Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules
aa r X i v : . [ m a t h . AG ] A ug Logarithmic Frobenius manifolds, hypergeometric systems andquantum D -modules Thomas Reichelt and Christian SevenheckMay 5, 2018
Abstract
We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered D -modules. We discuss in particular the logarithmic degeneration behavior at the large radius limitpoint, and express the mirror correspondence as an isomorphism of Frobenius manifolds with loga-rithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D -module, and a detailed study of a natural filtration definedon this differential system. We obtain a solution of the Birkhoff problem for lattices defined by thisfiltration and show the existence of a primitive form, which yields the construction of Frobeniusstructures with logarithmic poles associated to the mirror Laurent polynomial. As a final applica-tion, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski opensubset of the complexified K¨ahler moduli space of the variety. Contents D -modules and filtered Gauß-Manin systems 4 D -modules with logarithmic structure and good bases 20 D -modules on K¨ahler moduli spaces . . . . 213.2 Logarithmic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Logarithmic Frobenius structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 D -module and the mirror correspondence 39 J -function, Givental’s theorem and mirror correspondence . . . . . . . . . . . . . . . . . . 42 In this paper we study the differential systems that occur in the mirror correspondence for smooth toricweak Fano varieties. On the so-called A-side of mirror symmetry, which is mathematically expressedas the quantum cohomology of this variety, these systems has been known since quite some time as
Mathematics Subject Classification. D -module, toric variety, primitive form, quantum cohomology, Frobeniusmanifold, mirror symmetry, non-commutative Hodge structureTh.R. is supported by the DFG grant He 2287/2-2. Ch.S. is supported by a DFG Heisenberg fellowship (Se 1114/2-1).Both authors acknowledge partial support by the ANR grant ANR-08-BLAN-0317-01 (SEDIGA). uantum D -modules . A striking fact which makes their study attractive is that the integrability of thecorresponding connection encodes many properties of the quantum product, in particular, the associa-tivity, usually expressed by the famous WDVV-equations. It is well-known (see, e.g., [Man99]) that thequantum D -module (or first structure connection) is essentially equivalent to the Frobenius structuredefined by the quantum product on the cohomology space of the variety.The main subject of this paper is to establish the same kind of structures for the B -side, also calledthe Landau-Ginzburg model, of such a variety. This problem is related to more classical objects in thetheory of singularities of holomorphic or algebraic functions: namely, period integrals, vanishing cyclesand the Gauß-Manin connection in its various forms. A by-now well-known construction going back toK. Saito and M. Saito endows the semi-universal unfolding space of an isolated hypersurface singularity f : ( C n , → ( C ,
0) with a Frobenius structure. There are two main ingredients in constructing thesestructures: a very precise analysis of the Hodge theory of f , which is done using the the so-calledBrieskorn lattice, and which culminates in a solution of the Birkhoff problem (also called a good basis ofthe Brieskorn lattice). The second step is to show that there is a specific section of the Brieskorn lattice,called primitive and homogeneous (which is also known as the “primitive form”).However, these Frobenius manifolds will never appear as the mirror of the quantum cohomology of somevariety. Sabbah has shown in a series of papers (partly joint with Douai, see [Sab97], [Sab06], [DS03])that the above results can be adapted if one starts with an algebraic function f : U → C defined ona smooth affine variety U . Besides the isolatedness of the critical locus of f , one is forced to imposea stronger condition, known as tameness. Roughly speaking, it states that no change of the topologyof the fibres comes from critical points at infinity. The need for this condition reflects the fact thatthe Gauß-Manin system of such a function, and other related objects, are not simply direct sums ofthe corresponding local objects at the critical points. For tame functions, it is known that the Birkhoffproblem for the Brieskorn lattice always has a solution, similarly to the local case, one uses informationcoming from the Hodge theory of f to show this result. One the other hand, the existence of a primitive(and homogeneous) form is a quite delicate problem which is not known in general. It has been shown forcertain tame polynomials in [Sab06], for convenient and non-degenerate Laurent polynomials in [DS03](and later with different methods in [Dou05]) and also for some other particular cases of tame functions(e.g., [GMS09]). In any case, the outcome of these constructions is a germ of a Frobenius structureon the deformation space of a single function. The general construction in [DS03] does not give muchinformation on how these Frobenius manifolds vary for families of, say, Laurent polynomials. Notice alsothat the Frobenius structure associated to a Laurent polynomial (or even to a local singularity) is notat all unique, it depends on both the choice of a good basis and a primitive (and homogeneous) form.However, there is a canonical choice of a solution of the Birkhoff problem, predicted by the use of Hodgetheory (more precisely, it is defined by Deligne’s I p,q -splitting of the Hodge filtration associated to f ),but in general Frobenius structure coming from this solution will not behave well in families.For some special kind of Fano varieties like the projective spaces (see [Bar00]) or, more generally, forsome orbifolds like weighted projective spaces ([Man08], [DM09]), it is possible to find explicit solutionsto the Birkhoff problem and to carry out the construction of the Frobenius manifold rather directly.Then one may compare the Brieskorn lattices (or their extension using good bases) to the quantum D -module by an explicit identification of bases. This yields isomorphisms of Frobenius manifolds andeven some results on their degeneration behavior near the large radius limit (see [DM09]), but of coursethis method is limited if one wants to attack more general classes of examples.In the present paper, we obtain such an identification of Frobenius manifolds for all weak Fano toricmanifolds, using Givental’s I = J -theorem ([Giv98]). We do not rely on the results of [DS03], instead,we identify the family of Gauß-Manin systems attached to the Landau-Ginzburg model of our varietywith a certain hypergeometric D -module (also called Gelfand-Kapranov-Zelevinski-(GKZ)-system) by apurely algebraic argument. This makes available some known results and constructions from the theoryof these special D -modules, and we are able to deduce a finiteness and a duality statement for thefamily of Brieskorn lattices. The tameness assumption from above is used via an adaption of a result in[Ado94], who has calculated the characteristic variety of a hypergeometric D -module. In general, thistameness will hold on a Zariski open subspace of the parameter space, and we show that if our varietyis genuine Fano, then this is the whole parameter space. An important point in the construction is toextend the family of Brieskorn lattices on the K¨ahler moduli space of the variety to a certain partialcompactification including the large radius limit point. This compactification depends on a choice ofcoordinates on the complexified K¨ahler moduli space, that is, on a choice of a basis of nef classes of the2econd cohomology of our variety. Once we have this logarithmic extension, we can apply [Rei09] whichyields the construction of a logarithmic Frobenius manifold , that is, a Frobenius structure on a manifoldwhich is the complement of a normal crossing divisor, and such that both multiplication and metricare defined on the sheaf of logarithmic vector fields. At any point inside the K¨ahler moduli space, thisrestricts to a germ of a Frobenius manifold constructed in [DS03]. In this sense our mirror statement alsogeneralizes the equivalence of Frobenius structures (at fixed points of the K¨ahler moduli space) knownin particular cases like P n .Let us give a short overview on the content of this paper: In section 2 we study in some detail variousdifferential systems associated to toric data defined by a smooth toric weak Fano variety X Σ A (where Σ A is the defining fan), parts of the results hold even more generally for a given set of vectors in a lattice.In particular, we obtain an identification of a certain hypergeometric D -module with the Gauß-Maninsystem of a generic family of Laurent polynomials defined by the toric data, more precisely, with apartial Fourier-Laplace transformation of it (theorem 2.4). We next study a natural filtration of thisGauß-Manin system, prove a finiteness result (theorem 2.14) and show that it satisfies a compatibilitycondition with respect to the duality functor (proposition 2.18).The actual Landau-Ginzburg model is a subfamily of the family of generic Laurent polynomials studiedin section 2, parameterized by the K¨ahler moduli space, i.e., by a dim H ( X Σ A , C )-dimensional torus. Insection 3, we first identify the Gauß-Manin system of the Landau-Ginzburg model of X Σ A with a GKZ-system on the K¨ahler moduli space (corollary 3.3). In the second part of this section, we extend thismodule to a vector bundle with an integrable connection having logarithmic poles along the boundarydivisor of an appropriate compactification of the K¨ahler moduli space (theorem 3.7). From this objectwe can derive, using a a method which goes back to [Gue08], a specific basis defining a solution to theBirkhoff problem in family in the sense of [DS03]. This is a family of P -bundles which extends theGKZ- D -module mentioned above. An important new point is that this construction works taking intoaccount the logarithmic degeneration behavior near the large radius limit point. As a consequence, wecan construct a canonical logarithmic Frobenius manifold associated to the Landau-Ginzburg model of X Σ A , which has an algebraic structure on the subspace corresponding to the compactified K¨ahler modulispace (theorem 3.16). One may speculate that it restricts to the canonical Frobenius structure consideredin [DS03] in a small neighborhood of any point of the K¨ahler moduli space (question 3.17).In section 4 we first recall very briefly the construction of the quantum D -module of a projective variety,and then show that it is isomorphic, in the toric weak Fano case, to the family of P -bundles withconnection constructed in section 3. From this we deduce (theorem 4.11) an isomorphism of logarithmicFrobenius manifolds by invoking the main result from [Rei09].In the final section 5 we show (theorem 5.3), using the fundamental result from [Sab08] that the quan-tum D -module is equipped with the structure of a variation of pure polarized non-commutative Hodgestructures in the sense of [KKP08]. As there are several versions of this notion around, we briefly recallthe basic definitions and show how they apply in our context. This result strengthens a theorem ofIritani ([Iri09b]), who directly shows the existence of tt ∗ -geometry in quantum cohomology, however, heuses an asymptotic argument, whereas our approach gives the existence of an ncHodge structure wher-ever the small quantum product is convergent and the mirror map is defined. We also deduce from theconstruction of a logarithmic Frobenius manifold that this tt ∗ -geometry behaves quite nicely along theboundary divisor of the K¨ahler moduli space, namely, that the corresponding harmonic bundle is tamealong this divisor (theorem 5.5).We finish this introduction by some remarks on how our work relates to other papers concerning mirrorsymmetry for Fano varieties and hypergeometric differential systems: As already mentioned above, ourmain result relies on Givental’s I = J -theorem. It is certainly well-known to specialists (and it is brieflymentioned at some places in [Giv98] and also in subsequent papers) that the I -function is related tooscillating integrals and hence to the Fourier-Laplace transformation of the Gauß-Manin system of themirror Laurent polynomial, but to the best of our knowledge, a thourough treatment of these issuesis missing in the literature. More recently, Iritani has given in [Iri09a] an analytic description of thedifferential system associated to the Landau-Ginzburg model and discussed its relation to hypergeometric D -modules. He considers the more general case of toric weak Fano orbifolds, however, solutions to theBirkhoff problem resp. Frobenius structures are not treated in loc.cit. Passing through the analyticcategory one also looses the algebraic nature of the objects involved, which may be an obstacle in somesituations. As an example, one cannot apply the general results on formal decomposition of meromorphicbundles with connection from [Moc09] and [Moc08b] for non-algebraic bundles. Nevertheless, some of3he techniques used here are also present in [Iri09a], and at some points our presentation is (withoutexplicit mentioning) similar to that of loc.cit.Finally, let us notice that although one may think of an extension of some of our results (like those insection 2) to the orbifold case, there is a serious obstacle in the construction of a logarithmic Frobeniusstructure associated to the Landau-Ginzburg model of a weak Fano toric orbifold. This is mainly due tothe fact that the “limit” orbifold cup product does not satisfy an “ H -generation condition”, in contrastto the case of toric manifolds (see also the preprint [DM09] for a discussion of this phenomenon for thecase of weighted projected spaces). D -modules and filtered Gauß-Manin systems In this section we study Gauß-Manin systems associated to generic families of Laurent polynomials. Weshow that (a partial Fourier-Laplace transformation of) these D -modules always have a hypergeometricstructure, i.e., are isomorphic to (a partial Fourier-Laplace transformation of) a certain GKZ-system.Moreover, both Gauß-Manin systems and GKZ-systems carry natural filtrations by O -modules. For theGauß-Manin system, these are the so-called Brieskorn lattices, as studied, for more general polynomialfunctions, in [Sab06]. We show that the above identification also works at the level of lattices. Asan application, we prove that if the family of Laurent polynomials is associated to a fan of a smoothtoric weak Fano manifold, then outside a certain “bad part” of the parameter space, the family ofBrieskorn lattices is O -locally free. This will be needed later in the construction of Frobenius manifoldsassociated to these special families of Laurent polynomials. Finally, we study the holonomic dual of theGauß-Manin system and obtain (up to a shift of the homological degree) an isomorphism of this dualto the Gauß-Manin system itself. The way of constructing this isomorphism is purely algebraic, usinga resolution called Euler-Koszul complex of the hypergeometric D -module which is isomorphic to theGauß-Manin system. This proof differs from [Sab06] or [DS03], where the duality isomorphism is obtainin a topological way. We could also give a topological proof along the lines of the quoted papers, byusing a partial compactifications of the family of Laurent polynomials and a smoothness property atinfinity (see the proof of proposition 2.9 for a description of this partial compactification). However, ouralgebraic approach gives almost for free that the above mentioned filtration is compatible (up to a shift),with the duality isomorphism. This fact is also needed for the construction of Frobenius structures. We start with the following set of data: Let N be a finitely generated free abelian group of rank n ,for which we choose once and for all a basis which identifies it with Z n . Let a , . . . , a m be elementsof N , which we also see as vectors of Z n . We suppose that a , . . . , a m generates N , if we only have P ni =1 Q a i = N Q := N ⊗ Q , then some of our results can be adapted, see proposition 2.6 below. In orderto orient the reader, let us point out from the very beginning that the case we are mostly interestedin is when these vectors are the primitive integral generators of the rays of a fan Σ A in N R := N ⊗ R defining a smooth projective toric variety X Σ A which is weak Fano , that is, such that the anticanonicaldivisor − K X Σ A is numerically effective (nef). The Fano case, i.e., when − K X Σ A is ample is of particularimportance and will sometime be treated apart, as there are cases in which we obtain stronger statementsfor genuine Fano varieties. See also the proof of proposition 2.1, the proof of lemma 2.8 and the beginningof section 3 for toric characterizations of the weak Fano condition. We will abbreviate this case by sayingthat a , . . . , a m are defined by toric data . We write L for the module of relations between a , . . . , a m , i.e., l ∈ L ⊂ Z m iff P mi =1 l i a i = 0. We will denote by S the n -dimensional torus Spec C [ N ] with coordinates y , . . . , y n and by W ′ the m -dimensional affine space Spec C [ ⊕ mi =1 N a i ] with coordinates w , . . . , w m . Weare slightly pedantic in this latter definition in order to make a clear difference with the dual space,called W , which will appear later.An important point in the arguments used below will be to consider the following set of extendedvectors: Put e N := Z × N ∼ = Z n +1 , e a i := (1 , a i ) ∈ e N for all i = 1 , . . . , m and e a := (1 , ∈ e N . Write e A = ( e a , e a , . . . , e a m ). Notice that the module of relations of e A is isomorphic to L , any l = ( l , . . . , l m ) ∈ L gives in a unique way rise to the relation ( − P mi =1 l i ) e a + P mi =1 l i e a i = 0. By abuse of notation, we alsowrite L for the module of relations of e A . As another piece of notation, we put l := P mi =1 l i . Let V ′ = Spec C [ ⊕ mi =0 N e a i ] with coordinates w , . . . , w m and V the dual space, with coordinates λ , . . . , λ m .4e also need the m -dimensional torus S := Spec C [( ⊕ mi =1 Z a i ) ∨ ], with inclusion map j : S ֒ → W .Moreover, put b V := Spec C [ N e a ] × W and b T := Spec C [ N e a ] × S , we still denote the map b T ֒ → b V by j . We put τ = − w so that ( τ, λ , . . . , λ m ) gives coordinates on b V resp. b T . We will also write C τ forSpec C [ N e a ] and C ∗ τ for Spec C [ Z e a ]. Later we will consider algebraic D b V - (resp. D b T )-modules whichare localized along τ = 0, and in this case we also use the variable z := τ − . Sometimes we will implicitlyidentify such modules with their restriction to C ∗ τ × W resp. to C ∗ τ × S .The first geometric statement about these data is the following proposition. Proposition 2.1.
1. Consider the map k : S −→ W ′ ( y , . . . , y n ) ( w , . . . , w m ) := ( y a , . . . , y a m ) , where y a i := Q nk =1 y a ki k . Suppose that lies in the interior of Conv( a , . . . , a m ) , where for anysubset K ⊂ N , Conv( K ) denotes the convex hull of K in N R . Then k is a closed embedding.2. Suppose that a , . . . , a m are defined by toric data. In particular, the completeness of Σ A implies that is an interior point of Conv( a , . . . , a m ) . Let N e A = P mi =0 N e a i , then N e A is a normal semigroup,i.e. it satisfies e N ∩ C ( e A ) = N e A and positive, i.e., the origin is the only unit in N e A . Here for afinite set { e x , . . . , e x k } we write C ( { e x , . . . , e x k } ) for the cone P kj =1 R ≥ e x j . The associate semigroupring Spec C [ N e A ] is normal, Cohen-Macaulay and Gorenstein.Proof.
