = (id C z × (cid:37) ) + ∗ (cid:98) N (0 , , A (cid:48) holds. C z × (cid:37) ) is obviously non-characteristic for both of the modules ∗ (cid:99) M ( − c, , A (cid:48)(cid:48) and ∗ (cid:98) N (0 , , A (cid:48)(cid:48) as their singular locus is contained in( { , ∞} × KM ◦ ) ∪ (cid:0) P z × ( W ∗ \KM ◦ ) (cid:1) . Hence, the complexes (id C z × (cid:37) ) + ∗ (cid:99) M ( − c, , A (cid:48) and (id C z × (cid:37) ) + ∗ (cid:98) N (0 , , A (cid:48) have cohomology only in degreezero. Proof.
The proof of the first isomorphism is the same as [RS15, corollary 3.3]: Consider the cartesiandiagram (which is part of the diagram (60)) S × KM ◦ (cid:47) (cid:47) π (cid:15) (cid:15) Γ ∗ ∼ = S × W ∗ ϕ (cid:15) (cid:15) C λ × KM ◦ (cid:31) (cid:127) id C λ × (cid:37) (cid:47) (cid:47) V ∗ = C λ × W ∗ (61)then the base change property (Theorem 2.1) and the commutation of FL loc with inverse images showsthat FL loc KM ◦ ( H π + O S ×KM ◦ ) ∼ = (id C z × (cid:37) ) + G + | V ∗ , where G + is the D C λ × W -module introduced in subsection 3.1, and then one concludes using Proposition3.3.Concerning the second isomorphism, we use base change (with respect to the morphism id C λ × (cid:37) indiagram (61)) for proper direct images and exceptional inverse images. However, the latter ones equalordinary inverse images if the horizontal morphisms in the above diagram are non-characteristic for themodules in question. This is the case by Proposition 2.22, 2., so that we obtainFL loc KM ◦ (cid:0) H π † O S ×KM ◦ (cid:1) ∼ = (id C z × (cid:37) ) + FL locW (cid:0) H ϕ B, † O S × W (cid:1) | V ∗ = (id C z × (cid:37) ) + G †| V ∗ . The second part of corollary 5.5 (and the second part of Proposition 3.3) tells us that G † ∼ = (cid:99) M − ( c, , A (cid:48) .However, the isomorphism Ψ : ∗ (cid:98) N (0 , , A (cid:48) −→ ∗ (cid:99) M − (2 c, , A (cid:48) given by right multiplication with z c · λ m +1 · . . . · λ m + c (see equation (53)) shows that(id C z × (cid:37) ) + ∗ (cid:99) M − (2 c, , A (cid:48) ∼ = (id C z × (cid:37) ) + ∗ (cid:98) N (0 , , A (cid:48) so that finally we arrive at the desired equalityFL loc KM ◦ (cid:0) H π † O S ×KM ◦ (cid:1) ∼ = (id C z × (cid:37) ) + ∗ (cid:98) N (0 , , A (cid:48) . Next we show the analog of Proposition 3.21 for the morphism π . Lemma 6.5.
Let (cid:101) F : X aff × KM ◦ → C λ be the first component of the morphism π , then we have thefollowing isomorphism of R C z ×KM ◦ -modules z − c H n + c (Ω • X aff ×KM ◦ / KM ◦ (log D )[ z ] , zd − d (cid:101) F ) ∼ = (id C z × (cid:37) ) ∗ (cid:16) ∗ (cid:99) M ( − c, , A (cid:48) (cid:17) . (62) Proof.
In order to show the statement, notice that by definition H n + c (Ω • X aff × W ∗ /W ∗ (log D )[ z ] , zd − dF )is the cokernel of Ω n + c − X aff × W ∗ /W ∗ (log D )[ z ] zd − dF −→ Ω n + cX aff × W ∗ /W ∗ (log D )[ z ] , that is, the cokernel of an O C z × W ∗ -linear morphism between free (though not coherent) O C z × W ∗ -modules. Hence tensoring with O C z ×KM ◦ yields the exact sequenceΩ n + c − X aff ×KM ◦ / KM ◦ (log D )[ z ] zd − d (cid:101) F −→ Ω n + cX aff ×KM ◦ / KM ◦ (log D )[ z ] −→O C z ×KM ◦ ⊗ O C z × W ∗ H n + c (Ω • X aff × W ∗ /W ∗ (log D )[ z ] , zd − dF ) −→ H n + c (Ω • X aff ×KM ◦ / KM ◦ (log D )[ z ] , zd − d (cid:101) F ) = O C z ×KM ◦ ⊗ O C z × W ∗ H n + c (Ω • X aff × W ∗ /W ∗ (log D )[ z ] , zd − dF ) . Notice that the restriction functor ( O C z ×KM ◦ ⊗ O C z × W ∗ − ) is defined via the embedding (cid:37) : KM ◦ (cid:44) → W ∗ ,and hence involves the involution ι . Therefore the function (cid:101) F appears on the left hand side of the lastformula, whereas on the right hand side we have to put F .We know by Proposition 3.21 that z − c H n + c (Ω • X aff × W ◦ /W ◦ (log D )[ z ] , zd − dF ) ∼ = z − c ◦ (cid:99) M (0 , , A (cid:48) . On the other hand, we know from equation (38) that right multiplication by z c induces an isomorphism (cid:16) (cid:99) M (0 , , A (cid:48) (cid:17) | C z × W ◦ · z c −→ (cid:16) (cid:99) M ( − c, , A (cid:48) (cid:17) | C z × W ◦ which maps z − c ◦ (cid:99) M (0 , , A (cid:48) ⊂ (cid:16) (cid:99) M (0 , , A (cid:48) (cid:17) | C z × W ◦ isomorphically to ◦ (cid:99) M ( − c, , A (cid:48) ⊂ (cid:16) (cid:99) M (0 , , A (cid:48) (cid:17) | C z × W ◦ . Thedesired statement, i.e., Formula (62) follows as the restriction map (cid:37) : KM ◦ (cid:44) → W ∗ factors by definitionover W ◦ .Similarly to the last statement, we now give a geometric interpretation of the (restriction to C z × KM ◦ ofthe) modules ∗ (cid:98) N (0 , , A (cid:48) using the twisted relative logarithmic de Rham complex on X aff × KM ◦ . We needsome preliminary notations. Denote by ( − ) (cid:48) the duality functor in the category of locally free O C z ×KM ◦ -modules with meromorphic connection with poles along { } × KM ◦ , that is, if ( F , ∇ ) is an object of thiscategory, we put ( F , ∇ ) (cid:48) := ( H om O C z ×KM◦ ( F , O C z ×KM ◦ ) , ∇ (cid:48) ), where ∇ (cid:48) is the dual connection. Noticethat the R C z ×KM ◦ -modules from isomorphism (62) are actually objects of this category. Notice also thatthe duality functor in the category of R C z ×KM ◦ -modules (i.e., the functor E xt r +1 R C z ×KM◦ ( − , R C z ×KM ◦ ))restricts to ( − ) (cid:48) on the subcategory described above (this follows from [DS03, Lemma A.12]).As a piece of notation, for any complex manifold M we denote by σ the involution of C z × M definedby ( z, x ) (cid:55)→ ( − z, x ). Lemma 6.6.
There is an isomorphism of R C z ×KM ◦ -modules σ ∗ z n (cid:16) H n + c (Ω • X aff ×KM ◦ / KM ◦ (log D )[ z ] , zd − d (cid:101) F ) (cid:17) (cid:48) ∼ = −→ (id C z × (cid:37) ) ∗ (cid:16) ∗ (cid:98) N (0 , , A (cid:48) (cid:17) . Proof.
Consider the filtration on D C z × W resp. on D C z × W ∗ which extends the order filtration on D W (resp. on D W ∗ ) and for which z has degree − ∂ z has degree 2. Denote by G • the induced filtrationson the modules ∗ (cid:98) N (0 , , A (cid:48) and (cid:99) M ( − c, , A (cid:48) resp. on ∗ (cid:99) M ( − c, , A (cid:48) , in particular, we have G (cid:16) ∗ (cid:98) N (0 , , A (cid:48) (cid:17) = ∗ (cid:98) N (0 , , A (cid:48) and G (cid:16) (cid:99) M ( − c, , A (cid:48) (cid:17) = (cid:99) M ( − c, , A (cid:48) resp. G (cid:16) ∗ (cid:99) M ( − c, , A (cid:48) (cid:17) = ∗ (cid:99) M ( − c, , A (cid:48) .Similar to the proof of [RS15, Proposition 2.18, 3.], we consider the saturation of the filtration F • on M βA (cid:48)(cid:48) by ∂ − λ . More precisely, we first notice that Lemma 3.2 can be reformulated by saying that for any β (cid:48) = ( β (cid:48) , β (cid:48) , . . . , β (cid:48) n + c ) ∈ Z n + c , we have (cid:99) M βA (cid:48) = FL W (cid:16) M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] (cid:17) , where β = β (cid:48) + 1 and β i = β (cid:48) i for i = 1 , . . . , n + c and where we write M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] := D V [ ∂ − λ ] ⊗ D V M β (cid:48) A (cid:48)(cid:48) .Now we consider the natural localization morphism (cid:99) loc : M β (cid:48) A (cid:48)(cid:48) → M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] and we put F k M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] := (cid:88) j ≥ ∂ − jλ (cid:99) loc (cid:16) F k + j M β (cid:48) A (cid:48)(cid:48) (cid:17) . As we have F k M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] = im (cid:0) ∂ kλ C [ λ , λ , . . . , λ m + c ] (cid:104) ∂ − λ , ∂ − λ ∂ λ , . . . , ∂ − λ ∂ λ m + c (cid:105) (cid:1) in M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] , F k M β (cid:48) A (cid:48)(cid:48) [ ∂ − λ ] on (cid:99) M βA (cid:48) is precisely G k (cid:99) M βA (cid:48) . We conclude from [Sai89, formula2.7.5] and from the fact that Fourier-Laplace transformation commutes with the duality functor up tothe action of σ that( G l (cid:99) M − ( c, , A (cid:48) ) (cid:48) = H om O C z × W (cid:16) G l (cid:99) M − ( c, , A (cid:48) , O C z × W (cid:17) ! = σ ∗ G D l +( m + c +2) (cid:99) M (1 , , A (cid:48) , where G D • (cid:99) M (1 , , A (cid:48) is the filtration induced by the saturation of the filtration on M (0 , , A (cid:48)(cid:48) dual to the orderfiltration F • on M − ( c +1 , , A (cid:48)(cid:48) . By Theorem 5.4, 2. and by restriction to C z × W ∗ we obtain G D • ∗ (cid:99) M (1 , , A (cid:48) = G • + n − ( m + c +1) ∗ (cid:99) M (1 , , A (cid:48) . Hence ( G l ∗ (cid:99) M − ( c, , A (cid:48) ) (cid:48) = σ ∗ G l + n +1 ∗ (cid:99) M (1 , , A (cid:48) . Now we use the fact that for any k ∈ Z , the isomorphism (see equation (38)) · z k : (cid:99) M ( β ,β ) A (cid:48) ∼ = −→ (cid:99) M ( β − k,β ) A (cid:48) sends G k (cid:99) M ( β ,β ) A (cid:48) = z − k (cid:99) M ( β ,β ) A (cid:48) to G (cid:99) M ( β − k,β ) A (cid:48) = (cid:99) M ( β − k,β ) A (cid:48) . Therefore (setting l = 0) we have( G (cid:99) M − ( c, , A (cid:48) ) (cid:48) ∼ = σ ∗ G n +1 ∗ (cid:99) M (1 , , A (cid:48) = σ ∗ G n ∗ (cid:99) M (0 , , A (cid:48) . which implies G ∗ (cid:99) M − ( c, , A (cid:48) ∼ = (cid:16) σ ∗ G n ∗ (cid:99) M (0 , , A (cid:48) (cid:17) (cid:48) The isomorphism Ψ from Formula (53) satisfiesΨ : ∗ (cid:98) N (0 , , A (cid:48) ∼ = −→ z c · ∗ (cid:99) M − (2 c, , A (cid:48) ∼ = ∗ (cid:99) M − ( c, , A (cid:48) In conclusion, we obtain ∗ (cid:98) N (0 , , A (cid:48) ∼ = (cid:16) σ ∗ z − n · ∗ (cid:99) M (0 , , A (cid:48) (cid:17) (cid:48) ∼ = σ ∗ z n · (cid:16) ∗ (cid:99) M (0 , , A (cid:48) (cid:17) (cid:48) , and then the statement follows from Proposition 3.21 as the inverse image under id C z × (cid:37) ∗ commuteswith the functor ( − ) (cid:48) .Now we can construct a D C z ×KM ◦ -module from the non-affine Landau-Ginzburg model Π : Z ◦ X −→ C λ × KM ◦ that will ultimately give us the reduced quantum D -module. It will consist in a minimalextension of the local system of intersection cohomologies of the fibres of Π. Recall that M IC ( Z ◦ X ) is theintersection cohomology D -module of Z ◦ X , that is, the unique regular singular D Z ◦ -module supported on Z ◦ X which corresponds to the intermediate extension of the constant sheaf on the smooth part of Z ◦ X . Proposition 6.7.