1. The condition that the origin is a interior point of the convex hull of the vectors a i translatesinto the existence of a relation l = ( l , . . . , l m ) ∈ L ∩ Z m> between a , . . . , a m consisting of positiveintegers. On the other hand, the closure of the image of the map k is contained in the vanishinglocus of the so-called toric ideal I = Y i : l i < w − l i i − Y i : l i > w l i i ! l ∈ L ⊂ O W ′ . From the existence of l ∈ L ∩ Z m> we deduce that the function Q mi =1 w l i i − I . This showsthat for any point w = ( w , . . . , w m ) ∈ Im ( k ) ⊂ V ( I ) ⊂ W ′ , we have w i = 0, i.e., w ∈ Im ( k ).2. First we show the normality property: Consider any integer vector e x = ( x , x , . . . , x n ) ∈ C ( e A ) ∩ e N .We have C ( e A ) ∩ ( { } × N R ) = [ λ i ∈ R ≥ ; P mi =0 λ i =1 λ i e a i = { } × Conv( a , . . . , a m ) (1)Now define P (Σ A ) = [ h a i ,...,a in i∈ Σ A ( n ) Conv(0 , a i , . . . , a i n )We have the following reformulation of the weak Fano condition (see, e.g., [Wi´s02, page 268]): − K X Σ A is nef ⇐⇒ P (Σ A ) is convex.Hence by assumption we know that P (Σ A ) is convex. We claim that P (Σ A ) = Conv( a , . . . , a m ).The inclusion ⊂ follows from the fact 0 , a i , . . . , a i n ∈ Conv( a , . . . , a m ) for h a i , . . . , a i n i ∈ Σ A ( n ).The other inclusion follows from a , . . . , a m ∈ P (Σ A ) and the convexity of P (Σ A ). From the claimand equality (1) we get the following decomposition of the cone C ( e A ): C ( e A ) = [ h a i ,...,a in i∈ Σ A ( n ) C ( { e a , e a i , . . . , e a i n } )Using this decomposition, we see that e x lies in a cone C ( e a , e a j , . . . , e a j n ), that is, there are λ , λ j , . . . , λ j n ∈ R ≥ such that e x = λ e a + P nk =1 λ j k e a j k . Notice that e a , e a j , . . . , e a j n is Z -basis5f e N , as a j , . . . , a j n is a Z -basis of N which follows from the smoothness of Σ A . From thisfollows e x ∈ N e A . Notice also that the “exterior boundary” ∂C ( e a , e a j , . . . , e a j n ) ∩ ∂C ( e A ) equals P nk =1 R ≥ e a i k so that e x ∈ Int( C ( e A )) precisely iff the coefficient λ in the above sum is positive.From the fact that N e A is normal it follows that Spec C [ N e A ] is Cohen-Macaulay by a classical resultdue to Hochster ([Hoc72, theorem 1]). That N e A is positive is equally easy to see: it follows (see,e.g., [MS05, lemma 7.12]) from the fact that C ( e A ) is pointed, i.e., that the vectors ( e a i ) i =0 ,...,m arecontained in the half-space { e x ∈ R n +1 | e x > } .It remains to show that Spec C [ N e A ] is Gorenstein: We use [BH93, corollary 6.3.8] stating that thisproperty is equivalent, for normal positive semigroup rings, to the fact that that there is a vector e c ∈ Int( N e A ) with Int( N e A ) = e c + N e A. From the above proof of the normality of N e A we see that Int( N e A ) = e N ∩ Int( C ( e A )). On the otherhand, the map e N → e N which sends e x to e x + (1 ,
0) induces a bijection from C ( e A ) to Int( C ( e A )),this follows from the characterization of C ( e A ) given above.In order to state our first main result, we will associate (several variants of) a D -module) to the set ofvectors a , . . . , a m above. This construction is a special case of the well-known A-hypergeometric systems (also called hypergeometric D -modules or GKZ-systems). We recall first the general definition. Definition 2.2 ([GKZ90], [Ado94]) . Consider a lattice Z t and vectors b , . . . , b s ∈ Z t which we alsowrite as a matrix B = ( b , . . . , b s ) . Moreover, let β = ( β , . . . , β t ) ∈ C t . Write (as above) L for themodule of relations of B and D C s for the sheaf of rings of algebraic differential operators on C s (wherewe choose x , . . . , x s as coordinates). Define M βB := D C s / (cid:0) ( (cid:3) l ) l ∈ L + ( Z k ) k =1 ,...t (cid:1) , where (cid:3) l := Q i : l i < ∂ − l i x i − Q i : l i > ∂ l i x i Z k := P si =1 b ki x i ∂ x i + β k M βB is called hypergeometric system. We will use at several places in this paper the Fourier-Laplace transformation for algebraic D -modules.In order to introduce a convenient notation for this operation, let X be a smooth algebraic variety, and M a D C s × X -module, where we have coordinates ( x , . . . , x s ) on C s . Then we write FL y ,...,y s x ,...,x s M for the D ( C s ) ∨ × X -module, which is the same as M as a D X -module, and where y i acts as − ∂ x i and ∂ y i acts as x i , here y , . . . , y s are the dual coordinates on ( C s ) ∨ . One could also work with the functor FL y ,...,y s x ,...,x s ,where y i acts as ∂ x i and ∂ y i acts as − x i , this would lead to slightly uglier formulas. Definition 2.3.
Let D V , D b V and D b T be the sheaves of algebraic differential operators on V , b V and b T ,respectively.1. Consider the hypergeometric system M β e A associated to the vectors e a , e a , . . . , e a m . More explicitly, M β e A := D V / I , where I is the sheaf of left ideals in D V defined by I := D V ( (cid:3) l ) l ∈ L + D V ( Z k ) k ∈{ ,...,n } + D V E, where (cid:3) l := ∂ lλ · Q i : l i < ∂ − l i λ i − Q i : l i > ∂ l i λ i if l ≥ , (cid:3) l := Q i : l i < ∂ − l i λ i − ∂ − lλ · Q i : l i > ∂ l i λ i if l < ,Z k := P mi =1 a ki λ i ∂ λ i + β k ,E := P mi =0 λ i ∂ λ i + β , ere a i = ( a i , . . . , a ni ) when seen as a vector in Z n .2. Let c M β e A be the D b V -module FL w λ ( M β e A )[ τ − ] . In other words, c M β e A = D b V [ τ − ] / b I , where b I is theleft ideal generated by the Fourier-Laplace transformed operators b (cid:3) l , b Z k and b E , i.e., b (cid:3) l := τ l · Q i : l i < ∂ − l i λ i − Q i : l i > ∂ l i λ i = z − l · Q i : l i < ∂ − l i λ i − Q i : l i > ∂ l i λ i , b Z k := P mi =1 a ki λ i ∂ λ i + β k , b E := P mi =1 λ i ∂ λ i − τ ∂ τ − β , = P mi =1 λ i ∂ λ i + z∂ z − β .
3. Define c M β,loc e A := j ∗ c M β e A to be the restriction of c M β e A to b T . We will use the presentations D b T [ τ − ] / b I ′ and D b T [ τ − ] / b I ′′ of c M β,loc e A where b I ′ resp. b I ′′ is the sheaf of left ideals generated by b (cid:3) ′ l , b Z k and b E resp. b (cid:3) ′′ l , b Z k and b E , where b (cid:3) ′ l := z P i : li> l i · b (cid:3) l and b (cid:3) ′′ l := Y i : l i > ( z · λ i ) l i · b (cid:3) l , so that b (cid:3) ′ l = Y i : l i < ( z∂ λ i ) − l i − Y i : l i > ( z∂ λ i ) l i , and, using the formula λ ji ∂ jλ i = Q j − ν =0 ( λ i ∂ λ i − ν ) , b (cid:3) ′′ l = m Y i =1 λ l i i · Y i : l i < − l i − Y ν =0 ( zλ i ∂ λ i − νz ) − Y i : l i > l i − Y ν =0 ( zλ i ∂ λ i − νz ) . Notice that obviously b I ′ = b I ′′ , but we will later need the two different explicit forms of the generatorsof this ideal, for that reason, two different names are appropriate.4. Write M e A := M (1 , e A , c M e A := c M (1 , e A and c M loc e A := c M (1 , ,loc e A .In order to avoid too heavy notations, we will sometimes identify c M β e A resp. c M β,loc e A with the correspondingmodules over either C ∗ τ × W resp. C ∗ τ × S or P z × W resp. P z × S , here P z is P with defined by z = 0 . The first main result is a comparison of these D -modules to some Gauß-Manin systems associated tofamilies of Laurent polynomials. When this paper was written, a similar result appeared in [AS10]. Thetechniques of loc.cit. are not too far from those used in the proof of the next theorem, however, it seemsnot to be more efficient to translate their result into our situation than to give a direct proof. Theorem 2.4.
Let a , . . . , a m ∈ N such that P mi =1 Z a i = N . Consider the family of Laurent polynomials ϕ : S × W → C t × W defined by ϕ (( y , . . . , y n ) , ( λ , . . . , λ m )) = ( m X i =1 λ i y a i , λ ) = m X i =1 λ i n Y k =1 y a ki k , ( λ , . . . , λ m ) ! =: ( t, λ , . . . , λ m ) . Then there is an isomorphism φ : c M e A −→ FL τt ( H ϕ + O S × W )[ τ − ] =: G of D b V -modules. Before entering into the proof, let us recall the following well-known description of the Fourier-Laplacetransformation of the Gauß-Manin system. 7 emma 2.5.
Write ϕ = ( F, π ) , where F : S × W → C t , ( y, λ ) P mi =1 λ i y a i and π : S × W → W isthe projection. Then there is an isomorphism of D b V -modules G ∼ = H (cid:16) π ∗ Ω • + nS × W/W [ z ± ] , d − z − · dF ∧ (cid:17) , where d is the differential in the relative de Rham complex π ∗ Ω • S × W/W . The structure of a D b V -moduleon the right hand side is defined as follows ∂ z ( ω · z i ) := i · ω · z i − − z − 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 a i · ω · z i − , where ω ∈ Ω nS × W/W .Proof.
The identification of both objects as D b V / D W -modules is well-known (see, e.g., [DS03, proposition2.7], where the result is stated, for a proof, one uses [Sai89, lemma 2.4]). The proof of the formulas forthe action of the vector fields ∂ λ i can be found, in a similar situation, in [Sev11, lemma 7]. Proof of the theorem.
Throughout the proof, we will use the following notation: Let X be a smoothalgebraic variety, and f a meromorphic function on X with pole locus D := g − ( ∞ ) ⊂ X , then we denoteby O X ( ∗ D ) · e f the locally free O X ( ∗ D )-module of rank one with connection operator ∇ := d + df ∧ . The D X -module thus obtained has irregular singularities along D , notice that this irregularity locus may layin a boundary of a smooth projective compactification X of X if f ∈ O X . For any D X -module M , wewrite M · e f for the tensor product M ⊗ O X O X ( ∗ D ) · e f .Put T := Spec C [ e N ] with coordinates y , y , . . . , y n , and define e k : T −→ C ∗ × W ′ ⊂ V ′ ( y , y , . . . , y n ) (cid:0) w := y , ( w i := y · y a i ) i =1 ,...,m (cid:1) , where, as before, we write y a i for the product Q nk =1 y a ki k . It is an obvious consequence of the firstpoint of proposition 2.1, that e k is again a closed embedding from T to C ∗ × W ′ . Write moreover p forthe projection C ∗ τ × S × W ։ C ∗ τ × W . We identify T with C ∗ τ × S by the map ( y , y , . . . , y n ) ( − y , y , . . . , y n ) = ( τ, y , . . . , y n ).First we claim that G ∼ = H p + (cid:16) O C ∗ τ × S × W · e − τ P mi =1 λ i y ai (cid:17) . (2)As p is a projection, the direct image p + of any module is nothing but its relative de Rham complex, i.e. H p + (cid:16) O C ∗ τ × S × W · e − τ P mi =1 λ i y ai (cid:17) ∼ = H (cid:16) p ∗ Ω • + n C ∗ τ × S × W/ C ∗ τ × W , d − τ · dF ∧ (cid:17) , and this module is the same as G , using lemma 2.5. It follows from the projection formula ([HTT08,corollary 1.7.5]) that (cid:16) ( e k × id W ) + O T × W (cid:17) · e P mi =1 λ i w i = ( e k × id W ) + (cid:16) O T × W · e y P mi =1 λ i y ai (cid:17) . This can also be shown by a direct calculation, in fact, both modules are quotients of D C ∗ τ × W ′ × W . Nowconsider the following diagram C ∗ τ × W ′ × W π w w w w ooooooooooo π ' ' ' ' OOOOOOOOOOO T e k / / C ∗ τ × W ′ C ∗ τ × W, where π and π are the obvious projections. As π ◦ ( e k × id W ) = p , we obtain that H p + (cid:16) O S × C ∗ τ × W · e − τ P mi =1 λ i y ai (cid:17) = H π , + (cid:16) (( e k × id W ) + O T × W ) · e P mi =1 λ i w i (cid:17) .
8n the other hand, we obviously have that ( e k × id W ) + O T × W = π ∗ e k + O T , hence H π , + (cid:16) (( e k × id W ) + O T × W ) · e P mi =1 λ i w i (cid:17) = H π , + (cid:16) ( π ∗ e k + O T ) · e P mi =1 λ i w i (cid:17) , Now we use the following well-known description of the Fourier-Laplace transformation: H π , + (cid:16) (( π ) ∗ e k + O T ) · e P mi =1 λ i w i (cid:17) = FL − λ ,..., − λ m w ,...,w m (cid:16)e k + O T (cid:17) . We are thus left to show that the latter module equals c M e A . In order to do so, notice that the D T -module O T can be written as a quotient of D T . The natural choice would be to mod out the left ideal generatedby ( y k ∂ y k ) k =0 ,...,n , however, we will rather write O T = D T ( y ∂ y ) + ( y k ∂ y k + 1) k =1 ,...,n , (3)which we abbreviate as O T · Q nk =1 y − k . Now notice that e k is a closed embedding, hence a calculation sim-ilar to the proof of [SW09, proposition 2.1], using the ( D T , e k − D C ∗ τ × W ′ )-transfer bimodule D T → C ∗ τ × W ′ shows that the direct image e k + O T is given by e k + O T = D C ∗ τ × W ′ (cid:0)Q i : l i < ( w − w i ) − l i − Q i : l i > ( w − w i ) l i (cid:1) l ∈ L + ( P mi =1 a ki ∂ w i w i ) k =1 ,...,n + ( w ∂ w + P mi =1 ∂ w i w i ) . Now as w = − τ and ∂ λ i = − w i in FL − λ ,..., − λ m w ,...,w m e k + O T , we obtain that the latter module equals D C ∗ τ × W (cid:0)Q i : l i < ( τ − ∂ λ i ) − l i − Q i : l i > ( τ − ∂ λ i ) l i (cid:1) l ∈ L + ( P mi =1 a ki λ i ∂ λ i ) k =1 ,...,n + ( w ∂ w − P mi =1 λ i ∂ λ i ) . so that finallyFL − λ ,..., − λ m w ,...,w m e k + O T = D b V [ τ − ] τ l (cid:16)Q i : li< ∂ − liλi − Q i : li> ∂ liλi (cid:17) l ∈ L + ( P mi =1 a ki λ i ∂ λi ) k =1 ,...,n + ( τ∂ τ − P mi =1 λ i ∂ λi )= c M (1 , e A = c M e A . In the following proposition, we comment upon the more general case where the vectors a , . . . , a m onlygenerate N Q over Q . Let as before A = ( a , . . . , a m ) where a i are seen as vectors in Z n . Then it is awell-known fact that A can be factorized as B · C · B where B resp. B is in Gl( n, Z ) resp. Gl( m, Z )and C has the form e . . . 0 e n = e . . . e n · = D · E where e i are natural numbers called elementary divisors. Set A ′ := E · B , then A = B · D · A ′ and thecolumns of A ′ generate N over Z . Proposition 2.6.