1. Consider the local system L from Proposition 2.13. Then H α + M IC ( Z ◦ X ) ∼ = (id C λ × (cid:37) ) + (cid:0) M IC ( X ◦ , L ) ⊕ (cid:0) IH n + c − ( X ) ⊗ O V (cid:1)(cid:1) | V ∗ . Using the Riemann-Hilbert correspondence, the above isomorphism can be expressed in terms of themorphism Π as p H R Π ∗ IC ( Z ◦ X ) ∼ = (id C λ × (cid:37) ) − (cid:0) ( j X ◦ ) ! IC ( X ◦ , L ) ⊕ IH n + c − ( X ) (cid:1) | V ∗ , where p H denotes the perverse cohomology functor, where j X ◦ : X (cid:44) → V is the canonical closedembedding and where IH n + c − ( X ) is the constant sheaf on V with fibre IH n + c − ( X ) .2. We have isomorphisms of D C z ×KM ◦ -modules FL loc KM ◦ (cid:0) H α + M IC ( Z ◦ X ) (cid:1) ∼ = (id C z × (cid:37) ) + (cid:91) M IC ( X ◦ , L ) | V ∗ ∼ = (id C z × (cid:37) ) + im( (cid:101) φ ) , where (cid:101) φ : ∗ (cid:98) N (0 , , A (cid:48) −→ ∗ (cid:99) M ( − c, , A (cid:48) is the morphism introduced in Definition 6.1. roof.
1. As the inclusion Z ◦ (cid:44) → Z is open and hence non-characteristic for any D Z -module, the asser-tion to be shown follows from Proposition 2.22 (more precisely, from Formula (32)) and Proposition2.13.2. The first isomorphism is a direct consequence of the last point, using again the commutation ofFL loc with the inverse image and the fact that O V -free modules are killed by FL locW . The secondisomorphism follows from equation (54).For future use, we give names to the D -modules on the K¨ahler moduli space considered above. We alsodefine natural lattices inside them. Definition 6.8.
Define the following D C z ×KM ◦ -modules: QM A (cid:48) := (id C z × (cid:37) ) + (cid:16) ∗ (cid:98) N (0 , , A (cid:48) (cid:17) and QM IC A (cid:48) := (id C z × (cid:37) ) + (cid:16) im( (cid:101) φ ) (cid:17) . Define moreover QM A (cid:48) := (id C z × (cid:37) ) ∗ (cid:16) ∗ (cid:98) N (0 , , A (cid:48) (cid:17) and QM IC A (cid:48) := (id C z × (cid:37) ) ∗ (cid:16) (cid:101) φ (cid:16) ∗ (cid:98) N (0 , , A (cid:48) (cid:17)(cid:17) , where here the functor (id C z × (cid:37) ) ∗ is the inverse image in the category of holomorphic vector bundles on C z × KM ◦ with meromorphic connection (meromorphic along { } × KM ◦ ). We proceed by comparing the objects QM A (cid:48) and QM IC A (cid:48) just introduced to the twisted and the reducedquantum D -module from section 4. For the readers convenience, let us recall one of the main resultsfrom [MM11] which concerns the toric description of the twisted resp. reduced quantum D -modules. Theorem 6.9 ([MM11, Theorem 5.10]) . Let X Σ be as before, and suppose that L = O X Σ ( L ) , . . . , L c = O X Σ ( L c ) are ample line bundles on X Σ such that − K X Σ − (cid:80) cj =1 L j is nef. Put again E := ⊕ cj =1 L j . Forany L ∈
Pic( X Σ ) with c ( L ) = (cid:80) ra =1 d a p a ∈ L ∨ A , we put (cid:98) L = (cid:80) ra =1 zd a q a ∂ q a ∈ R C z ×KM . Define the leftideal J of R C z ×KM by J := R C z ×KM ( Q l ) l ∈ L A (cid:48) + R C z ×KM · (cid:98) E , where Q l := (cid:81) i ∈{ ,...,m } : l i > l i − (cid:81) ν =0 (cid:16) (cid:98) D i − νz (cid:17) (cid:81) j ∈{ ,...,c } : l m + j > l m + c (cid:81) ν =1 (cid:16) (cid:98) L j + νz (cid:17) − q l · (cid:81) i ∈{ ,...,m } : l i < − l i − (cid:81) ν =0 (cid:16) (cid:98) D i − νz (cid:17) (cid:81) j ∈{ ,...,c } : l m + j < − l m + c (cid:81) ν =1 (cid:16) (cid:98) L j + νz (cid:17) , (cid:98) E := z ∂ z − (cid:98) K V ( E ∨ ) . Here we write D i ∈ Pic( X Σ ) for a line bundle associated to the torus invariant divisor D i , where i =1 , . . . , m . Notice that the ideal J was called G in [MM11, Definition 4.3].Moreover, let Quot be the left ideal in R C z ×KM generated by the following set G := { P ∈ R C z ×KM | (cid:98) c top · P ∈ J } , where (cid:98) c top := (cid:81) cj =1 (cid:98) L j . We define P := R C z ×KM /J resp. P res := R C z ×KM /Quot and denote by P = R C z ×KM / J resp. P res = R C z ×KM / Q uot the corresponding R C z ×KM -modules. Notice that wehave J ⊂ Q uot, hence there is a canonical surjection P (cid:16) P res .Put B ∗ ε := { q ∈ ( C ∗ ) r | < | q | < ε } ⊂ KM ◦ , then there is some ε such that the following diagram iscommutative and the horizontal morphisms are isomorphisms of R C z × B ∗ ε -modules. P | C z × B ∗ ε ∼ = (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (id C z × Mir) ∗ (QDM( X Σ , E )) ξ (cid:15) (cid:15) (cid:15) (cid:15) P res | C z × B ∗ ε ∼ = (cid:47) (cid:47) (id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) ere Mir: B ∗ ε → H ( X Σ ) × U is the mirror map , as described in [Giv98b, Theorem 0.1] (see also[CG07, Corollary 5 and the remark thereafter]). Recall that U ⊂ H ( X Σ , C ) / πiH ( X Σ , Z ) ∼ = ( C ∗ ) r isthe convergency domain of the twisted quantum product, i.e., the quantum D -modules QDM( X Σ , E ) and QDM( X Σ , E ) are defined on C z × H ( X Σ , C ) × U (see subsection 4.1). We now define another quotient Q res of P which is better suited to our approach and which turns outto be isomorphic to P res resp. to ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) in some neighborhood of q = 0. Definition 6.10.
Let K be the following ideal in R C z ×KM : K := { P ∈ R C z ×KM | ∃ p ∈ Z , k ∈ N such that k (cid:89) i =0 (cid:98) c p + itop P ∈ J } , where (cid:98) c itop := (cid:81) cj =1 ( (cid:98) L j + i ) . Define Q res := R C z ×KM /K and denote by Q res be the corresponding R C z ×KM -module. Proposition 6.11.
Using the notations from above, we have the following isomorphisms: P res | C z × B ∗ ε (cid:39) Q res | C z × B ∗ ε (cid:39) ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) . Proof.
First notice that we have a surjective morphism P res (cid:16) Q res because the generating set G of Quot is contained in the ideal K . If we can construct a well-defined morphism Q res | C z × B ∗ ε → ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) (63)such that the following diagram P res | C z × B ∗ ε (cid:47) (cid:47) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) Q res | C z × B ∗ ε (cid:117) (cid:117) ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) commutes, the proposition follows. In order to construct the morphism (63) we recapitulate the con-struction from [MM11] of the morphisms P | C z × B ∗ ε → ( id C z × Mir) ∗ (QDM( X Σ , E )) resp. P res | C z × B ∗ ε → ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) . It relies on a certain multivalued section L tw in End(QDM( X Σ , E )) having the property that L tw z − µ z c ( T X ) − c ( E ) is a fundamental solution of QDM( X Σ , E ) (see again [Giv98b] and [CG07]). We use the formulation from[MM11, Proposition 2.17]. Moreoverm we also need the multi-valued section J tw having the propertythat J tw := ( L tw ) − X Σ , E ) . Finally, we are going to use the cohomological multi-valued section I := q T/z (cid:88) d ∈ H ( X, Z ) q d A d ( z ) , where A d ( z ) := c (cid:89) i =1 (cid:81) d Li m = −∞ ([ L i ] + mz ) (cid:81) m = −∞ ([ L i ] + mz ) (cid:89) θ ∈ Σ(1) (cid:81) m = −∞ ([ D θ ] + mz ) (cid:81) d θ m = −∞ ([ D θ ] + mz ) ,q T/z := e z (cid:80) ra =1 T a log( q a ) ,d θ := (cid:82) d D θ and d L i := (cid:82) d c ( L i ) and which has asymptotic development I = F ( q )1 + O ( z − ). Theaforementioned mirror theorem of Givental ([Giv98b, Theorem 0.1] and [CG07, Corollary 5]), which weuse it in the version stated in [MM11, Theorem 5.6], says that I ( q, z ) = F ( q ) · J tw (Mir( q ) , z ) . R C z × B ∗ ε −→ ( id × Mir) ∗ (QDM( X Σ , E )) , (64) P ( z, q, zq∂ q , z ∂ z ) (cid:55)→ L tw (Mir( q ) , z ) z − µ z c ( T X ) − c ( E ) P ( q, z, z∂ q i , z ∂ z ) z − c ( T X )+ c ( E ) z µ F ( q ) J tw (Mir( q ) , z )= L tw (Mir( q ) , z ) z − µ z c ( T X ) − c ( E ) P ( q, z, z∂ q i , z ∂ z ) z − c ( T X )+ c ( E ) z µ I ( q, z )the proof of its surjectivity can be found in the proof [MM11, Theorem 5.10].The morphism above descends to P | C z × B ∗ ε by the fact that P ( q, z, zq∂ q , z ∂ z ) z − c ( T X )+ c ( E ) z µ I = 0 for P ∈ J . If one composes the morphism (64) with the quotient morphism ξ , then this descends to a morphism P res | C z × B ∗ ε → ( id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) , (65)which follows from P ( q, z, zq∂ q , z ∂ z ) z − c ( T X )+ c ( E ) z µ I ∈ ker ( m c top ) for P ∈ Q uot (66)and the fact that L tw preserves ker( m c top ) (cf. [MM11, Lemma 2.31]).As explained above, the proposition will follow if the morphism (65) descends to Q res | C z × B ∗ ε , i.e. we haveto show that P ( q, z, zq∂ q , z ∂ z ) z − c ( T X )+ c ( E ) z µ I ∈ ker ( m c top ) for P ∈ K . (67)We will adapt the proof of (66) from [MM11, Lemma 5.21] to our situation. First notice that z − c ( T X )+ c ( E ) z µ I = (cid:88) d ∈ H ( X, Z ) q T + d z − c ( T X )+ c ( E ) − (cid:82) d ( c ( T X ) − c ( E )) A d (1) . Now let P ( q, z, q∂ q , z ∂ z ) ∈ K and decompose it: P ( q, z, q∂ q , z ∂ z ) = (cid:88) d (cid:48) ∈ H ( X, Z ) finite q d (cid:48) P d (cid:48) ( z, z∂ q , z∂ z ) . This gives P ( q, z, q∂ q , z ∂ z ) z − c ( T X )+ c ( E ) z µ I = (cid:88) d ∈ H ( X, Z ) q T + d z − c ( T X )+ c ( E ) − (cid:82) d ( c ( T X ) − c ( E )) B d ( z ) , where B d ( z ) := (cid:88) d (cid:48) ∈ H ( X, Z ) finite P d (cid:48) (cid:18) z, z ( T + d ) , z ( − c ( T X ) + c ( E ) − (cid:90) d ( c ( T X ) − c ( E ))) (cid:19) A d − d (cid:48) (1) . Similarly to loc. cit., the statement (67) will follow from the fact that c top B d ( z ) = 0 for all d ∈ H ( X, Z ).Because P ∈ K , there exists p ∈ Z and k ∈ N such that (cid:32) k (cid:89) i =0 (cid:98) c p + itop (cid:33) P ( q, z, zq∂ q , z ∂ z ) z − c ( T X ⊗E ∨ ) z µ I = 0 , which gives (cid:88) d ∈ H ( X, Z ) q T + d z − c ( T X )+ c ( E ) − d T X ⊗E∨ k (cid:89) i =0 c (cid:89) j =1 z ([ L j ] + d L j + p + i ) B d ( z ) = 0 . z, q ) ∈ C ∗ z × W the term q T + d z − c ( T X )+ c ( E ) − d T X ⊗E∨ is invertible, so we deduce that k (cid:89) i =0 c (cid:89) j =1 ([ L j ] + d L j + p + i ) B d ( z ) = 0 ∀ d ∈ H ( X, Z ) . Let J d := { j ∈ { , . . . , c } | ∃ i ∈ { , . . . , k } with d L j + p + j = 0 } and notice that for every j there is atmost one i ∈ { , . . . , k } such that d L j + p + i = 0. Because cup-product with [ L j ] + l is an automorphismof H ∗ ( X, C ) for every l (cid:54) = 0, we conclude that (cid:89) j ∈ J d [ L j ] B d ( z ) = 0 ∀ d ∈ H ( X, Z ) , which in turn shows that c top B d ( z ) = ( (cid:81) cj =1 [ L j ]) B d ( z ) = 0 for all d ∈ H ( X, Z ).The next proposition compares the R -modules from Theorem 6.9 and Definition 6.10 with QM A (cid:48) and QM IC A (cid:48) . Proposition 6.12.
We have isomorphisms of R C z ×KM ◦ -modules P | C z ×KM ◦ ∼ = QM A (cid:48) and Q res | C z ×KM ◦ ∼ = QM IC A (cid:48) . Proof.
The first isomorphism follows from a similar argument as [RS15, Proposition 3.2], namely, thesection (cid:37) = i ◦ (cid:37) (cid:48) : KM (cid:44) → W ∗ , ( q , . . . , q r ) (cid:55)→ ( λ = q m , . . . , λ m = q m m , λ m +1 = − q m m +1 , . . . , λ m + c = − q m m + c )can be used to construct an isomorphism θ : F × KM −→ W ∗ , ( f , . . . , f n + c , q , . . . , q r ) (cid:55)→ ( q m y a , . . . , q m m y a m , − q m m +1 y a m +1 , . . . , − q m m + c y a m + c )with inverse θ − : W ∗ −→ F × KM , ( λ , . . . , λ m + c ) (cid:55)→ ( f j = ( − (cid:80) m + ci = m +1 c ij λ c j , q a = ( − (cid:80) m + ci = m +1 l ia λ l a ) , where L = ( l a ) resp. M = ( m i ) are the matrices which were introduced above Definition 6.3 and C = ( c j )is a ( m + c ) × ( n + c )- matrix such that the following equations are fulfilled (cf. Section 2.4): M · L = I r , B · C = I n + c , B · L = 0 , M · C = 0 , C · B + L · M = I m + c . Under this coordinate change the module ∗ (cid:98) N (0 , , A (cid:48) has the following presentation: R C z × F ×KM / (( Q l ) l ∈ L + ( (cid:98) E ) + ( (cid:98) E (cid:48) k ) k =1 ,...,n + c )with Q l and (cid:98) E as in Definition 6.9 and (cid:98) E (cid:48) k := f k ∂ k for k ∈ { , . . . , n + c } .Its module of global sections can be described simply by forgetting ∂ f k , i.e. we have the followingdescription C [ z, f ± , . . . , f ± n + c , q ± , . . . , q ± r ] (cid:104) z ∂ z , z∂ q , . . . , z∂ q r (cid:105) (( Q l ) l ∈ L + ( (cid:98) E )) . (68)Notice that the map (cid:37) can be factorized as θ ◦ i θ with i θ : KM −→ F × KM , ( q , . . . , q r ) (cid:55)→ (1 , . . . , , q , . . . , q r )64hus the inverse image of (68) with respect to i θ is given by C [ z, q ± , . . . , q ± r ] (cid:104) z ∂ z , z∂ q , . . . , z∂ q r (cid:105) (( Q l ) l ∈ L + ( (cid:98) E )) , which is exactly the definition of the module P from Theorem 6.9.Concerning the second isomorphism, the associated sub- R C z × F ×KM -module corresponding to (cid:98) K N fromLemma 6.2 can be described by { P ∈ C [ z, f ± , . . . , f ± n + c , q ± , . . . , q ± r ] (cid:104) z ∂ z , z∂ q , . . . , z∂ q r (cid:105) | ∃ p ∈ Z , k ∈ N s.t. k (cid:89) i =0 (cid:98) C p + itop P ∈ (( Q l ) l ∈ L +( (cid:98) E )) } , where (cid:98) C ktop := m + c (cid:89) i = m +1 (( n + c (cid:88) j =1 c ij f j ∂ j + r (cid:88) a =1 l ia q a ∂ a ) + l ) , = m + c (cid:89) i = m +1 (( n + c (cid:88) j =1 c ij f j ∂ j + (cid:98) D i ) + k )for k ∈ Z . It is easy to see that its inverse image under ( id C z × i θ ) is given by { P ∈ C [ z, q ± , . . . , q ± r ] (cid:104) z ∂ z , z∂ q , . . . , z∂ q r (cid:105) | ∃ p ∈ Z , k ∈ N s.t. k (cid:89) i =0 (cid:98) c p + itop P ∈ (( Q l ) l ∈ L + ( (cid:98) E )) } , which is exactly the definition of the ideal K in Definition 6.10. Thus, the second isomorphism follows.Combining Proposition 6.12, Theorem 6.9, Lemma 6.4 and 6.6 as well as Proposition 6.7, we obtain thefollowing mirror statement. Theorem 6.13.