We have the following isomorphism FL τt ( H ϕ + O S × W )[ τ − ] ≃ M j ∈ I n c M (1 ,j /e ,...,j n /e n ) e A . where j = ( j , . . . , j n ) ∈ N n and I n = Q nk =1 ([0 , e k − ∩ N ) ⊂ N n . roof. First notice that the morphism ϕ can be factorized into ϕ ′ ◦ (Φ × id S ), where Φ is the automorphismof S defined by B ∈ Gl( n, Z ). Hence ϕ + O S × W = ϕ ′ + O S × W , so that we can assume that B = id Z n ,i.e., that A = D · A ′ . Now one checks that the arguments in the proof of theorem 2.4 showing thatFL λ ,...,λ m w ,...,w m (cid:16)e k + O T (cid:17) ≃ FL τt ( H ϕ + O S × W )[ τ − ] are still valid under the more general hypothesis that A = D · A ′ where only the columns of A ′ do generate N over Z . Hence we need to compute the moduleFL λ ,...,λ m w ,...,w m (cid:16)e k + O T (cid:17) .The factorization of A corresponds to a factorization e k = e k ′ ◦ c , where c : ( y , y , . . . , y n ) ( y , y e , . . . y e n n )is a covering map and e k ′ is a closed embedding defined by the matrix A ′ . Let us first compute the directimage of O T under c . To do so, we look at the one-dimensional case, i.e. a map c k : y k y e k k . We have c k, + O C ∗ ≃ c k, + D C ∗ / ( y k ∂ y k ) ≃ e k − M j =0 D C ∗ / ( y k ∂ y k + 1 − j/e k ) , and moreover c + O T = O C ∗ ⊠ c , + O C ∗ ⊠ . . . ⊠ c n, + O C ∗ so that we get c + O T ≃ M j ∈ I n D T y ∂ y + ( y k ∂ y k + 1 − j k /e k ) k =1 ,...,n . In the next step we compute the direct image under the closed embedding e k ′ . Similar as above, weobtain for the direct image e k ′ + (cid:18) D T y ∂ y + ( y k ∂ y k + 1 − j k /e k ) k =1 ,...,n (cid:19) = D C ∗ τ × W ′ (cid:0)Q i : l i < ( w − w i ) − l i − Q i : l i > ( w − w i ) l i (cid:1) l ∈ L + ( P mi =1 a ki ∂ w i w i − j k /e k ) k =1 ,...,n + ( w ∂ w + P mi =1 ∂ w i w i )(4)The Fourier-Laplace transformation in the variables w , . . . , w m yieldsFL − λ ,..., − λ m w ,...,w m (cid:18)e k ′ + (cid:18) D T y ∂ y + ( y k ∂ y k + 1 − j k /e k ) k =1 ,...,n (cid:19)(cid:19) = c M β e A where β = (1 , j /e , . . . , j n /e n ). Taking the direct sum this givesFL τt ( H ϕ + O S × W )[ τ − ] ≃ FL − λ ,..., − λ m w ,...,w m (cid:16)e k + O T (cid:17) = M j ∈ I n c M (1 ,j /e ,...,j n /e n ) e A . In the following proposition, we collect some properties of the hypergeometric D -modules introducedabove. An important tool will be the notion of non-degeneracy of a Laurent polynomial, recall (see, e.g.,[Kou76] or [Ado94]) that f : ( C ∗ ) t → C , f = µ x b s + . . . + µ s x b s is called non-degenerate if for any properface τ of Conv(0 , b , . . . , b s ) ⊂ R t not containing 0, f τ = P b i ∈ τ µ i x b i has no critical points in ( C ∗ ) t . Proposition 2.7. M β e A (resp. c M β e A , c M β,loc e A ) is a coherent and holonomic D V -module (resp. D b V -module, D b T -module). Moreover, M β e A has only regular singularities, included at infinity.2. Let as before F : S × W → C t , ( y , . . . , y n , λ , . . . , λ m ) P mi =1 λ i · y a i . Define S := { ( λ , . . . , λ m ) ∈ S | F ( − , λ ) is non-degenerate with respect to its Newton polyhedron } . Moreover, consider the following extended family e F : T × V −→ C (( y , y , . . . , y n ) , ( λ , λ , . . . , λ m )) y · (cid:0) λ + P mi =1 λ i · y a i (cid:1) nd put V := { ( λ , λ , . . . , λ m ) ∈ C × S | e F ( − , λ ) is non-degenerate with respect to its Newton polyhedron } . Both S and V are Zariski open subspaces of S resp. C × S (as well as of W resp. V ). We have(a) The characteristic variety of the restriction of M β e A to V is the zero section of T ∗ V , i.e., M β e A is smooth on V .(b) Suppose that a , . . . , a m are defined by toric data and moreover, that the the projective variety X Σ A is genuine Fano, i.e., that its anti-canonical class is ample (and not only nef ). Then V \ V ⊂ ∆( F ) ∪ S mi =1 { λ i = 0 } ⊂ V , where ∆( F ) := (cid:8) ( − t, λ , . . . , λ m ) ∈ V | F ( − , λ ) − ( t ) is singular (cid:9) is the discriminant of the family − F .(c) The restriction of c M β,loc e A to C ∗ τ × S is smooth.3. Suppose that a , . . . , a m are defined by toric data. Then the generic rank of both M β e A and c M β e A is equal to n ! · vol(Conv( a , . . . , a m ))) = ( n + 1)! · vol(Conv( e , e a , . . . , e a m )) , where the volume of ahypercube [0 , t ⊂ R t is normalized to one, and where e denotes the origin in Z n +1 . Before entering into the proof, we need the following lemma.
Lemma 2.8.
Suppose that a , . . . , a m are the primitive integral generators of the rays of a fan Σ A defining a smooth toric Fano manifold X Σ A . Then the family F : S × S → C t is non-degenerate forany ( λ , . . . , λ m ) ∈ S .Proof. If X Σ A is Fano, then it is well known (see, e.g., [CK99, lemma 3.2.1]) that Σ A is the fan over theproper faces of Conv( a , . . . , a m ). Let τ be a face of codimension n + 1 − s and σ the corresponding s -dimensional cone over τ . As Σ A is regular, the primitive generators a τ , . . . , a τ s are linearly independent.We have to check that F τ ( λ, y ) = λ τ y a τ + . . . + λ τ s y a τs has no singularities on S for any ( λ τ , . . . , λ τ s ) ∈ ( C ∗ ) s . The critical point equations y k ∂ y k F τ = 0 canbe written in matrix notation as ( a τ ) ( a τ ) . . . ( a τ s ) ... ... ...( a τ ) n ( a τ ) n . . . ( a τ s ) n · λ τ · y a τ ... λ τ s · y a τs = 0 . This matrix has maximal rank and therefore can only have the trivial solution, contradicting the factthat ( λ τ , . . . , λ τ s ) ∈ ( C ∗ ) s and y ∈ S . Hence there is no solution at all and F is non-degenerate for all λ ∈ S . Proof of the proposition.
1. The holonomicity statement for M β e A is [Ado94, Theorem 3.9] (or eventhe older result [GKZ90, Theorem 1], as the vectors e a , e a , . . . , e a m lie in an affine hyperplaneof e N ). Then also c M β e A and c M β,loc e A are holonomic as this property is preserved under (partial)Fourier-Laplace transformation. The regularity of M β e A has been shown, e.g., in [Hot98, section 6].2. (a) This is shown in [Ado94, lemma 3.3].(b) By lemma 2.8, e F τ := P i : e a i ∈ τ λ i Q nk =0 y e a ki k can have a critical point in T only in the case that τ = Conv( e a , e a , . . . , e a m ), i.e., we have the following system of equations y ∂ y e F = y ( λ + P mi =1 λ i · Q nk =1 y a ki k ) ! = 0 , (cid:18) y k ∂ y k e F = y P mi =1 λ i · a ki Q nk =1 y a ki k ! = 0 (cid:19) k =1 ,...,n . The first equation yields λ = − t , where t denotes the value of the family F , and the secondone is the critical point equation for F . 11c) We know that char( c M β e A ) is included in the variety cut out by the ideal (cid:16) σ ( b (cid:3) l ) (cid:17) l ∈ L + ( σ ( b Z k )) k =1 ,...,n + σ ( b E ) . Write y resp. µ i for the cotangent coordinates on T ∗ ( C ∗ τ × S ) corresponding to z resp. λ i .As σ ( b E ) = zy + P ni =1 λ i µ i , it suffices to show that the sub-variety of C ∗ τ × T ∗ S defined bythe ideal (cid:16) σ ( b (cid:3) l ) (cid:17) l ∈ L + ( σ ( b Z k )) k =1 ,...,n equals the zero section. Write β = ( β , β ′ ) with β ′ ∈ N C . Notice that for any l ∈ L , if l = 0,then either σ ( b (cid:3) l ) or σ ( z l b (cid:3) l ) belongs to C [ µ , . . . , µ m ] and equals the symbol of one of theoperators defining M β ′ A . Similarly, if l = 0, then already (cid:3) l itself is independent of z andequal to an operator from M β ′ A . This shows that [Ado94, lemma 3.1 to lemma 3.3] holds for c M β e A , and hence c M β e A is smooth on C ∗ τ × S .3. For the D V -module M β e A this is [Ado94, corollary 5.21] as Spec C [ N e A ] is Cohen-Macaulay by propo-sition 2.1, 2., notice that the Cohen-Macaulay condition is needed only for the ring Spec C [ N e A ],not for any of its subrings as the only face τ occurring in loc.cit. that does not contain the originis the one spanned by the vectors e a , e a . . . , e a m .Similarly, [Ado94, corollary 5.21] shows that the generic rank of M β ′ A equals n ! · vol(Conv( a , . . . , a m )):Here we have to use the fact that all cones σ ∈ Σ A are smooth, so that the semigroup rings gener-ated by their primitive integral generators are normal and Cohen-Macaulay. Now it follows fromthe calculation of the characteristic variety from 2(c) that this is then also the generic rank of c M β e A .For later purpose, we need a precise statement on the regularity resp. irregularity of the module c M β e A ,at least in the case of main interest where a , . . . , a m are defined by toric data. As a preliminary step,we show in the following proposition a finiteness result for the singular locus of M β e A . Proposition 2.9.
Suppose that a , . . . , a m are defined by toric data. Let p : V → W be the projectionforgetting the first component. Then for any λ = ( λ , . . . , λ m ) ∈ S , there is a small analytic neighborhood U λ ⊂ S ,an such that the restriction p | ∆( F ) an ∩ p − ( U λ ) : ∆( F ) an ∩ p − ( U λ ) −→ U λ is finite, i.e., proper with finite fibres. In particular p | ∆( F ) ∩ p − ( S ) : ∆( F ) ∩ p − ( S ) → S is finite.Proof. Write P λ for the restriction p | ∆( F ) an ∩ p − ( U λ ) . The quasi-finiteness of P λ is obvious, as for any λ ∈ S , F ( − , λ ) has only finitely many critical values. Hence we need to show that P λ is proper. Take anycompact subset K in U λ . Suppose that P − λ ( K ) is not compact, then it must be unbounded in V ∼ = C m +1 for the standard metric. Hence there is a sequence ( λ ( i )0 , λ ( i ) ) ∈ P − λ ( K ) with lim i →∞ | λ ( i )0 | = ∞ , as K isclosed and bounded in W ∼ = C m . Consider the projection π : V → P ( V ) = Proj C [ λ , λ , . . . , λ m ], then(possibly after passing to a subsequence), we have lim i →∞ π ( λ ( i )0 , λ ( i ) ) = (1 : 0 : . . . : 0).In order to construct a contradiction, we will need to consider a partial compactification of the family F , or rather of the morphism ϕ : S × S → C t × S . This is done as follows (see, e.g., [DL91] and[Kho77]): Write X B for the projective toric variety defined by the polytope Conv( a , . . . , a m ) (underthe assumption that X Σ A is weak Fano, this is a reflexive polytope in the sense of [Bat94]) then X B embeds into P ( V ′ ) and contains the closure of the image of the morphism k from proposition 2.1. Write Z = { P mi =0 λ i · w i = 0 } ⊂ P ( W ′ ) × P ( W ) for the universal hypersurface and put Z B := ( X B × P ( W )) ∩ Z .Consider the map π : X B × ( C t × S ) → X B × P ( W ), let e Z B := π − ( Z B ), and write φ for the restrictionof the projection X B × ( C t × S ) ։ C t × S to e Z B . Then φ is proper, and restricts to ϕ on S × S ∼ =Γ ϕ ⊂ e Z B . There is a natural stratification of X B by torus orbits and this gives a product stratificationon X B × ( C t × S ). Now consider the restriction φ ′ of φ to e Z ′ B := φ − ( C t × S ), then one checks thatthe non-degeneracy of F on S is equivalent to the fact that Z cuts all strata of ( X B \ S ) × ( C t × S )12ransversal. Hence we have a natural Whitney stratification Σ on (the analytic space associated to) e Z ′ B .If we write Crit Σ ( φ ′ ) for the Σ-stratified critical locus of φ ′ , i.e., Crit Σ ( φ ′ ) := S Σ α ∈ Σ Crit( φ ′| Σ α ), thenwe have Crit Σ ( φ ′ ) = Crit( ϕ ′ ), where ϕ ′ := ϕ | S × S . On the other hand, Whitney’s (a)-condition impliesthat Crit Σ ( φ ′ ) is closed in e Z ′ B , and so is Crit( ϕ ′ ).Now consider the above sequence ( λ ( i )0 , λ ( i ) ) ∈ P − λ ( K ) ⊂ ∆( F ) an , then the fact that the projection fromthe critical locus of ϕ to the discriminant is onto shows that there is a sequence (( w ( i )0 , w ( i ) ) , ( λ ( i )0 , λ ( i ) ) ∈ Crit( ϕ ′ ) ⊂ S × K projecting under ϕ ′ to ( λ ( i )0 , λ ( i ) ). Consider the first component of the sequence π (( w ( i )0 , w ( i ) ) , ( λ ( i )0 , λ ( i ) )), then this is a sequence ( w ( i )0 , w ( i ) ) in X B which converges (after passing possiblyagain to a subsequence) to a limit (0 : w lim1 , . . . , w lim m ) (this is forced by the incidence relation P mi =0 w i λ i =0), in other words, this limit lies in X B \ S . However, we know that lim i →∞ (( w ( i )0 , w ( i ) ) , ( λ ( i )0 , λ ( i ) ) existsin Crit Σ ( φ ′ ) as the latter space is closed. This is a contradiction, as we have seen that φ is non-singularoutside S × ( C t × S ), i.e., that Crit Σ ( φ ′ ) = Crit( ϕ ′ ) ⊂ S × S .Now the regularity result that we will need later is the following. Lemma 2.10.
Consider c M β e A as a D P z × W -module, where W is a smooth projective compactification of W . Then c M β e A is regular outside ( { z = 0 } × W ) ∪ ( P z × ( W \ S )) .Proof. It suffices to show that any λ = ( λ , . . . , λ m ) ⊂ S has a small analytic neighborhood U λ ⊂ S ,an such that the partial analytization c M β,loc e A ⊗ O C ∗ τ × S O anU λ [ τ, τ − ] is regular on C τ × U λ (but not at τ = ∞ ).This is precisely the statement of [DS03, theorem 1.11 (1)], taking into account the regularity of M β e A (i.e., proposition 2.7, 1.), the fact that on C λ × U λ , the singular locus of M β e A coincides with ∆( F )(see the proof of proposition 2.7, 2(b)) as well as the last proposition (notice that the non-characteristicassumption in loc.cit. is satisfied, see, e.g., [Pha79, page 281]). The next step is to study natural lattices that exist in G and in c M β e A . To avoid endless repetition ofhypotheses, we will assume throughout this subsection that our vectors a , . . . , a m are defined by toricdata. In order to discuss lattices in c M β e A , we start with definition. Definition 2.11.
1. Consider the ring R := C [ λ ± , . . . , λ ± m , z ] h z∂ λ , . . . , z∂ λ m , z ∂ z i , i.e. the quotient of the free associative C [ λ ± , . . . , λ ± m , z ] -algebra generated by z∂ λ , . . . , z∂ λ m , z ∂ z by the left ideal generated by the relations [ z∂ λ i , z ] = 0 , [ z∂ λ i , λ j ] = δ ij z, [ z ∂ z , λ i ] = 0 , [ z ∂ z , z ] = z , [ z∂ λ i , z∂ λ j ] = 0 , [ z ∂ z , z∂ λ i ] = z · z∂ λ i . Write R for the associated sheaf of quasi-coherent O C z × S -algebras which restricts to D C ∗ τ × S on { ( z = 0 } . We also consider the subring R ′ := C [ λ ± , . . . , λ ± m , z ] h z∂ λ , . . . , z∂ λ m i of R , and theassociated sheaf R ′ . The inclusion R ′ ֒ → R induces a functor from the category of R -modules tothe category of R ′ -modules, which we denote by For z ∂ z (“forgetting the z ∂ z -structure”).2. Choose β ∈ e N C , consider the ideal I := R ( b (cid:3) ′ l ) l ∈ L + R ( z · b Z k ) k =1 ,...,n + R ( z · b E ) in R and write c M β,loc e A for the quotient R / I . We have For z ∂ z ( c M β,loc e A ) = R ′ / (( b (cid:3) ′ l ) l ∈ L + ( z · b Z k ) k =1 ,...,n ) , and therestriction of c M β,loc e A to C ∗ τ × S equals c M β,loc e A . Again we put c M loc e A := c M (1 , ,loc e A . Corollary 2.12.
Consider the restriction of the isomorphism φ from theorem 2.4 to C ∗ τ × S .1. φ sends the class of the section in c M loc e A to class of the (relative) volume form ω := dy /y ∧ . . . ∧ dy n /y n ∈ Ω nS × S /S . . The morphism φ maps c M loc e A isomorphically to G := π ∗ Ω nS × S /S [ z ]( zd − dF ∧ ) π ∗ Ω n − S × S /S [ z ] . Proof.
1. Following the identifications in the proof of theorem 2.4, this is evident, if one takes intoaccount that due to the choice in formula (3), we have actually computed G | C ∗ τ × S = FL τt (cid:18) H ϕ + O S × S y · . . . · y n (cid:19) = FL τt (cid:18) H ϕ + D S × S D S × S ( y k ∂ y k + 1) k =1 ,...,n (cid:19)
2. First notice that due to 1. and the formulas in lemma 2.5, we have φ (cid:16) c M loc e A (cid:17) ⊂ G . To see thatit is surjective, take any representative s = P i ≥ ω ( i ) z i of a class in G . As an element of G , s has a unique preimage under φ , which is an operator P ∈ c M loc e A and we have to show that actually P ∈ c M loc e A . By linearity of φ , it is sufficient to do it for the case where ω (0) = 0. There is aminimal k ∈ N such that z k P ∈ c M loc e A , and then the class of z k P in c M loc e A /z · c M loc e A does notvanish. Suppose that k >
0, then the class of φ ( z k P ) = z k s vanishes in G /zG , which contradictsthe next lemma. Hence k = 0 and P ∈ c M loc e A . Lemma 2.13.