Let X Σ and L , . . . , L c be as in Theorem 6.9. Consider the affine resp. non-affineLandau-Ginzburg models π = ( (cid:101) F , q ) : X aff × KM ◦ → C λ × KM ◦ , π : S × KM ◦ → C λ × KM ◦ and Π : Z ◦ X (cid:44) → Z ◦ α −→ C λ × KM ◦ associated to ( X Σ , L , . . . , L c ) . Let B ∗ ε ⊂ KM ◦ be the punctured ball fromTheorem 6.9. Then there are isomorphisms of D C z × B ∗ ε -modules FL loc KM ◦ (cid:0) H π † O S ×KM ◦ (cid:1) | C z × B ∗ ε ∼ = (id C z × Mir) ∗ (QDM( X Σ , E )) ( ∗ ( { } × B ∗ ε )) , FL loc KM ◦ (cid:0) H α + M IC ( Z ◦ X ) (cid:1) | C z × B ∗ ε ∼ = (id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) ( ∗ ( { } × B ∗ ε )) and an isomorphism of R C z × B ∗ ε -modules σ ∗ z n · (cid:16) H n + c (Ω • X aff ×KM ◦ / KM ◦ (log D )[ z ] , zd − d (cid:101) F ) (cid:17) (cid:48)| C z × B ∗ ε ∼ = (id C z × Mir) ∗ (QDM( X Σ , E )) . The following corollary is the promised Hodge theoretic application of the above main theorem.
Corollary 6.14.
There exists a variation of non-commutative pure polarized Hodge structures ( F , L Q , iso , P ) on KM ◦ (see [KKP08], [HS10] or [Sab11] for the definition) such that F ( ∗ ( { } × B ∗ ε )) ∼ = (id C z × Mir) ∗ (cid:0) QDM( X Σ , E ) (cid:1) ( ∗ ( { } × B ∗ ε )) . (69) Proof.
Using Theorem 6.13, this is a direct consequence of [Sai88, Th´eor`eme 1] and [Sab08, Corollary3.15].It would of course be desirable to remove the localization with respect to { }× B ∗ ε from the above theorem.We conjecture that the corresponding statement still holds, however, we cannot give a complete proof ofthis for the moment as we are not able to control the Hodge filtration on M IC ( Z ◦ X ). More precisely, weexpect the following to be true. 65 onjecture 6.15.
1. Write F H • H α + M IC ( Z ◦ X ) for the Hodge filtration on H α + M IC ( Z ◦ X ) , whichunderlies a pure Hodge module due to[Sai88, Th´eor`eme 1], and which has weight n + c + ( m − n ) = m + c . Let F H • [ ∂ − λ ] be the saturation of F H • as in the proof of Lemma 6.6 and write G H • for theinduced filtration on FL KM ◦ ( H α + M IC ( Z ◦ X )) . Then under the isomorphism of Proposition 6.7,2., we have that G H •− ( m + c ) FL KM ◦ ( H α + M IC ( Z ◦ X )) ∼ = z • · QM IC A (cid:48) . Notice that the bundle F which was used in the isomorphism from corollary 6.14 is nothing but theobject G H − ( m + c ) FL KM ◦ ( H α + M IC ( Z ◦ X )) .2. The isomorphism (69) holds without localization, i.e., there is an isomorphism of R C z × B ∗ ε -modules (cid:16) G H − ( m + c ) FL KM ◦ ( H α + M IC ( Z ◦ X )) (cid:17) | C z × B ∗ ε ∼ = (id C z × Mir) ∗ QDM( X Σ , E ) . As a consequence, the reduced quantum D -module underlies a variation of non-commutative Hodgestructures. This conjecture, if proved, should be seen as a first step towards establishing the existence of a veryspecial geometric structure on the cohomology space of the complete intersection subvariety Y ⊂ X Σ ,known as tt ∗ -geometry (see [CV91, CV93] or [Her03] for a modern account). Its existence is known forthe quantum cohomology of nef toric manifolds themselve (this follows from [RS15, Theorem 5.3], seealso [Iri09b]). For (non-toric) complete intersections one needs of course to consider its total quantumcohomology, not just the ambient part, but at least on this part the above conjecture would give thedesired result.Comparing Theorem 6.13 with Lemma 6.4 one may wonder whether the module FL loc KM ◦ (cid:0) H π + O S ×KM ◦ (cid:1) also has an interpretation as a mirror object. This is actually the case, namely, it corresponds to the so-called Euler − -twisted quantum D -module (whereas the object QDM( X Σ , E ) from Definition 4.3 wouldbe the Euler -twisted quantum D -module in this terminology). The Euler − -twisted quantum D -moduleencodes the so-called local Gromov-Witten invariants of the dual bundle E ∨ and is denoted byQDM( E ∨ ) (see [Giv98a, Theorem 4.2]). There is a non-degenerate pairing between QDM( X Σ , E ) and (cid:0) id C z × ( h ◦ f ) (cid:1) ∗ QDM( E ∨ ) (this is the non-equivariant limit of the quantum Serre theorem from [CG07,Corollary 2]) where f , h ∈ C [[ H ∗ ( X Σ , C ) ∨ ]] n are maps. The existence of this pairing has been proved inthe recent paper [IMM16]. However, in the formulation of this result, all objects are defined on the totalcohomology space, i.e., correspond to the big (twisted) quantum product. Nevertheless, we are able toobtain a mirror theorem for local Gromov-Witten invariants.Consider the situation of Theorem 6.13, in particular, let E := ⊕ cj =1 L j . As L j are nef bundles and henceglobally generated, also E is globally generated and therefore convex . Let QDM( E ∨ ) be the ( Euler − )-twisted quantum D -module governing local Gromov-Witten invariants, that is, integrals over the modulispace M ,l,d ( V ( E ∨ )) of stable maps to the total space V ( E ∨ ) (notice that M ,l,d ( V ( E ∨ )) is compactunless d = 0). Theorem 6.16.