1. The quotient c M loc e A /z · c M loc e A is the sheaf of commutative O S -algebras associatedto C [ λ ± , . . . , λ ± m , µ , . . . , µ m ]( Q l i < µ − l i i − Q l i > µ l i i ) l ∈ L + ( P mi =1 a ki λ i µ i ) k =1 ,...,n
2. The induced map [ φ ] : c M loc e A /z · c M loc e A −→ G /zG ∼ = π ∗ Ω nS × S /S /d y F ∧ π ∗ Ω n − S × S /S is an isomorphism.Proof.
1. Letting µ i be the class of z∂ λ i in c M loc e A /z · c M loc e A , we see that the commutator [ µ i , λ i ]vanishes in this quotient.2. This can be shown along the lines of [Bat93, theorem 8.4]. Namely, consider the morphism of C [ λ ± , . . . , λ ± ]-algebras ψ : C [ λ ± , . . . , λ ± m , µ , . . . , µ m ] −→ C [ λ ± , . . . , λ ± m , y ± , . . . , y ± n ] µ i y a i From the completeness and smoothness of Σ A we deduce that ψ is surjective. Moreover, we have ker ( ψ ) = ( Q l i < µ − l i i − Q l i > µ l i i ) l ∈ L (for a proof, see, e.g., [MS05, theorem 7.3]), and obviously ψ ( P mi =1 a ki λ i µ i ) = y k ∂ y k F for all k = 1 , . . . , n . One easily checks that the induced map ψ : C [ λ ± , . . . , λ ± m , µ , . . . , µ m ]( Q l i < µ − l i i − Q l i > µ l i i ) l ∈ L + ( P mi =1 a ki λ i µ i ) k =1 ,...,n −→ C [ λ ± , . . . , λ ± m , y ± , . . . , y ± n ]( y k ∂ y k F ) k =1 ,...,n coincides with the map [ φ ] induced by φ , notice that C [ λ ± , . . . , λ ± m , y ± , . . . , y ± n ]( y k ∂ y k F ) k =1 ,...,n ∼ = π ∗ Ω nS × S /S dF ∧ π ∗ Ω n − S × S /S . by multiplication with the relative volume form dy /y ∧ . . . ∧ dy n /y n .14ollowing the terminology of [Sab06] and [DS03] (going back to [Sai89], and, of course, to [Bri70]), wecall G (and, using the last result, also c M loc e A ) the (family of) Brieskorn lattice(s) of the morphism ϕ .For the case of a single Laurent polynomial F λ := ϕ ( − , λ ) : S → C , it follows from the results of [Sab06]that the module Ω nS [ z ] / ( zd − dF λ ∧ )Ω n − S [ z ] is C [ z ]-free provided that λ ∈ S , recall that S denotes theZariski open subset of S of parameter values λ such that F ( − , λ ) is non-degenerate with respect to itsNewton polyhedron. However, this does not directly extend to a finiteness (and freeness) result for theBrieskorn lattice G of the family ϕ : S × S → C t × S . We can now prove this freeness using corollary2.12. Theorem 2.14.
The module O C z × S ⊗ O C z × S c M loc e A (and hence also the module O C z × S ⊗ O C z × S G )is O C z × S -locally free.Proof. The main argument in the proof is very much similar to the proof of proposition 2.7, 2.c). It isactually sufficient to show that O C z × S ⊗ O C z × S c M loc e A is O C z × S -coherent. Namely, we know that therestriction O S ⊗ O S (cid:16) c M loc e A /z · c M loc e A (cid:17) equals the Jacobian algebra of ϕ | S × S , which is O S -locally freeof rank equal to the Milnor number of ϕ | S × S , that it, equal to n ! · vol(Conv( a , . . . , a m )), see [Kou76,th´eor`eme 1.16]. Moreover, the restriction O C ∗ τ × S ⊗ O C z × S c M loc e A = O C ∗ τ × S ⊗ O C ∗ τ × S c M e A is locally freeof the same rank and equipped with a flat structure, so that c M loc e A ⊗ O C z × S O C z × S can only have thesame rank everywhere, provided that it is coherent.It will be sufficient to show the coherence of N := O C z × S ⊗ O C z × S For z ∂ z ( c M loc e A ) only, as this is thesame as O C z × S ⊗ O C z × S c M loc e A when considered as an O C z × S -module. Let us denote by F • the naturalfiltration on R ′ defined by F k R ′ := P ∈ R ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = X | α |≤ k g α ( z, λ )( z∂ λ ) α · . . . · ( z∂ λ m ) α r . This filtration induces a filtration F • on N which is good, in the sense that F k R ′ · F l N = F k + l N .Obviously, for any k , F k N is O C z × S -coherent, so that it suffices to show that the filtration F • becomeeventually stationary. The ideal generated by the symbols of all operators in the ideal defining N , thatis, by the highest order terms with respect to the filtration F • , cut out a subvariety of C z × T ∗ S , andit suffices to show that this subvariety equals C z × S , then by the usual argument the filtration F • stabilizes for some sufficiently large index. However, for any of the operators b (cid:3) ′ l and b Z k in I ′ , its symbolwith respect to the above filtration F • is precisely the same as the symbol of b (cid:3) l , b Z k with respect to theordinary filtration on c M loc e A , hence, the same argument as in the proof of proposition 2.7, 2.c) (that is,the arguments in [Ado94, lemma 3.1 to lemma 3.3]) shows that the above mentioned subvariety is thezero section C ∗ × S . In this section, we discuss the holonomic dual of the hypergeometric system M e A , from which we deducea self-duality property of the module c M e A . Moreover, we study the natural good filtration on M e A byorder of operators, and show that it is preserved, up to shift, by the duality isomorphism. We obtain aninduced filtration on c M loc e A by O S [ z ]-modules (which is not a good filtration on this module). Its zerothstep turns out to coincide with the lattice c M loc e A considered in the last subsection. This shows that weobtain a non-degenerate pairing on c M loc e A , a fact that we will need later in the construction of Frobeniusstructures.We start by describing the holonomic dual of the D V -module M e A . This description is based on the localduality theorem for the Gorenstein ring Spec C [ N e A ]. If we were only interested in the description of thisdual module, we could simply refer to [Wal07, proposition 4.1], however, as we need later a more refinedversion taking into account filtrations, we recall the techniques using Euler-Koszul homology that leadsto this duality result.We suppose throughout this section that the vectors a , . . . , a m are defined by toric data.15 heorem 2.15.
1. For any holonomic left D V -module N , write D N for the left D V -module associ-ated to the right D V -module E xt m +1 D V ( N , D V ) , where we use, as V is an affine space, the canonicalidentification O V ∼ = Ω m +1 V given by multiplying functions with the volume form dλ ∧ . . . ∧ dλ m .Then we have D M β e A = M − β +(1 , e A , in particular D M e A = M (0 , e A ,
2. We have the following isomorphisms of holonomic left D b V - (resp. D b T )-modules c M e A ∼ = ι ∗ D c M e A c M loc e A ∼ = ι ∗ D c M loc e A , here ι : b V → b V resp. ι : b T → b T is the automorphism sending ( z, λ , . . . , λ m ) to ( − z, λ , . . . , λ m ) . Before giving the proof of this result, we need to introduce some notations. The basic ingredient for theproof is an explicit resolution of M β e A by the so-called Euler-Koszul complex. We recall the descriptionof this complex from [MMW05]. In order to be consistent with the notations used in loc.cit., we willrather work with rings and modules than with sheaves. Therefore, put R = C [ w , . . . , w m ] and S = R/I where I is the toric ideal of e a , e a , . . . , e a m , i.e., the ideal generated by w l · Q i : l i < w − l i i − Q i : l i > w l i i for any l ∈ L with l ≥ Q i : l i < w − l i i − w l · Q i : l i > w l i i for any l ∈ L with l < Z n +1 -graded, where deg( w i ) := − e a i ∈ Z n +1 (more invariantly, they are e N -graded),notice that the homogeneity of I follows from the fact that L is the kernel of the surjection Z m +1 ։ e N given by the matrix e A . We write D = Γ( V, D V ) for the ring of algebraic differential operators on V .However, using the Fourier-Laplace isomorphism D ∼ = Γ( V ′ , D V ′ ) given by ∂ λ i
7→ − w i and λ i ∂ w i ,we can also view D as the ring of differential operators on the dual space, and we shall do so if D -modules are considered as R -modules. We have a natural Z n +1 -grading on D defined by deg( λ i ) = e a i and deg( ∂ λ i ) = − e a i , and the Fourier-Laplace isomorphism gives rise to an injective Z n +1 -graded ringhomomorphism R ֒ → D sending w i to − ∂ λ i . Again in order to match our notations with those from[MMW05], let us put E := P mi =0 λ i ∂ λ i ∈ D and E k := P mi =1 a ki λ i ∂ λ i ∈ D for all k = 1 , . . . , n . Let P beany Z n +1 -graded D -module, and α ∈ C n arbitrary, then by putting ( E k − α k ) ◦ y := ( E k − α k − deg k ( y ))( y )for k = 0 , . . . , n and for any homogeneous element y ∈ P and by extending C -linearly, we obtain a D -linear endomorphism of P . We also have that the commutator [( E i − α i ) ◦ , ( E j − α j ) ◦ ] vanishes for any i, j ∈ { , . . . , m } . Hence we can define the Euler-Koszul complex K • ( E − α, P ), a complex of Z n +1 -graded left D -modules, to be the Koszul complex of the endomorphisms ( E − α ) ◦ , . . . , ( E n − α n ) ◦ on P . Notice that here E is an abbreviation for the vector ( E , E , . . . , E n ) and should not be confusedwith the single vector field P mi =0 λ i ∂ λ i + β used in the definition of the modules M β e A . The definitionof the Euler-Koszul complex applies in particular to the case P := D ⊗ R T , where T is a so-called toric R -module (see [MMW05, definition 4.5]), in which case we also write K • ( E − α, T ) for the Euler-Koszulcomplex. Similarly one defines the Euler-Koszul cocomplex, denoted by K • ( E − α, P ) resp. K • ( E − α, T ),where K i ( E − α, P ) = K n +1 − i ( E − α, P ) and the signs of the differentials are changed accordingly. Inparticular, we have H i ( K • ( E − α, P )) = H n +1 − i ( K • ( E − α, P )). We will mainly use the constructionof the Euler-Koszul complex resp. cocomplex in the case of the toric R -module S , or for shifted version S ( e c ), where e c ∈ Z n +1 .The main result on the Euler-Koszul homology and holonomic duality that we need is the following.For any D -module M , consider a D -free resolution L • ։ M , then we write D M for the complex of left D -modules associated to Hom D ( L • , D ). 16 emma 2.16 ([MMW05, theorem 6.3]) . Put ε e a := P mk =0 e a i ∈ Z n +1 . Then there is a spectral sequence E p,q = H q ( K • ( E + α, Ext pR ( S, ω R ))( − ε e a ) = ⇒ H p + q D (cid:0) H p + q − ( m +1) ( K • ( E − α, S )) (cid:1) − . (5) Here ( − ) − is the auto-equivalence of D -modules induced by the involution λ i
7→ − λ i and ∂ λ i
7→ − ∂ λ i .Notice that it is shown in [MMW05, lemma 6.1] that Ext pR ( S, ω R ) is toric. Notice also that the dualizingmodule ω R is nothing but the ring R , placed in Z n +1 -degree ε e a (see, e.g., [MS05, definition 12.9 andcorollary 13.43] or [BH93, corollary 6.3.6] for this). In our situation, the relevant
Ext -group occurring in the spectral sequence of this lemma is actuallyrather simple to calculate, as the next result shows.
Lemma 2.17.
There is an isomorphism of Z n +1 -graded R -modules Ext m − nR ( S, ω R ) ∼ = ω S ∼ = S ((1 , .Proof. First it follows from a change of ring property that
Ext m − n ( S, ω R ) = ω S (see [BH93, proposition3.6.12]). We are thus reduced to compute a canonical module for the ring S . Remark that S is nothingbut the semigroup ring C [ N e A ] from proposition 2.1 (see again [MS05, theorem 7.3]), and its canonicalmodule is the ideal in S generated by the monomials corresponding to the interior points of N e A . Wehave seen in proposition 2.1, 2., that the set of these interior points is given as (1 ,
0) + N e A , i.e., we havethat ω S = S ((1 , S is a quotient of R = C [ w , w , . . . , w m ] and that deg( w i ) = − e a i . Proof of the theorem.
In order to use lemma 2.16 for the computation of the holonomic dual of M β e A ,write M β e A := H ( V, M β e A ) and notice that the homology group H ( K • ( E − α, S )), seen both as a R -module and a D -module, is nothing but Γ( V ′ , FL w ,...,w m λ ,...,λ m ( M − α e A )). Hence by putting α := − β , we havean equality D M β e A = H m +1 ( D H ( K • ( E + β, S ))) − of D -modules. Notice that the duality functor and theFourier-Laplace transformation commutes only up to a sign (see, e.g., [DS03, paragraph 1.b]), for thisreason, the right hand side of the last formula is twisted by the involution ( − ) − .1. As the ring S is Cohen-Macaulay, Ext pR ( S, ω R ) can only be non-zero if p = codim R ( S ) = m +1 − ( n + 1) = m − n . This implies that the spectral sequence (5) degenerates at the E -term,so that E m − n,q = H m − n + q D (cid:0) H ( m − n )+ q − ( m +1) ( K • ( E − α, S )) (cid:1) − . On the other hand, we deducefrom lemma 2.17 that E m − n,q = H q ( K • ( E + α, S ((1 , ∼ = H n +1 − q ( K • ( E + α + (1 , , S ))(1 , . where we have used the equality K • ( E + α, S ( e c )) = K • ( E + α + e c, S )( e c )of complexes of Z n +1 -graded D -modules. As noticed in [MMW05, remark 6.4] the CM-property of S also implies that the Euler-Koszul complex K • ( E − α, S ) can only have homology in degree zero,hence E m − n,q = 0 unless q = n + 1. This is consistent with the fact that due to the holonomicityof M β e A , the right hand side of the spectral sequence (5) can only be non-zero for p + q = m + 1.Summarizing, we obtain an isomorphism of Z n +1 -graded D -modules H m +1 ( D H ( K • ( E − α, S ))) − = H ( K • ( E + α + (1 , , S ))(1 , , from which we deduce an isomorphism of sheaves of D V -modules (recall that α = − β ) D M β e A ∼ = M − β +(1 , e A , as required.2. Put β = (1 , ∈ Z n +1 ∼ = e N , then it follows from 1. that we have a morphism φ : M β e A −→ D M β e A m a · ∂ λ D [ ∂ − λ ] for the partial (polynomial) microlocalization C [ λ , λ , . . . , λ m ] h ∂ λ , ∂ − λ , ∂ λ , . . . , ∂ λ m i .Then φ induces an isomorphism D [ ∂ − λ ] ⊗ D M β e A ∼ = → D [ ∂ − λ ] ⊗ D ( D M β e A ). On the other hand, it followsfrom the D -flatness of D [ ∂ − λ ] that Ext m +1 D [ ∂ − λ ] ( M, D [ ∂ − λ ]) = Ext m +1 D ( M, D ) ⊗ D [ ∂ − λ ] for any left D -module M , hence we obtain an isomorphism D [ ∂ − λ ] ⊗ D M β e A ∼ = −→ D ( D [ ∂ − λ ] ⊗ D M β e A )where the symbol D on the right hand side denotes the composition of Ext m +1 D [ ∂ − λ ] ( − , D [ ∂ − λ ]) withthe transformation of right D [ ∂ − λ ] to left D [ ∂ − λ ]-modules. Performing a partial Fourier-Laplacetransformation, we obtain an isomorphism (still denoted by φ ) φ : FL τλ ( M β e A )[ τ − ] ∼ = −→ FL τλ (cid:16) D ( D [ ∂ − λ ] ⊗ D M β e A ) (cid:17) , which is given by right multiplication with τ = z − . On the other hand, it is known (see, e.g.,[DS03, paragraph 1.f]) that for any D [ ∂ − λ ]-module N , we have an isomorphism FL τλ ( D N ) ∼ = ι ∗ D (FL τλ ( N )) which gives usFL τλ ( M β e A )[ τ − ] ∼ = −→ ι ∗ D (cid:16) FL τλ ( D [ ∂ − λ ] ⊗ D M β e A ) (cid:17) ∼ = ι ∗ D (cid:16) FL τλ ( M β e A )[ τ − ] (cid:17) from which we deduce the isomorphism c M e A ∼ = ι ∗ D c M e A of D b V -modules resp. the isomorphism c M loc e A ∼ = ι ∗ D c M loc e A of D b T -modules.The next step is to investigate a natural good filtration defined on the sheaf M β e A . We write M β e A := H ( V, M β e A ) which is isomorphic to H ( K • ( E + β, S )) as a D -module. Proposition 2.18.
1. Write F • for the natural filtration on D by order of ∂ λ i -operators and denotethe induced filtration on M β e A also by F • . There is a resolution L • of M β e A by free D -modules whichis equipped with a strict filtration F L • • and we have a filtered quasi-isomorphism ( L • , F L • ) ։ ( M β e A , F • ) .2. Consider the case β = (1 , , i.e., M β e A = M e A . Write F D • D M e A for the dual filtration of F • M e A , i.e., D ( M e A , F • ) = ( D M e A , F D • ) (see, e.g., [Sai94, page 55]), then we have F k M (0 , e A = F D k − n +( m +1) D M e A .
3. For any β ∈ Z n +1 , F • M β e A induces a filtration G β • by O C z × S -modules on the D b T -module c M β,loc e A and we have an isomorphism of O C z × S -modules G c M β,loc e A ∼ = c M β,loc e A , in particular G c M loc e A ∼ = c M loc e A . Moreover, for any k , O C z × S ⊗ O C z × S G k c M loc e A is O C z × S -locally free.For β = (1 , we obtain from the dual filtration F D • on D M e A a filtration G D • by O C z × S -moduleson c M (0 , e A .4. Consider the isomorphism φ : FL w λ ( M e A )[ τ − ] = c M e A −→ ι ∗ FL w λ ( D M e A )[ τ − ] = c M (0 , e A from the proof of theorem 2.15, 2., which is given by multiplication with z − . Then we have φ ( G • ) = G D • + m +2 − n c M (0 , b A . roof.
1. The free resolution L • ։ M β e A is obtained as in the proof of [MMW05, theorem 6.3] asthe total complex Tot K • ( E + β, F • ) of a resolution of the Euler-Koszul complex obtained from a R -free Z n +1 -graded resolution F • of S . In particular, this resolution is Z -graded for the gradingof R = C [ w , w , . . . , w m ] = C [ ∂ λ , ∂ λ , . . . , ∂ λ m ] for which deg( w i ) = deg( ∂ λ i ) = 1. On theother hand, the differentials of the Euler-Koszul complex are constructed from linear differentialoperators. Hence by putting on each term of the above total complex (which is D -free) a filtrationwhich is on each factor of such a module the order filtration on D , shifted appropriately, we obtaina strict resolution of ( M β e A , F • ).2. From the construction of the resolution L • ։ M e A , from point 1., we see that L k = 0 for all k > m + 1 (notice that we write this resolution such that d : L k → L k − so that M e A = H ( L • , d ))and L m +1 = D . We have seen that the filtration on L m +1 is the order filtration on D , shiftedappropriately and we have to determine this shift. It is the sum of the length of the Euler-Koszulcomplex (i.e., n + 1) and the degree (with respect to the grading of R for which deg( ∂ λ i ) = 1) of Ext n − mR ( S, ω R ). The latter is equal to m , which is the first component of the difference betweenthe canonical degree of R (i.e., ε A ) and the canonical degree of S (i.e., (1 , L m +1 is F •− ( n + m +1) D . Now by definition (see, e.g., [Sai94, page 55]), we have D ( M e A , F • ) = H m +1 Hom D (cid:0) ( L • , F L • • ) , (( D ⊗ Ω m +1 V ) ∨ , F •− m +1) D ⊗ (Ω m +1 V ) ∨ ) (cid:1) and this implies the formula for F D • D M e A .3. We will consider the ∂ − λ -saturation of the filtration steps F k M e A . More precisely, consider again M e A [ ∂ − λ ] := D [ ∂ − λ ] ⊗ D M e A , and the natural localization morphism c loc : M e A → M e A [ ∂ − λ ]. Put F k M e A [ ∂ − λ ] := P j ≥ ∂ − jλ c loc( F k + j M e A ). Then we easily see that F k M e A [ ∂ − λ ] = Im (cid:0) ∂ kλ C [ λ , λ , . . . , λ m ] h ∂ − λ , ∂ − λ ∂ λ , . . . , ∂ − λ ∂ λ m i (cid:1) in M e A [ ∂ − λ ] . The filtration F • M e A [ ∂ − λ ] induces a filtration G • on c M loc e A = Γ( b T , c M loc e A ), with G k c M loc e A = Im (cid:0) z − k C [ z, λ ± , . . . , λ ± m ] h z∂ λ , . . . , z∂ λ m , z ∂ z i (cid:1) in c M loc e A Hence we obtain a filtration G • on the sheaf c M loc e A and we have G c M loc e A = c M loc e A , as required.Moreover, z k · : G p c M loc e A ∼ = −→ G p − k c M loc e A , and it follows from theorem 2.14 that O C z × S ⊗ O C z × S G c M loc e A , and hence all O C z × S ⊗ O C z × S G p c M loc e A are O C z × S -locally free. Notice however that G • is in general not a good filtration on c M loc e A , as ∂ z G k c M loc e A ⊂ G k +2 c M loc e A whereas ∂ λ i G k c M loc e A ⊂ G k +1 c M loc e A .Concerning the filtration G D • , notice that due to the definition of F k M e A [ ∂ − λ ], the strictly filteredresolution of ( M e A , F • ) from part 2 from above yields a strictly filtered resolution of the filteredmodule ( M e A [ ∂ − λ ] , F • M e A [ ∂ − λ ]), and the dual complex is then also strictly filtered and defines afiltration G D • on D ( M e A [ ∂ − λ ]), which is nothing but the ∂ − λ -saturation of the dual filtration F D • from point 2. from above. Hence we obtain a filtration G • by O C z × S -modules on D c M e A = c M (0 , e A .4. This is a direct consequence of 2. and 3.As a consequence, we obtain the existence of a non-degenerate pairing on the lattice c M loc e A consideredabove. Corollary 2.19.
1. There is a non-degenerate flat ( − n -symmetric pairing P : (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) ⊗ ι ∗ (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) → O C ∗ τ × S . . We have that P ( c M loc e A , c M loc e A ) ⊂ z n O C z × S , and P is non-degenerate on O C z × S ⊗ O C z × S c M loc e A ,i.e., it induces a non-degenerate symmetric pairing [ z − n P ] : " O S ⊗ O S c M loc e A z · c M loc e A ⊗ " O S ⊗ O S c M loc e A z · c M loc e A → O S . Proof.
1. The statement can be reformulated as the existence of an isomorphism ψ : (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) ∼ = −→ ι ∗ (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) ∗ where ( − ) ∗ denotes the dual meromorphic bundle with its dual connection. We deduce from[DS03, lemma A.11] (see also [Sai89, 2.7]) that D ( O C ∗ τ × S ⊗ O C τ × S c M loc e A )( ∗ ( { , ∞} × S )) =( O C ∗ τ × S ⊗ O C τ × S c M loc e A ) ∗ . On the other hand, theorem 2.15, 2. gives an isomorphism O C ∗ τ × S ⊗ O C τ × S c M loc e A ∼ = ι ∗ D ( O C ∗ τ × S ⊗ O C τ × S c M loc e A ) so that the latter module is already localized, i.e., equal to( O C ∗ τ × S ⊗ O C τ × S c M loc e A ) ∗ , which gives the existence of the isomorphism ψ from above.2. We have seen in point 1. that the duality isomorphism φ = z − · : FL w λ ( M e A )[ τ − ] −→ FL w λ ( D M e A )[ τ − ]yields an isomorphism ψ : (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) ∼ = −→ ι ∗ (cid:16) O C ∗ τ × S ⊗ O C τ × S c M loc e A (cid:17) ∗ of meromorphic bundles with connection. Now it follows from [Sai89, formula 2.7.5] that we have H om O C z × S (cid:16) O C z × S ⊗ O C z × S G k c M loc e A , O C z × S (cid:17) = O C ∗ τ × S ⊗ O C z × S G D k +( m +2) c M (0 , ,loc e A . Hence by proposition 2.18, 4. from above we conclude that ψ sends the module O C z × S ⊗ O C z × S G c M loc e A = O C z × S ⊗ O C z × S c M loc e A isomorphically into H om O C z × S (cid:16) O C z × S ⊗ O C z × S G − n c M loc e A , O C z × S (cid:17) = z n H om O C z × S (cid:16) O C z × S ⊗ O C z × S G c M loc e A , O C z × S (cid:17) = z n H om O C z × S (cid:16) O C z × S ⊗ O C z × S c M loc e A , O C z × S (cid:17) , which is equivalent to the statement to be shown. D -modules with logarithmic structure and good bases In this section we apply the results of section 2 to study hypergeometric D -modules on a subtorus ofthe m -dimensional torus S . We suppose that our vectors a , . . . , a m are defined by toric data. In thissituation, the subtorus is defined as S := Spec C [ L ], where, as before, L is the module of relationsbetween a , . . . , a m . Following standard terminology, we call this torus the complexified K¨ahler modulispace of X Σ A . We will consider a subfamily of Laurent polynomials of the morphism ϕ : S × W → C t × W from the last section, parameterized by S and we will show that the associated Gauß-Manin system alsohas a hypergeometric structure.For a good choice of coordinates on S embedding it into some affine space C r , we will construct anextension of this hypergeometric modules to a certain lattice with logarithmic poles along the boundary20ivisor C r \ S . This “ D -module with logarithmic structure” will play a crucial role in the next section:on the one hand, we will see that it equals the so-called Givental connection defined by the quantumcohomology of the variety X Σ A , on the other hand, we will use it to construct logarithmic Frobeniusstructures and express the mirror correspondence in terms of them. For that purpose, we will show thatthis logarithmic extension is still a free module, and can be extended to a family of trivial bundle over P × C r (or at least outside the locus where the family of mirror Laurent polynomials is degenerate) onwhich the connection extends with a logarithmic pole at infinity. This structure is the key ingredient toconstruct a logarithmic Frobenius manifold, this will be done in section 4. D -modules on K¨ahler mod-uli spaces We briefly recall the situation considered in the beginning of the last section, with the more specificassumption that now the input data we are working with are of toric nature. Hence, let again N be afree abelian group of rank n which we identify with Z n by chosing a basis. Let Σ A ⊂ N R = N ⊗ R be a fan defining a smooth projective toric weak Fano variety X Σ A . We write Σ A (1) for the set of rays(i.e., one dimensional cones) of Σ A , we will often denote such a ray by v i . As before, a , . . . , a m are theprimitive integral generators of the rays v , . . . , v m in Σ A (1). Consider the exact sequence0 −→ L m −→ Z Σ A (1) ∼ = Z m A −→ N −→ . (6)Applying the functor Hom Z ( − , C ∗ ) yields1 −→ S = Spec C [ N ] ∼ = ( C ∗ ) n −→ ( C ∗ ) Σ A (1) ∼ = ( C ∗ ) m q −→ S := Spec C [ L ] ∼ = L ∨ ⊗ C ∗ −→ . (7)The middle torus ( C ∗ ) Σ A (1) is naturally dual to S = Spec C [ λ ± , . . . , λ ± m ], however, we will from now onidentify both (as well as the corresponding affine spaces W and W ′ ), so that we denote ( C ∗ ) Σ A (1) alsoby S . Notice that the composition of the first map of the exact sequence (7) with the open embedding( C ∗ ) m ֒ → C m is nothing but the map k from proposition 2.1, which was shown to be closed. Recallthat for smooth toric varieties, L ∨ equals the Picard group Pic( X Σ A ). Inside L ∨ R := L ∨ ⊗ R we have theK¨ahler cone K Σ A , which consists of all classes [ a ] ∈ L ∨ R such that a , seen as a piecewise linear functionon N R (linear on each cone of Σ A ) is convex. The interior K A of the K¨ahler cone are the strictly convexpiecewise linear functions on N R . Write D i for the torus invariant divisors of X Σ A associated to the raygenerated by a i , then the anti-canonical divisor of X Σ A is ρ = P mi =1 [ D i ] ∈ L ∨ . Recall that X Σ A is Fanoresp. weak Fano iff ρ ∈ K A resp. ρ ∈ K Σ A . We will choose a basis of L ∨ consisting of classes p , . . . , p r (with r = m − n ) which lie in K Σ A and such that ρ lies in the cone generated by p , . . . , p r . This identifies S with ( C ∗ ) r , and we write q , . . . , q r for the coordinates defined by this identification.The next definition describes one of the main objects of study of this paper. Definition 3.1.
Consider the linear function W = w + . . . + w m : S → C t . The Landau-Ginzburgmodel of the toric weak Fano variety X Σ A is the restriction of the function W to the fibres of the torusfibration q : S ∼ = ( C ∗ ) m → S ∼ = ( C ∗ ) r . We will also sometimes call the morphism ( W, q ) : S ∼ = ( C ∗ ) m −→ C t × S ∼ = C t × ( C ∗ ) r a Landau-Ginzburg model. Notice that the choice of the basis p , . . . , p r (and hence the choice of coordi-nates on S ) are part of the data of the Landau-Ginzburg model, which would otherwise only depend onthe set of rays Σ(1) , but not on the fan Σ itself. The choice of a basis p , . . . , p r of L ∨ also determines an open embedding S ֒ → C r . An important issuein this section will be to extend the various data defined by the Landau-Ginzburg model of X Σ A over theboundaray divisor C r \ S . As a side remark, notice that the K¨ahler cone of a toric Fano variety doesnot need to be simplicial, the simplest example being the toric del Pezzo surface obtained by blowing upthree points in P in generic position. Hence the above chosen basis of L ∨ does not necessarily generatethe K¨ahler cone.Using the dual basis ( p ∨ a ) a =1 ,...,r of L , the above map m is given by a matrix ( m ia ) with columns m a and hence the torus fibration q : ( C ∗ ) m ։ ( C ∗ ) r is given by q ( w , . . . , w m ) = ( q a = w m a :=21 mi =1 w m ia i ) a =1 ,...,r . We will also consider the product map (id z , q ) : P z × S ։ P × S as well as itsrestriction to C ∗ z × S . Choose moreover a section g : L ∨ → Z m of the projection Z m ։ L ∨ , which isgiven in the chosen basis p , . . . , p r of L ∨ by a matrix ( g ia ) with rows g i , so that P mi =1 g ia m ib = δ ab . Themap g induces a section of the fibration q , still denoted by g , which is given as g : S −→ S ( q , . . . , q r ) ( w i := q g i := Q ra =1 q g ia a ) i =1 ,...,m Obviously, g also gives a splitting of the fibration q , see diagram (8) below. Let us notice that the section g can be chosen such that the entries of the matrix g ia are non-negative integers. For this, recall (see,e.g., [CK99, section 3.4.2]) the description of the K¨ahler cone as the intersection of cones in L ∨ ⊗ R eachof which is generated by the images under Z m ։ L ∨ of some of the standard generators of Z m (theso-called anti-cones associated to the cones σ ∈ Σ A ). Hence, the chosen basis ( p a ) a =1 ,...,r of L ∨ whichconsists of elements of K Σ A can be expressed in the generators of any of these cones, and the coefficientsare exactly the entries of the matrix ( g ia ), hence, non-negative. It follows that the section g : S → S extends to a map g : C r → W = Spec C [ w , . . . , w m ], although the projection map q : S ։ S cannotbe extended over the boundary S mi =1 { w i = 0 } ⊂ W . In what follows, we will always assume that g isconstructed in such a way.Write S := g − ( S ) = { ( q , . . . , q r ) ∈ S | f W := P mi =1 q g i y a i is Newton non-degenerate } . Finally, wedefine e g = (id z , g ) : P z × S → P z × S , which is a section of the above projection map (id z , q ). Proposition 3.2.
The embedding e g is non-characteristic for c M loc e A on P z × S . Moreover, the inverseimage e g + c M loc e A is given as the quotient of D C τ × S [ τ − ] / e I , where e I is the left ideal generated by e (cid:3) l := Y a : p a ( l ) > q p a ( l ) a Y i : l i < − l i − Y ν =0 ( r X a =1 m ia zq a ∂ q a − νz ) − Y a : p a ( l ) < q − p a ( l ) a Y i : l i > l i − Y ν =0 ( r X a =1 m ia zq a ∂ q a − νz ) for any l ∈ L and by the single operator z∂ z + r X a =1 ρ ( p ∨ a ) q a ∂ q a . Notice that p a is a linear form on L and that we have P mi =1 g ia l i = P i,b g ia ( m ib p b ( l )) = p a ( l ) ∈ Z .Proof. The non-characteristic condition is evident as the singular locus of c M loc e A , seen as a D P z × S -moduleis contained in ( { , ∞} × S ) ∪ ( P z × ( S \ S )). In order to calculate the inverse image, consider thefollowing diagram S − | | yyyyyyyyyyyyyyyyyyyyyyyyyyy q " " EEEEEEEEEEEEEEEEEEEEEEEEEEE S × S π / / S ( y , . . . , y n , q , . . . , q r ) (cid:31) / / ( q , . . . , q r ) (8)where the coordinate change Φ is given asΦ( y, q ) := (cid:0) w i = q g i · y a i (cid:1) i =1 ,...,m As the diagram commutes, the q -component of Φ − is q a = w m a . Putting e Φ : S × C τ × S → C τ × S ,( y, τ, q ) ( τ, Φ( y, q )) and similarly e π : S × C τ × S → C τ × S , ( y, τ, q ) ( τ, q ), we consider the module22 Φ + c M loc e A which is (using the presentation c M loc e A = D b T [ τ − ] / b I ′′ ) equal to the quotient of D S × C τ × S [ τ − ]by the left ideal generated by Q a : p a ( l ) < q p a ( l ) a e (cid:3) l = r Q a =1 q p a ( l ) a Q i : l i < − l i − Q ν =0 ( P ra =1 m ia zq a ∂ q a − νz ) − Q i : l i > l i − Q ν =0 ( P ra =1 m ia zq a ∂ q a − νz ) e Z k = y k ∂ y k e E = z∂ z + P ra =1 ( P mi =1 m ia ) q a ∂ q a = z∂ z + P ra =1 ρ ( p ∨ a ) q a ∂ q a In other words, we have e Φ + c M loc e A = C [ z ± , y ± , . . . , y ± n , q ± , . . . , q ± r ] h ∂ z , ∂ q , . . . , ∂ q r i ( e (cid:3) l ) l ∈ L + e E Obviously, the map e g is given in the new coordinates by e g ( τ, q ) := ( τ, , q ) ∈ C τ × S × S , hence weobtain e g + c M loc e A = C [ z ± , q ± , . . . , q ± r ] h ∂ z , ∂ q , . . . , ∂ q r i ( e (cid:3) l ) l ∈ L + e E As a consequence of this lemma, and using the comparison result in theorem 2.4, we can interpret thisreduced GKZ-system as a Gauß-Manin-system.