Let again X Σ and L , . . . , L c be as in Theorem 6.9. There is some convergency neigh-borhood B ∗ ε (cid:48) , an isomorphism of D C z × B ∗ ε (cid:48) -modules FL loc KM ◦ (cid:0) H π + O S ×KM ◦ (cid:1) | C z × B ∗ ε (cid:48) ∼ = (cid:0) id C z × Mir (cid:48) (cid:1) ∗ (QDM( E ∨ )) ( ∗ ( { } × B ∗ ε (cid:48) )) and an isomorphism of R C z × B ∗ ε (cid:48) -modules H n + c (Ω • X aff ×KM ◦ / KM ◦ (log D )[ z ] , zd − d (cid:101) F ) | C z × B ∗ ε (cid:48) ∼ = σ ∗ z n · (cid:0) id C z × Mir (cid:48) (cid:1) ∗ QDM( E ∨ ) . Here
Mir (cid:48) is some base change involving the above mentioned maps f , h as well as the base change Mir .Proof.
It is actually sufficient to show the second statement as the first follows by applying the localizationfunctor ( − ) ⊗ O C z × B ∗ ε (cid:48) ( ∗ ( { } × B ∗ ε (cid:48) )) (This follows by using Propostion 3.21, Remark 3.16, 2. as well asProposition 3.3 together with Lemma 3.4).It follows from [IMM16, Theorem 3.14] that there exists a non-degenerate pairing(QDM( X Σ , E )) | C z × B ∗ ε (cid:48) ⊗ (cid:0) id C z × ( h ◦ f ) (cid:1) ∗ (cid:16) QDM( E ∨ ) | C z × B ∗ ε (cid:48) (cid:17) −→ O C z × B ∗ ε (cid:48) R C z × B ∗ ε (cid:48) -module structures of theobjects on the left hand side. As has been pointed out above, this statement is given in loc.cit. forthe big quantum D -modules, hence, one has to check that (cid:0) id C z × ( h ◦ f ) (cid:1) ∗ (cid:16) QDM( E ∨ ) | C z × B ∗ ε (cid:48) (cid:17) is still avector bundle on C z × B ∗ ε (cid:48) . From the definition of the map h (see [IMM16, Proposition 3.11]) it is clearthat it restricts to an invertible map h : H ( X Σ , C ) → H ( X Σ , C ). We claim that f restricts to a map f : H ( X Σ , C ) −→ H ( X Σ , C ) ⊕ H ( X Σ , C )so that the pullback f ∗ γ of any class γ ∈ H ( X Σ , C ) is still an element of H ( X Σ , C ). This can be seenas follows: From [IMM16, Proof of Lemma 3.2], we know that f ( τ ) = h (cid:88) α =0 (cid:88) d ∈ H ( X Σ , C ) ,n ≥ (cid:104) T α , (cid:101) , τ, τ, . . . , τ (cid:124) (cid:123)(cid:122) (cid:125) n − times , (cid:105) ,n +3 ,d T α (70)According to the definition 4.1, the correlator (cid:104) T α , (cid:101) , τ τ, . . . , τ (cid:124) (cid:123)(cid:122) (cid:125) n − times , (cid:105) ,n +3 ,d is non-zero only if the degreedeg( T α ) + ( n + 1) + deg ( e ( E ,n +3 ,d (2))) equals the dimension of the moduli space [ M ,n +3 ,d ( X )], i.e.,the number dim( X Σ ) + (cid:82) d c ( X ) + n . Under the assumption of the theorem, E ,n +3 ,d is represented bya vector bundle, which is of rank (cid:82) d c ( E ) + rank( E ). Hence E ,n +3 ,d (2), being the kernel of the map E ,n +3 ,d → ev ∗ ( E ) is a bundle of rank (cid:82) d c ( E ), so that we see that (cid:104) T α , (cid:101) , τ τ, . . . , τ (cid:124) (cid:123)(cid:122) (cid:125) n − times , (cid:105) ,n +3 ,d (cid:54) = 0 iffdeg( T α ) + 1 = dim( X ) + (cid:90) d c ( X Σ ) − (cid:90) d c ( E ) = dim( X ) + (cid:90) d c ( − K X Σ − c (cid:88) j =1 L j ) ≥ dim( X )where the last inequality holds due to the assumptions on X Σ and E . We conclude for any class T α occurring in formula (70) the following holds: either its degree is at most 1 or its coefficient is zero. Thismeans nothing else than im ( f ) ⊂ H ( X Σ , C ) ⊕ H ( X Σ , C ).Hence we can deduce from [IMM16, Theorem 3.14] that there is an isomorphism (cid:0) id C z × ( h ◦ f ) (cid:1) ∗ (cid:16) QDM( E ∨ ) | C z × B ∗ ε (cid:48) (cid:17) ∼ = (cid:0) (QDM( X Σ , E )) (cid:48) (cid:1) | C z × B ∗ ε (cid:48) of R C z × B ∗ ε (cid:48) -modules, and then the desired statement follows from the third line in the displayed formulain Theorem 6.13. Remark:
In view of [Giv98a, corollary 4.3], one may conjecture that Mir (cid:48) is the identity if the number c of line bundles defining the bundle E is strictly bigger than 1. However, at this moment, we do nothave any further evidence for this conjecture.The following consideration shows that the main Theorem 6.13 can also be considered as a generalizationof mirror symmetry for Fano manifolds themselves, as presented in our previous paper (see [RS15,Proposition 4.10]). Namely, let us consider the case where the number c of line bundles on the toricvariety X Σ is zero. Then we have A (cid:48) = A , and the duality morphism φ from Definition 5.6 is φ : M − ( c +1 , , A (cid:48)(cid:48) = M ( − , A (cid:48)(cid:48) −→ M (0 , , A (cid:48)(cid:48) = M (0 , A (cid:48)(cid:48) and is induced by right multiplication by ∂ λ . In particular, the induced morphism (cid:98) φ is simply the identityon (cid:99) M (0 , A (cid:48) . In particular, we have that im ( (cid:101) φ ) ∼ = (cid:99) M (0 , A (cid:48) so that QM IC A (cid:48) ∼ = QM A (cid:48) and QM IC A (cid:48) ∼ = QM A (cid:48) .On the other hand, the reduced quantum D -module QDM( X Σ , E ) is nothing but the quantum D -moduleof the variety X Σ , so that we deduce from Theorem 6.13 that we have an isomorphism of D C z × B ∗ ε -modulesFL loc KM ◦ (cid:0) H π + O S ×KM ◦ (cid:1) | C z × B ∗ ε ∼ = (id C z × Mir) ∗ (QDM( X Σ )) ( ∗ ( { } × B ∗ ε )) . R C z × B ∗ ε -modules H n (Ω • S ×KM ◦ / KM ◦ [ z ] , zd − d (cid:101) F ) | C z × B ∗ ε ∼ = (id C z × Mir) ∗ QDM( X Σ , E ) . This isomorphism is the restriction of the isomorphism in [RS15, Proposition 4.10] to C z × B ε (see also[Iri09a, Proposition 4.8]), notice that the neighborhood B ε is called W in [RS15]. Hence we see that ourmain Theorem 6.13 contains in particular the mirror correspondence for smooth toric nef manifolds, atleast on the level of R C z × B ε -modules.One may conclude from the above observation that Landau-Ginzburg models, either affine or compact-ified, appear to be the right point of view to study various type of mirror models of (the quantumcohomology of) smooth projective manifolds, including Calabi-Yau, Fano and more generally nef ones.The preprint [GKR12] where varieties of general types and their mirrors are investigated, also seem toconfirm this observation. It would certainly be fruitful to apply our methods to varieties with positiveKodaira dimension to refine the results from loc.cit.68 ndex of NotationObjects E toric vector bundle 44 E ∨ dual toric vector bundle 47FL locW localized FL-transformation with basis W FL X Fourier-Laplace transformation with basis X G + Gauß-Manin system 33 G † compactly supported Gauß-Manin system 33 K ◦ X Σ K¨ahler cone of X Σ K X Σ nef cone of X Σ M βB global sections of GKZ-system 18 M βB GKZ-system 18 M (cid:101) β (cid:101) B homogeneous GKZ-system 18 (cid:99) M ( β ,β ) B global sections of Fourier-Laplace transformed GKZ-system 33 (cid:99) M ( β ,β ) B Fourier-Laplace transformed GKZ-system 33 ∗ (cid:99) M ( β ,β ) B GKZ-system restricted to torus 40 ∗ (cid:99) M ( β ,β ) B lattice restricted to torus 41 ◦ (cid:99) M ( β ,β ) B GKZ-system restricted to set of good parameters 41 ◦ (cid:99) M ( β ,β ) B lattice restricted to set of good parameters 41 M IC ( X ) minimal extension of structure sheaf 22 M IC ( X , L ) minimal extension of flat bundle 22 (cid:91) M IC ( X ◦ , L ) Fourier-Laplace transformed minimal extension 35 ∗ (cid:98) N ( β ,β ) A (cid:48) shifted version of ∗ (cid:99) M ( β ,β ) A (cid:48) ◦ (cid:98) N ( β ,β ) A (cid:48) shifted version of ◦ (cid:99) M ( β ,β ) B X , E ) twisted Quantum D -module of X X , E ) reduced Quantum D -module of X R Radon transformation 20 R cst constant Radon transformation 20 R ◦ open Radon transformation 20 R ◦ c compact, open Radon transformation 20 R C z × M Rees-ring 17 R (cid:48) C z × M restricted Rees-ring 17 69 aps and Spaces g torus embedding 17 KM ◦ set of good parameters inside K¨ahler moduli space 56 KM K¨ahler moduli space 56Mir mirror map 62Π non-affine Landau-Ginzburg model 57 π affine Landau-Ginzburg model on torus 57 π affine Landau-Ginzburg model on X aff ϕ B family of Laurent polynomials 20 (cid:37) embedding of K¨ahler moduli space 56 S torus 17 W ◦ set of good parameters 38 X compactification of S X aff partial compactification of S Z universal hyperplane 20 Z ◦ X hyperplane sections of X restricted to good parameters 56 References [Ado94] Alan Adolphson,
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