Corollary 3.3.
Consider the (restriction of the) Landau-Ginzburg model ( W, q ) : S → C t × S . Thenthere is an isomorphism of D C τ × S -modules D C τ × S [ τ − ] / e I ∼ = FL τt ( H (( W, q ) + O S ))[ τ − ] Proof.
First notice that due to diagram (8) we have an isomorphism H (( W, q ) + O S ) ∼ = H (( f W , π ) + O S × S ) , recall that f W ( y, q ) = m X i =1 y a i q g i = m X i =1 n Y k =1 y a ki k ! · r Y a =1 q g ia a ! . Consider the following cartesian diagram S × S ϕ ′ :=( f W ,π ) (cid:15) (cid:15) / / S × S ϕ (cid:15) (cid:15) C t × S
02 (id C t ,g ) / / U = C t × S (9)Now we use the base change properties of the direct image (see, e.g., [HTT08, section 1.7]), from whichwe obtain that (id C t , g ) + H ( ϕ + O S × S ) ∼ = H ( ϕ ′ + O S × S ) . This gives FL τt (id C t , g ) + H ( ϕ + O S × S )[ τ − ] ∼ = FL τt H ( ϕ ′ + O S × S )[ τ − ] , and as we have FL τt (cid:16) (id C t , g ) + H ( ϕ + O S × S ) (cid:17) ∼ = e g + FL τt H ( ϕ + O S × S ) ,
23e finally obtain e g + FL τt (cid:16) H ( ϕ + O S × S ) (cid:17) [ τ − ] = FL τt (cid:16) H ( ϕ ′ + O S × S ) (cid:17) [ τ − ] , from which the desired statement follows using proposition 3.2 and theorem 2.4.As a consequence of the last result, we have the following easy corollary concerning the the family ofBrieskorn lattices resp. the holonomic duality for the Gauß-Manin-system of the Landau-Ginzburg model( W, q ). Corollary 3.4.
1. The D C τ × S -module QM loc e A := O C τ × S ⊗ O C τ × S ( D C τ × S [ τ − ] / e I ) is equipped withan increasing filtration G • by O C z × S -modules. Moreover, for any k ∈ N , G k QM loc e A is O C z × S -locally free of rank n ! · vol(Conv( a , . . . , a m ) .2. Write QM loc e A for the O C z × S -module G QM loc e A , then this is the restriction to C z × S of the sheafassociated to the module C [ z, q ± , . . . , q ± r ] h z∂ q , . . . , z∂ q r , z ∂ z i ( e (cid:3) l ) l ∈ L + ( z ∂ z + P ra =1 ρ ( p ∨ a ) zq a ∂ q a ) .
3. There is a non-degenerate flat ( − n -symmetric pairing P : QM loc e A ⊗ ι ∗ QM loc e A → O C ∗ τ × S . P ( QM loc e A , QM loc e A ) ⊂ z n O C z × S , and P is non-degenerate on QM loc e A .Proof. As we have seen, the closed embedding e g | C z × S : C z × S ֒ → C z × S is non-characteristic for O C z × S ⊗ O C z × S c M loc e A . It is actually nothing else but the inverse image in the category of meromorphicbundles with connections. Hence the increasing filtration O C z × S ⊗ ( z −• c M loc e A ) on c M loc e A by locally free O C z × S -modules pulls back to an increasing filtration G • on QM loc e A by locally free O C z × S -modules, thezeroth term of which is given by the formula in 2. All other statements follow from proposition 2.18. In this subsection, we first construct a logarithmic extension of the hypergeometric system QM loc e A onthe K¨ahler moduli space. Recall from the last subsection that S is a Zariski open subspace of S :=Spec C [ L ] consisting of points q such that the Laurent polynomial f W ( − , q ) : S → C t is non-degenerate.Recall also that we have chosen a basis p , . . . , p r of L ∨ of nef classes, i.e., classes lying in the K¨ahlercone K ⊂ L ∨ R . The corresponding coordinates on S are q , . . . , q r , and define an embedding of S into C r .Write ∆ S := S \ S and denote by ∆ S the closure of ∆ S in C r . Finally, put S := C r \ ∆ S . We willwrite Z a for the divisor { q a = 0 } in both C r and S , and we define Z = S ra =1 Z a which is a simplenormal crossing divisor in C r resp. S . Lemma 3.5.
1. The origin of C r is contained in S .2. If X Σ A is Fano (i.e., ρ ∈ K A ), then ∆ S = ∅ , and, hence, S = C r .3. If ∆ S = ∅ , then there is a ball B := B r (0) ⊂ ( S ) an with radius equal to r := inf {| q | : q / ∈ ∆ S } .We set B := ( C r ) an if ∆ S = ∅ .Proof.
1. This has been shown in [Iri09a, appendix 6.1].2. This follows from lemma 2.8.3. This is clear from 1. 24e proceed with a construction which is a variant of the arguments used in the proof of theorem 2.14,however, now we also take into account the logarithmic structure along Z . We first define the appropriatenon-commutative algebras, and then show that they are actually locally free O -modules, possibly aftera further restriction to some Zariski open subset of S . Definition 3.6.
1. Consider the ring e R := C [ q , . . . , q r , z ] h zq ∂ q , . . . , zq r ∂ q r , z ∂ z i i.e., the quotient of the free C [ q , . . . , q r , z ] -algebra generated by zq ∂ q , . . . , zq r ∂ q r , z ∂ z by the leftideal generated by the relations [ zq a ∂ q a , z ] = 0 , [ zq a ∂ q a , q b ] = δ ab zq a , [ z ∂ z , q a ] = 0 , [ z ∂ z , z ] = z , [ zq a ∂ q a , zq b ∂ q b ] = 0 , [ z ∂ z , zq a ∂ q a ] = z · zq a ∂ q a Write e R for the associated sheaf of quasi-coherent O C z × C r -algebras, which restricts to D C ∗ τ × S on { ( q a = 0) a =1 ,...,r , z = 0 } .We also consider the subring e R ′ := C [ q , . . . , q r , z ] h zq ∂ q , . . . , zq r ∂ q r i of e R , and the associatedsheaf e R ′ . The inclusion e R ′ ֒ → e R induces a functor from the category of e R -modules to the categoryof e R ′ -modules, which we denote by For z ∂ z (“forgetting the z ∂ z -structure”).2. Let e I be the ideal in e R generated by the operators e (cid:3) l for any l ∈ L and by z ∂ z + P ra =1 ρ ( p ∨ a ) zq a ∂ q a and consider the quotient e R / e I . We have For z ∂ z ( e R / e I ) = ( e R ′ / ( e (cid:3) l ) l ∈ L ) and e R / e I equals QM loc e A on C z × S (and hence equals QM loc e A on C ∗ τ × S ). The basic finiteness result about the module QM e A is the following. Theorem 3.7.
There is a Zariski open subset U of S containing the origin in C r such that the module QM e A := O C z × U ⊗ O C z × C r e R / e I is O C z × U -coherent. If X Σ A is Fano, i.e., if ρ ∈ K (Σ A ) , then one canchoose U to be C r (which equals S in this case).There is a connection operator ∇ : QM e A −→ QM e A ⊗ z − Ω C z × U (log (( { } × U ) ∪ ( C z × Z ))) extending the D C ∗ τ × ( U ∩ S ) -structure on ( QM loc e A ) | C ∗ τ × ( U ∩ S ) .Proof. The arguments used here have some similarities with the proof of theorem 2.14. We first supposethat X Σ A is Fano, then we have to show that QM e A is O C z × C r -coherent. We will actually show thecoherence of For z ∂ z ( QM e A ), which is sufficient, as QM e A and For z ∂ z ( QM e A ) are equal as O C z × C r -modules. Consider the natural filtration on e R ′ given by order of operators, i.e., the filtration F • e R ′ givenon global sections by F k C [ q , . . . , q r , z ] h zq ∂ q , . . . , zq r ∂ q r i := P | P = X | s |≤ k g s ( z, q )( zq ∂ q ) s · . . . · ( zq r ∂ q r ) s r . This filtration induces a filtration F • on For z ∂ z ( QM e A ) which is good in the sense that F k e R ′ · F l For z ∂ z ( QM e A ) = F k + l For z ∂ z ( QM e A ) . We have a natural identification gr F • ( e R ′ ) = π ∗ O C z × T ∗ C r (log D ) where T ∗ C r (log D ) is the total space of the vector bundle associated to the locally free sheaf Ω C r (log D )and π : C z × T ∗ C r (log D ) ։ C z × C r is the projection. It will be sufficient to show that the subvariety C z × S of C z × T ∗ C r (log D ) cut out by the symbols of all operators e (cid:3) l for l ∈ L equals C z × C r , then bythe usual argument the filtration F • will become eventually stationary, and we conclude by the fact that25ll F k For z ∂ z ( QM e A ) are O C z × C r -coherent. For the proof, we will use some elementary facts from toricgeometry, namely, the notion of primitive collections and primitive relations (see [Bat91] and [CvR09]).Suppose that l ∈ L corresponds to a primitive relation in the sense of loc.cit., then it follows that p a ( l ) ≥ a = 1 , . . . , r , as a primitive relation lies in the Mori cone of X Σ A and as p a is a nef class,i.e., by definition it takes non-negative values on effective cycles. On the other hand, as X Σ A is Fano, wehave that l = ρ ( l ) >
0, remember that ρ = P mi =1 D i is the anti-canonical divisor which by definition liesin the interior of the K¨ahler cone. Hence, P i : l i > l i > P i : l i < − l i , moreover, for a primitive relation, wehave l i = 1 for all i such that l i > σ ( e (cid:3) l ) = Y i : l i =1 r X a =1 m ia σ ( zq a ∂ q a ) ! , Now identify T ∗ C r (log Z ) with the trivial bundle C r × X where X is the vector space dual to the spacegenerated by ( σ ( zq a ∂ q a )) a =1 ,...,r . Then the last equation shows that the variety S alluded to above is ofthe form C r × Y red , for some possibly non-reduced subvariety Y ⊂ X . We need to show that Y red = { } .First it is clear that Y is homogeneous so that it suffices to show that its Krull dimension is zero. Recallfrom [Ful93, section 5.2, page 106] that the classical cohomology ring of X Σ A with complex coefficientsis presented as H ∗ ( X Σ A , C ) = C [( v i ) i =1 ,...,m ]( P mi =1 a ki v i ) k =1 ,...,n + (cid:0) v i · . . . · v i p (cid:1) (10)where the tuple v i , . . . , v i p runs over all primitive collections in Σ A (1). However, it follows from theexactness of the sequence (6) that the spectrum of this ring equals the above subspace Y , in particularthe latter must be fat point, supported at the origin in the space V . This shows that the variety S isthe zero section of T ∗ C r (log D ), as required.Now suppose only that X Σ A is weak Fano, i.e., ρ ∈ K Σ A . Then it may happen that for a primitive relation l , we have l = ρ ( l ) = 0, which implies that σ ( e (cid:3) l ) = r Y a =1 q p a ( l ) a Y i : l i < r X a =1 m ia σ ( zq a ∂ q a ) ! − l i − Y i : l i =1 r X a =1 m ia σ ( zq a ∂ q a ) ! , as p a ( l ) ≥ l . This shows that the fibre of S over the point q = 0 , . . . , q r = 0is again the reduced space of the spectrum of the cohomology algebra of X Σ A , i.e, it is only the originin the fibre of T ∗ C r (log D ) over q = 0. In particular, the projection map S ։ C r is quasi-finite in aZariski open neighborhood U of 0 ∈ C r . On the other hand, by its very definition, S is homogeneouswith respect to the fibre variables. Hence on U , S is the zero section of the projection T ∗ U (log D ) ։ U ,as required.The statement concerning the connection follows directly from the definition of QM e A , namely, QM e A isinvariant under the operators ∇ zq a ∂ qa for a = 1 , . . . , r and ∇ z ∂ z .The next step is to discuss the restriction ( QM e A ) | C z ×{ q =0 } , this is a coherent O C z -module that wedenote by E . It turns out that it is actually locally free, from which we deduce the freeness of QM e A and certain extension properties of the pairing P from corollary 3.4, 3. Lemma 3.8.
1. There is a canonical isomorphism α : O C z ⊗ H ∗ ( X Σ A , C ) ∼ = −→ E, hence, E is O C z -free of rank µ := n ! · vol(Conv( a , . . . , a m )) . It comes equipped with a connection ∇ res ,q : E −→ E ⊗ z − Ω C z induced by the residue connection of ∇ on ( QM e A ) | C ∗ τ ×{ q =0 } along S ra =1 { q a = 0 } .Let i : C z ֒ → C z × U be the inclusion and write π : i − ( QM e A ) ։ E for the canonical projection.Let F = π ( C [ zq ∂ q , . . . , zq r ∂ q r ]) ⊂ E , where we denote abusively by C [ zq ∂ q , . . . , zq r ∂ q r ] the sheafassociated to the the image of this ring in Γ( C z × U, QM e A ) . Then α (1 ⊗ H ∗ ( X Σ A , C )) = F .The restriction E | z =0 = ( QM e A ) | (0 , is canonically isomorphic, as a finite-dimensional commuta-tive algebra, to the cohomology ring ( H ∗ ( X Σ A , C ) , ∪ ) . . QM e A is O C z × U -free of rank µ .3. Write QM e A for the restriction ( QM e A ) | C ∗ τ × U . Then for any a ∈ { , . . . , r } , the residue endomor-phisms zq a ∂ q a ∈ E nd O C ∗ τ (cid:0) ( QM e A ) | C ∗ τ ×{ } (cid:1) = E | C ∗ τ are nilpotent.4. There is a non-degenerate flat ( − n -symmetric pairing P : QM e A ⊗ ι ∗ QM e A → z n O C z × U , i.e., P is flat on C ∗ τ × ( U ∩ S ) , and the induced pairings P : ( QM e A /z · QM e A ) ⊗ ( QM e A /z · QM e A ) → z n O U and P : ( QM e A /q a · QM e A ) ⊗ ι ∗ ( QM e A /q a · QM e A ) → z n O C z × Z a are non-degenerate.5. The induced pairing P : E ⊗ ι ∗ E → z n O C z restricts to a pairing P : F × F → z n C . The pairing z − n P on F coincides, under the identification made in 1., with the Poincar´e pairing on H ∗ ( X Σ A , C ) up to a non-zero constant.Proof.
1. In order to construct the map α notice first that we have (cid:16) e (cid:3) l (cid:17) |{ q =0 } = Q i : l i > Q l i − ν =0 ( P ra =1 m ia zq a ∂ q a − νz ) if p a ( l ) ≥ a = 1 , . . . , r Q i : l i < Q − l i − ν =0 ( P ra =1 m ia zq a ∂ q a − νz ) if p a ( l ) ≤ a = 1 , . . . , r O C z -modules E = (cid:0) For z ∂ z ( QM e A ) (cid:1) | C z ×{ q =0 } ∼ = C [ z, zq ∂ q , . . . , zq r ∂ q r ] (cid:18)(cid:26)(cid:16) e (cid:3) l (cid:17) |{ q =0 } | l ∈ Eff X Σ A ∩ L (cid:27)(cid:19) , where Eff X Σ A ⊂ L R is the Mori cone of X Σ A . Notice that if l ∈ L eff := Eff X Σ A ∩ L , then any (cid:16) e (cid:3) l (cid:17) |{ q =0 } contains Q i : l i ≥ ( P ra =1 m ia zq a ∂ q a ) as a factor. The Mori cone can be characterized asfollows (see, e.g., the discussion in [CK99, 3.4.2]):Eff X Σ A = X σ ∈ Σ A ( n ) C σ , (11)where C σ is the cone generated by elements l ∈ L with l i ≥ R ≥ a i is not a ray of σ .It follows that whenever l ∈ L eff \{ } , then the set { a i | l i ≥ } cannot generate a cone in Σ A , forotherwise − l would also lie in Eff X Σ A , and thus l = 0. As a consequence, for any l ∈ L eff \{ } , theelement ( e (cid:3) l ) |{ q =0 } contains a factor Q i ∈ I ( P ra =1 m ia zq a ∂ q a ) where P i ∈ I R ≥ a i / ∈ Σ A .Now consider the case where l is primitive, in particular, l ∈ L eff . Then ( e (cid:3) l ) |{ q =0 } is equal to Q i ∈ I ( P ra =1 m ia zq a ∂ q a ), where { a i | i ∈ I } is a primitive collection. As any set of rays { a j | j ∈ J } which does not generate a cone contains a primitive collection, we conclude from the abovediscussion that E is equal to C [ z, zq ∂ q , . . . , zq r ∂ q r ] (cid:18)(cid:26)(cid:16) e (cid:3) l (cid:17) |{ q =0 } | l primitive (cid:27)(cid:19) ∼ = C [ z ] ⊗ C [ zq ∂ q , . . . , zq r ∂ q r ] (cid:0)Q i ∈ I ( P ra =1 m ia zq a ∂ q a ) (cid:1) I , where the index set I in the denominator of the right hand side runs over all subsets of { , . . . , m } such that { a i | i ∈ I } is a primitive collection.Now to define α we use again the presentation of H ∗ ( X Σ A , C ) from formula (10). We concludefrom the above discussion that putting α ( v i ) := P ra =1 m ia zq a ∂ q a yields a well-defined map O C z ⊗ H ∗ ( X Σ A , C ) → E , which is obviously surjective. We have seen in theorem 3.7 that QM e A is coherent,and its generic rank is that of QM loc e A , i.e., µ . On the other hand, O C z ⊗ H ∗ ( X Σ A , C ) is O C z -freeof rank µ , hence by semi-continuity and comparison of rank, we obtain that α is an isomorphism.Then we also have that α ( H ∗ ( X Σ A , C )) = F . The pole order property of the connection operator ∇ res,q on E follows from the pole order properties of ∇ on QM e A as stated in theorem 3.7.27. This is now a standard argument: For any I ⊂ { , . . . , r } , put e Z I := T a ∈ I Z a and consider therestriction ( QM e A ) | C ∗ τ × Z I , where Z I := e Z I \ (cid:16)S J ) I e Z J (cid:17) . This restriction is equipped with thestructure of a D C ∗ τ × Z I -module, so that it must be locally free. Hence it suffices to show freeness of QM e A in a neighborhood of 0 ∈ C z × U . But this is clear after from point 1.: The dimension ofthe fibre at 0 is n ! · vol(Conv( a , . . . , a m )), which is also the rank on C z × S . Hence it can neitherbe smaller nor bigger at any point in a neighborhood of the origin in C z × U .3. Using the isomorphism α from 1., the residue endomorphism [ zq a ∂ q a ] equals Id O C ∗ τ ⊗ ( D a ∪ − ) ∈E nd O C ∗ τ ( E | C ∗ τ ) from which its nilpotency follows easily.4. Using the O C z × U -freeness of QM e A and point 5. above, this can be shown by an argument similarto [HS10, lemma 3.4]. Namely, consider the canonical V -filtration (denoted by V • ) on QM loc e A alongthe normal crossing divisor Z . Then the last point shows that we have V QM loc e A = QM e A (recallthat QM e A is the restriction of QM e A to C ∗ τ × U ), hence, gr V ( QM e A ) = ( QM e A ) | C ∗ τ ×{ } . This impliesimmediately (see [HS10, proof of lemma 3.4 and formula 3.4]) that P extends in a non-degenerateway to QM e A . Hence we obtain a non-degenerate pairing on the restriction ( QM e A ) | ( C z × U ) \ ( { }× Z ) .However, as { } × Z has codimension two in C z × U , P necessarily extends to a non-degeneratepairing on QM e A , as required.5. The non-degenerate pairing P : E ⊗ ι ∗ E → z n O C z restricts to a pairing P : F × F → z n O C z .Let us show that it actually takes values in z n C on F . Set r i = dim H i ( X Σ A , C ) and choose ahomogeneous basis w , = 1 , w , , . . . , w r , , . . . , w ,n − , . . . , w r n − ,n − , w ,n where w i,k ∈ H k ( X Σ A , C ) and which is adapted to the Lefschetz decomposition. Recall that theHard Lefschetz theorem says the following: H m ( X Σ A , C ) = M i L i H m − i ( X Σ A , C ) p , where H n − k ( X Σ A , C ) p = ker ( L k +1 : H n − k ( X Σ A , C ) → H n + k +2 ( X Σ A , C )) and the map L is equal tocup-product with c ( X Σ A ). It follows from equation 12 that z ∇ res ,q∂ z ( w i,k ) = k · w i,k + 1 z r k +1 X m =1 Θ m,i,k w m,k +1 for k < n ,z ∇ res ,q∂ z ( w ,n ) = n · w ,n , where Θ m,i,k := ( ˇ A ) u,v with u = m + P kl =1 r l and v = i + P k − l =1 r l and ˇ A is the matrix with respectto the basis w , , . . . , w ,n of the endomorphism − c ( X Σ A ) ∪ . The first claim is that P ( w i,k , w j,l ) = c ikjl z k + l with c ikjl ∈ C . Using the fact that P takes values in z n O C z on E , this implies in particular P ( w i,k , w j,l ) = 0 for k + l < n . We have z∂ z P ( w ,n , w ,n ) = 2 nP ( w ,n , w ,n ) ∈ z n O C z , thus it follows that P ( w ,n , w ,n ) = c · z n for some c ∈ C . Now assume that we have P ( w i,s , w j,t ) = c isjt z s + t for c isjt ∈ C and s + t ≥ d + 1. We have for k + l = dz∂ z P ( w i,k w j,l ) = P ( k · w i,k + 1 z ( r k +1 X m =1 Θ m,i,k w m,k +1 ) , w j,l )+ P ( w i,k , l · w j,l + 1 z ( r l +1 X m =1 Θ m,j,l w m,l +1 ))= ( k + l ) P ( w i,k , w j,l ) + c · z d for some c ∈ C , where the last equality follows from the inductive assumption for d + 1 and d + 2. Thus we have( z∂ z − d ) P ( w i,k w j,l ) = 0 , P ( w i,k w j,l ) − c · z d ∈ z d C . This shows the first claim, i.e. P ( w i,k , w j,l ) = c ijkl z k + l for k + l ≥ n and P ( w i,k , w j,l ) = 0 for k + l < n .As a second step we want to show P ( w i,k , w j,l ) = 0 for k + l > n . We prove this by descending induc-tion, beginning with the case k + l = 2 n . We first introduce some notation. We say w i,k is primitiveif it is not of the form − c ( X Σ A ) ∪ v for some v ∈ H k − ( X Σ A , C ). We say q w i,k ∈ H k − q ( X Σ A , C )is a q -th primitive of w i,k if ( − c ( X Σ A )) q ∪ q w i,k = w i,k . The Hard Lefschetz Theorem tells us thatfor 2 k ≥ n the element w i,k is never primitive.As the base case we have to prove P ( w ,n , w ,n ) = 0. Let w ,n be a first primitive of w ,n . Wehave 0 = ( z∂ z − (2 n − P ( w ,n , w ,n ) = P (( n − · w ,n , w ,n ) + P ( 1 z w ,n , w ,n )+ P ( w ,n , n · w ,n ) − (2 n − P ( w ,n , w ,n )= 1 z P ( w ,n , w ,n ) . Now assume P ( w i,k , w j,l ) = 0 for k + l ≥ s + 1. We will prove P ( w i,k , w j,l ) = 0 for k + l = s bydescending induction on k . Notice that by ( − w -symmetry we only have to prove this for k ≥ l .The base case is to show that P ( w ,n , w j,s − n ) = 0 for j ∈ { , . . . , r s − n } (recall that n +1 ≤ s < n ).We have to distinguish two cases:I. case: w j,s − n is not primitive. Thus there exists w j,s − n with − c ( X Σ A ) ∪ w j,s − n = w j,s − n . Wecalculate0 = ( z∂ z − ( s − P ( w ,n , w j,s − n )= P ( n · w ,n , w j,s − n ) + P ( w ,n , ( s − n − w j,s − n ) + P ( w ,n , z w j,s − n ) − ( s − P ( w ,n , w j,s − n )= − z P ( w ,n , w j,s − n ) . II. case: w j,s − n is primitive. This means that w j,s − n ∈ H s − n ( X Σ A , C ) p = ker (cid:0) c ( X Σ A ) n − s +1 : H s − n ( X Σ A , C ) → H n − s +2 ( X Σ A , C ) (cid:1) . We have 0 =( z∂ z − ( s − P ( w ,n , w j,s − n )= P (( n − · w ,n , w j,s − n ) + P ( 1 z w ,n , w j,s − n ) + P ( w ,n , ( s − n ) · w j,s − n )+ P ( w ,n , z ( − c ( X Σ A )) ∪ w j,s − n ) − ( s − P ( w ,n , w j,s − n )= P ( 1 z w ,n , w j,s − n ) + P ( w ,n , z ( − c ( X Σ A )) ∪ w j,s − n ) , which gives P ( w ,n , w j,s − n ) = P ( w ,n , c ( X Σ A ) ∪ w j,s − n ). Notice that 3 n − s < n . Because w ,n has an n − th -primitive (this follows from the Hard Lefschetz theorem: c ( X Σ A ) n : H ( X Σ A , C ) ≃ −→ H n ( X Σ A , C )), we can repeat this step 3 n − s + 1 times to get P ( w n , w j,s − n ) = P ( (3 n − s +1) w ,n , ( − c ( X Σ A )) n − s +1 ∪ w j,s − n ) = 0 . This shows the second case.We now assume that P ( w i,k , w j,l ) = 0 for k ≥ t + 1 and k + l = s as well as P ( w i,k , w j,l ) = 0 for k + l ≥ s + 1. We have to prove P ( w i,t , w j,s − t ) = 0 for i ∈ { , . . . r t } and j ∈ { , . . . , r t − s } and t ≥ s − t (the last restriction is allowed because of the ( − w -symmetry of P ).29. case: w j,s − t is not primitive: Thus there exists w j,s − t with − c ( X Σ A ) ∪ w j,s − t = w j,s − t . Wecalculate0 =( z∂ z − ( s − P ( w i,t , w j,s − t )= P ( t · w i,t , w j,s − t ) + P ( 1 z ( − c ( X Σ A ) ∪ w i,t ) , w j,s − t ) + P ( w i,t , ( s − t − · w j,s − t )+ P ( w i,t , z w j,s − t ) − ( s − P ( w i,t , w j,s − t )= P ( 1 z ( − c ( X Σ A ) ∪ w i,t ) , w j,s − t ) + P ( w i,t , z w j,s − t )= P ( w i,t , z w j,s − t ) . Notice that P ( c ( X Σ A ) ∪ w i,t , w j,s − t ) vanishes because c ( X Σ A ) ∪ w i,t is a linear combination of { w i,t +1 } and P ( w i,t +1 , w j,s − t ) vanishes for every i ∈ { , . . . , r t +1 } by the induction hypothesis.II. case: w j,s − t is primitive. This means that w j,s − t ∈ H s − t ( X Σ A , C ) p = ker (cid:0) c ( X Σ A ) n +2 t − s +1 : H s − t ( X Σ A , C ) → H n − s +2 t +2 ( X Σ A , C ) (cid:1) . Notice that w i,t has a (2 t − n )-th primitive and we have 2 t − n ≥ n + 2 t − s + 1, because of s ≥ n + 1. We calculate0 =( z∂ z − ( s − P ( w i,t , w j,s − t )= P (( t − · w i,t , w j,s − t ) + P ( 1 z w i,t , w j,s − t ) + P ( w i,t , ( s − t ) · w j,s − t )+ P ( w i,t , z ( − c ( X Σ A )) ∪ w j,s − t ) − ( s − P ( w i,t , w j,s − t )= P ( 1 z w i,t , w j,s − t ) + P ( w i,t , z ( − c ( X Σ A )) ∪ w j,s − t )which gives P ( w i,t , w j,s − t ) = P ( w i,t , ( − c ( X Σ A )) ∪ w j,s − t ). As w i,t has a (2 t − n )-th primitive wecan repeat this step n + 2 t − s + 1 times to get P ( w i,t , w j,s − t ) = P ( n +2 t − s +1 w i,t , ( − c ( X Σ A )) n +2 t − s +1 ∪ w j,s − t ) = 0 . This finishes the induction over t . Thus we have shown that P ( w i,k , w j,l ) = 0 if k + l = s ≥ n + 1and k ≥ l . The case k ≤ l follows by symmetry and this finishes the induction over s . This meansthat the pairing P : F × F −→ z n O C z takes values in z n C .It remains to show that the pairing z − n P coincides, under the isomorphism α : 1 ⊗ H ∗ ( X Σ A , C ) → F and possibly up to a non-zero constant, with the Poincar´e pairing on the cohomology algebra.First notice that by construction, z − n P , seen as defined on H ∗ ( X Σ A , C ) is multiplication invariant,i.e., P ( a, b ) = P (1 , a ∪ b ) for any two classes a, b ∈ H ∗ ( X Σ A , C ). This can be deduced from theflatness of P on QM loc e A , more precisely, by considering the restriction of P defined on the familyof commutative algebras QM e A /z · QM e A . Notice however that this argument holds a priori onlymodulo z , and in order to obtain the multiplication invariance of z − n P on 1 ⊗ H ∗ ( X Σ A , C ) onefirst needs to know that it takes constant values on that space. It suffices now to show that P (1 , a )equals the value of the Poincar´e pairing on 1 and a . But as we have seen above, P (1 , a ) can only benon-zero if a ∈ H n ( X Σ A , C ), so that the P on H ∗ ( X Σ A , C ) is entirely determined by the non-zerocomplex number P (1 , PD ([ pt ])). Remark:
The value of the pairing P at the point (0 , ∈ C z × U is determined, by the above argument,up to multiplication by a non-zero complex number. In order to simplify the statements of the subsequentresults, we will without further mentioning assume that this number is chosen such that P correspondsunder the above identifications exactly to the Poincar´e pairing on H ∗ ( X Σ A , C ). Such a choice is alwayspossible by changing the morphism φ : M e A = M (1 , e A → D M e A = M (0 , e A from the proof of theorem 2.15by multiplication by a non-zero complex number (and these are the only non-trivial morphisms between30hese two modules, due to [Sai01, theorem 3.3(3)]).We now show how to construct a specific basis of QM e A defining an extension to a family of trivial P parameterized by an analytic neighborhood of the origin in U and such that the connection has alogarithmic pole at z = ∞ . As already mentioned in the introduction, the method goes back to [Gue08],namely, we first construct an extension of E = ( QM e A ) | C z ×{ } to P z × { } and then show that it can beextended to a family of P -bundles restricting to QM e A outside z = ∞ . At any point q near the origin in U this yields a solution to the Birkhoff problem (in other words, a good base in the sense of [Sai89]) ofthe restriction of ( QM loc e A ) | C z ×{ q } , but it also gives an extension of the whole family QM e A taking intoaccount the logarithmic degeneration behavior at D . Proposition 3.9.
Consider the O C z -module E with the connection ∇ res ,q and the subspace F ⊂ E fromlemma 3.8.1. The connection operator ∇ res ,q : E → z − · E sends F into z − F ⊕ z − · F .2. Let b E := O P z ×{ } · F be an extension of E to a trivial P -bundle. Then the connection ∇ res ,q hasa logarithmic pole at z = ∞ with spectrum (i.e., set of residue eigenvalues) equal to the (algebraic)degrees of the cohomology classes of H ∗ ( X Σ A , C ) . This logarithmic extension corresponds to anincreasing filtration F • on the local system E an , ∇ res ,q | C ∗ τ by subsystems which are invariant underthe monodromy of ∇ res ,q . Let j τ : C ∗ τ ֒ → ( P z \{ } ) , and put E ∞ := ψ τ j τ, ! ( E an ) ∇ res ,q | C ∗ τ , where ψ τ is Deligne’s nearby cycle functor. Then F • is defined on E ∞ , and there is an isomorphism H ( P z , b E ) = F → E ∞ .3. Write N a for the nilpotent part of the monodromy of ( QM loc e A ) an, ∇ around C ∗ τ × Z a , then N a actson E ∞ and satisfies N a F • ⊂ F •− .4. The pairing P on E extends to a non-degenerate pairing P : b E ⊗ O P ι ∗ b E → O P ( − n, n ) , where O P ( a, b ) is the subsheaf of O P ( ∗{ , ∞} ) consisting of meromorphic functions with a pole of order a at and a pole of order b at ∞ .Proof.
1. Let w , . . . , w µ be a C -basis of F which consists of monomials in zq a ∂ q a . We will show that( z ∇ res ,qz )( w ) = w · ( A + zA ∞ ) , (12)where A , A ∞ ∈ M ( µ × µ, C ) and that the eigenvalues of A ∞ are exactly the set (counted with multi-plicity) of the (algebraic) degrees of the cohomology classes of X Σ A . First notice that under the iden-tification of H ∗ ( X Σ A , C ) with the quotient C [( v i ) i =1 ,...,m ] / (cid:0) ( P mi =1 a ki v i ) k =1 ,...,n + ( v i · . . . · v i p ) (cid:1) informula (10), a ray v i is mapped to the cohomology class in H ( X Σ A , C ) of the torus invariant divisorit determines.From the definition of QM loc e A we see that( z ∇ res ,qz )( zq b i ∂ q bi ) k i = ( z ∂ z ) · ( zq b i ∂ q bi ) k i = ( zq b i ∂ q bi ) k i · ( z ∂ z ) + k i · z · ( zq b i ∂ q bi ) k i = (cid:20) − r P a =1 ρ ( p ∨ a ) zq a ∂ q a (cid:21) · ( zq b i ∂ q bi ) k i + k i · z · ( zq b i ∂ q bi ) k i (13)Hence A ∞ is diagonal with eigenvalues equal to the algebraic cohomology degrees of H ∗ ( X Σ A , C ).As a by-product of the above calculation, we also see that the endomorphism of E/z · E representedby the matrix A is the multiplication with − c ( X Σ A ), and hence, is nilpotent. With a little morework, this shows that ∇ res ,q has a regular singularity at z = 0 on E . However, as we are not goingto use this fact in the sequel, we will not give the complete proof here. In any case, we see that[ A ∞ , A ] = A . 31. Formula (12) and formula (13) show that the connection ∇ res ,q has a logarithmic pole at z = ∞ on b E with residue eigenvalues equal to the algebraic cohomology degrees of the cohomology classes of H ∗ ( X Σ A , C ). The correspondence between logarithmic extensions of flat bundles over a divisor andfiltrations on the corresponding local system is a general fact, see, e.g., [Sab02, III.1.ab] or [Her02,lemma 7.6 and lemma 8.14]. The isomorphism F → E ∞ is explicitly given by multiplication by z − A ∞ · z − A .3. We have seen in the proof of theorem 3.8, 4., that E | C ∗ τ ∼ = gr V QM e A as flat bundles. N a naturallyacts on the latter one, and is flat with respect to the residue connection ∇ res ,q , hence it acts on E an , ∇ res ,q | C ∗ τ and thus on E ∞ . Under the identification of 2., the filtration F • is induced by F p = X | k |≥− p C (cid:0) ( zq ∂ q ) k · . . . · ( zq r ∂ q r ) k r (cid:1) . Notice that the only non-trivial filtration steps are those for p ∈ [ − n, z − ∇ z − = − z ∇ z on b E at z = ∞ (see formula (13) above). By definition, N a ,seen as defined on F is simply the multiplication by zq a ∂ q a , from which it follows that N a F • ⊂ F •− .4. This follows directly from lemma 3.8, 4. and from the definition of b E .The next result gives an extension of QM e A to a family of trivial P -bundles, possibly after restrictingto a smaller open subset inside U . Proposition 3.10.
There is an analytic open subset U ⊂ U an still containing the origin of C r and aholomorphic bundle [ QM e A → P z × U (notice that here b signifies an extension to z = ∞ , this shouldnot be confused with notation for the partial Fourier-Laplace transformation used before) such that1. ( [ QM e A ) | C z × U ∼ = ( QM an e A ) | C z × U ( [ QM e A ) | P z ×{ } ∼ = b E [ QM e A is a family of trivial P z -bundles, i.e., [ QM e A = p ∗ p ∗ ( [ QM e A ) , where p : P z × U → U is theprojection.4. The connection ∇ has a logarithmic pole along b Z on [ QM e A , where b Z is the normal crossing divisor ( { z = ∞} ∪ S ra =1 { q a = 0 } ) ∩ P z × U .5. The given pairings P : QM e A ⊗ ι ∗ QM e A → z n O C z × U and P : b E ⊗ O P z ι ∗ b E → O P z ( − n, n ) extend toa non-degenerate pairing P : [ QM e A ⊗ O P z × U ι ∗ [ QM e A → O P z × U ( − n, n ) , where the latter sheaf isdefined as in point 4. of proposition 3.9.6. The residue connection ∇ res ,τ : [ QM e A /τ · [ QM e A −→ [ QM e A /τ · [ QM e A ⊗ Ω {∞}× U (log( {∞} × Z )) . has trivial monodromy around {∞} × Z and any element of F ⊂ H ( P z × U , [ QM e A ) is horizontalfor ∇ .Proof. Recall that QM e A is the restriction of QM e A to C ∗ τ × U . The strategy of the proof is to show thatthere is a holomorphic bundle d QM e A on ( P z \{ } ) × B (where B is the analytic neighborhood of 0 ∈ C r which was defined in lemma 3.5) which is an extension of ( QM an e A ) | C ∗ τ × B over z = ∞ with a logarithmicpole along b Z an ∩ ( P z × B ) and such that the bundle obtained by gluing this extension to QM e A is afamily of trivial P z -bundles, possibly after restricting to some open subset P z × U of P z × B .A logarithmic extension of ( QM loc e A ) an | C ∗ τ × ( B ∩ S an ) over b Z an ∩ ( P z × B ) is given by a Z r +1 -filtration onthe local system L = ( QM loc e A ) an, ∇ which is split iff the extension is locally free (see [Her02, lemma8.14]). In our situation, the bundle QM e A already yields a logarithmic extension over C ∗ τ × Z and we are32eeking a bundle d QM e A → ( P z \{ } ) × B restricting to QM an e A on C ∗ τ × B . Moreover, the Z r -filtration P • corresponding to QM an e A is trivial, as this bundle is a Deligne extension due to lemma 3.8, 4. It followsthat if we choose an extra single filtration F • on L (this will be the one which define the extension d QM e A over { z = ∞} ), then the corresponding Z r +1 -filtration e P • := ( F • , P • ) will automatically be split. Write L ∞ for the space ψ τ ( ψ q ( . . . ( ψ q r ( j ! L ) . . . ))), where j : C ∗ τ × ( U \ Z ) an ֒ → ( P z \{ } ) × U an i.e., L ∞ is thespace of multivalued flat sections of QM loc e A . The basic fact used in order to construct F • is that we have L ∞ = ψ τ j τ, ! (cid:16) ( QM an e A ) ∇ res ,q | C ∗ τ ×{ } (cid:17) . This is again due to lemma 3.8, 4. More precisely, we have already seenthat V QM loc e A = QM e A , i.e., gr V ( QM e A ) = ( QM e A ) | C ∗ τ ×{ } = E | C ∗ τ , where V • is the canonical V -filtrationon QM loc e A along the normal crossing divisor Z , and then the statement follows from the comparisontheorem for nearby cycles.Now we have already constructed an extension of ( QM e A ) | C ∗ τ ×{ } to ( P z \{ } ) × { } : namely, the chartat z = ∞ of the bundle b E from proposition 3.9, and we have seen in point 3 of this proposition that itis encoded by a filtration F • on ψ τ j τ, ! (cid:16) ( QM an e A ) ∇ res ,q | C ∗ τ ×{ } (cid:17) . Hence we obtain a filtration F • on L ∞ thatwe are looking for. As explained above, this yields a split Z r +1 -filtration e P • giving rise to a bundle d QM e A → (cid:0) ( P z \{ } ) × B (cid:1) with logarithmic poles along b Z an ∩ ( P z × B ), and by construction this bundlerestricts to QM e A on C ∗ τ × B and to b E | ( P z \{ } ) ×{ } on ( P z \{ } ) × { } . Hence we can glue d QM e A and QM an e A on C ∗ τ × B to a holomorphic P z × B -bundle. Its restriction to P z × { } is trivial, namely, it is thebundle b E constructed in proposition 3.9. As triviality is an open condition, there exists an open subset(with respect to the analytic topology) U ⊂ B such that the restriction of this bundle to P z × U , whichwe call [ QM e A , is fibrewise trivial, i.e., satisfies [ QM e A = p ∗ p ∗ [ QM e A . This shows the points 1. to 4.Concerning the statement on the pairing, notice that the flat pairing P defined on L ∞ gives rise toa pairing on ψ τ j τ, ! (cid:16) ( QM an e A ) ∇ res ,q | C ∗ τ ×{ } (cid:17) . Then the pole order property of P on b E at z = ∞ can beencoded by an orthogonality property of the filtration F • with respect to that pairing (the one definedon ψ τ j τ, ! (cid:16) ( QM an e A ) ∇ res ,q | C ∗ τ ×{ } (cid:17) ) see, e.g., [Her03, theorem 7.17 and definition 7.18]. Hence the very sameproperty must hold for P and F • , seen as defined on L ∞ , so that we conclude that we obtain P : [ QM e A ⊗ O P z × U ι ∗ [ QM e A → O P z × U ( − n, n ), as required.Finally, let us show the last statement: It follows from the correspondence between monodromy in-variant filtrations and logarithmic poles used above that the residue connection ∇ res ,τ along z = ∞ on [ QM e A /z − [ QM e A has trivial monodromy around Z if for any a = 1 , . . . , r , the nilpotent part N a ofthe monodromy of ∇ on the local system ( QM loc e A ) ∇ kills gr F • , i.e., N a F • ⊂ F •− . Now by the aboveidentification, we can see F • as defined on ψ τ j τ, ! (cid:16) ( QM an e A ) ∇ res ,q | C ∗ τ ×{ } (cid:17) , and then N a F • ⊂ F •− has beenshown in proposition 3.9, 3. It follows directly from the above construction that all elements of F , seenas global sections over P z × U are horizontal for ∇ res ,τ . Remark:
If the algebraic subset ∆ S = S \ S , i.e., the subspace on which the Laurent polynomial W ( − , q ) : S → C t is degenerate, is a divisor, then additional monodromy phenomena may occur. Forthis reason, the bundle QM e A cannot in general be extended as an algebraic bundle over a Zariski opensubset of P × U . Such an extension a priori can only be defined on some covering space of a Zariskiopen subset of P × U . The choice of this covering space depends on the structure of the fundamentalgroup of U , which is not a priori known. We therefore restrict ourselves to the construction of an analyticextension parameterized by the ball B . Notice however that if X Σ A is Fano, then [ QM e A exists as analgebraic family of P z -bundles on some Zariski open subset of C r .At this point it is convenient to introduce the so-called I -function of the toric variety X Σ A . We followthe definition of Givental (see [Giv98]), and relate this function to the hypergeometric module QM loc e A discussed above. Definition 3.11.
Define I resp. e I to be the H ∗ ( X Σ A , C ) -valued formal power series I = e δ/z · X l ∈ L q l · m Y i =1 Q ν = −∞ ([ D i ] + νz ) Q l i ν = −∞ ([ D i ] + νz ) ∈ H ∗ ( X Σ A , C )[ z ][[ q , . . . , q r ]][[ z − , t , . . . , t r ]] . esp. e I = z − ρ · z µ · I . Here we have set q l = Q ra =1 q p a ( l ) a and ( t , . . . , t r ) are the coordinates on H ( X Σ A , C ) induced by the basis ( p , . . . , p r ) of L ∨ which were chosen at the beginning of subsection 3.1. Notice that δ = P ra =1 t a p a is a cohomology class in H ( X Σ A , C ) . Later we will set q i = e t i for i = 1 , . . . , r . As before ρ = P mi =1 [ D i ] ∈ L ∨ is the anti-canonical class of X Σ A and we write µ ∈ Aut( H ∗ ( X Σ A , C )) for the gradingautomorphism which take the value k · c on a homogeneous class c ∈ H k ( X Σ A , C ) . We collect the main properties of the I -function that we will need in the sequel. Most of the statementsof the next proposition are well-known, but rather scattered in the literature. Proposition 3.12.
1. We have e I = Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L q l · z − l Q mi =1 Γ( D i + l i + 1) , (14) where Γ( T X Σ A ) := Q mi =1 Γ(1 + D i ) . Moreover, e − δ/z · I, z ρ · e − δ · e I ∈ H ∗ ( X Σ A , C )[[ q , . . . , q r , z − ]] , (15) that is, these series are univalued and have no poles in { z = ∞} ∪ S ra =1 { q a = 0 } .2. I has the development I = 1 + γ ( q , . . . , q r ) · z − + o ( z − ) where γ = δ + γ ′ ( q , . . . , q r ) lies in δ + H ( X Σ A , C )[[ q , . . . , q r ]] . If X Σ A is Fano, then γ ′ = 0 .3. There is an open neighborhood S of in C r,an such that both e − δ/z · I and z ρ · e − δ · e I are elementsin H ∗ ( X Σ A , C ) ⊗ O an C ∗ τ × S ∗ , where S ∗ := S ∩ S . In particular, if we put κ := q · e γ ′ then κ lies in ( O anS ) r and defines a coordinate change on S . Notice that in the Fano case, κ is the identity, ingeneral it is called the mirror map . It will reappear in theorem 4.7 and proposition 4.10.4. Write π : ( ^ C ∗ τ × S ∗ ) an → ( C ∗ τ × S ∗ ) an for the universal cover, then for any linear function h ∈ ( H ∗ ( X Σ A , C )) ∨ , we have h ◦ e I ∈ H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ S ol • ( QM loc e A ) (cid:17) = H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ H om D C ∗ τ × S ∗ ( QM loc e A , O C ∗ τ × S ∗ ) (cid:17)
5. For all h ∈ ( H ∗ ( X Σ A , C )) ∨ , if h ◦ e I = 0 , then h = 0 , in other words, e I yields a fundamental systemof solutions of ( QM loc e A ) | C ∗ τ × S ∗ .Proof.
1. From z µ · δ/z = δ · z µ and z µ · D i /z = D i · z µ we deduce z − ρ · z µ · I = z − ρ · e δ · X l ∈ L q l · m Y i =1 Q ν = −∞ z ([ D i ] + ν ) Q l i ν = −∞ z ([ D i ] + ν )= e δ · z − ρ · X l ∈ L q l · Y l i ≥ Γ( D i + 1)Γ( D i + 1) · Q l i ν =1 z ( D i + ν ) · Y l i < Γ( D i + 1) · Q ν = l i +1 z · ( D i + ν )Γ( D i + 1)= e δ · z − ρ · m Y i =1 Γ( D i + 1) · X l ∈ L q l Y l i ≥ D i + l i + 1) · z l i · Y l i < z − l i Γ( D i + l i + 1) . The identity Q mi =1 Γ( D i + 1) = Γ( T X Σ A ) yields e I = Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L q l · z − l Q mi =1 Γ( D i + l i + 1)For the second point, notice first that e I = Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L ∩ Eff X Σ A q l · z − l Q mi =1 Γ( D i + l i + 1) , X Σ A ⊂ L R denotes the Mori cone of classes of effective curves in X Σ A . Indeed, wewill see that for any l outside L eff = L ∩ Eff X Σ A , the term q l · z − l Q mi =1 Γ( D i + l i +1) vanishes in H ∗ ( X Σ A , C ).Assume the contrary, and first notice that for l i < D i + l i +1) is divisible by D i . For q l · z − l Q mi =1 Γ( D i + l i +1) to be non-zero, there must be a maximal cone σ containing the set of all a i suchthat l i <
0, as otherwise the term Q i : l i < D i which occurs as a factor in q l · z − l Q mi =1 Γ( D i + l i +1) is zeroin H ∗ ( X Σ A , C ). We use again (see formula (11)) that Eff X Σ A = P σ ∈ Σ A ( n ) C σ , where C σ is thecone generated by elements l = ( l , . . . , l m ) with l i ≥ R ≥ a i is not a ray of σ . Thus l ∈ C σ ⊂ Eff X Σ A , which shows the claim. Now remember from the proof of theorem 3.7 thatfor all l ∈ L eff we have l ≥ X Σ A is weak Fano, hence, z − l has no poles at z = ∞ . Moreover,by the same argument p a ( l ) is non-negative for l ∈ L eff , which gives that q l has no poles along ∪ ra =1 { q a = 0 } . Hence we obtain e − δ/z · I, z ρ · e − δ · e I ∈ H ∗ ( X Σ A , C )[[ q , . . . , q r , z − ]].2. After what has been said before, it is evident that the I -function can be written as I = e δ/z · X l ∈ L eff q l · z − l · m Y i =1 Q ν = −∞ (cid:16) [ D i ] z + ν (cid:17)Q l i ν = −∞ (cid:16) [ D i ] z + ν (cid:17) . Let us calculate the first terms in the z − -development of this expression: The constant termcan only get contributions from elements l ∈ L eff with l = 0. The zero relation l = 0 gives thecohomology class 1, on the other hand, for any l = 0 with l = 0, there must be at least one i ∈ { , . . . , m } with l i <
0, and then constant coefficient in the product Q ν = −∞ (cid:16) [ Di ] z + ν (cid:17)Q liν = −∞ (cid:16) [ Di ] z + ν (cid:17) gets afactor ν = 0, i.e., is zero. By a similar argument, the coefficient γ of the z − -term cannot havea H ( X Σ A , C )-component. One also sees immediately that γ has no components in H > ( X Σ A , C ).Hence we are left to show that γ ( q , . . . , q r ) = δ + γ ′ ( q , . . . , q r ). We have I = (cid:0) δ/z + o ( z − ) (cid:1) · X l ∈ L eff q l · z − l · m Y i =1 Q ν = −∞ (cid:16) [ D i ] z + ν (cid:17)Q l i ν = −∞ (cid:16) [ D i ] z + ν (cid:17) . For the coefficient γ , we have a contribution from the δ/z -term in the first factor, and if X Σ A isFano, this is the only term as then l > l ∈ L eff \{ } . In the weak Fano case, any l ∈ L eff \{ } with l = 0 give some extra contribution from the [ D i ] /z -terms, but this part is multiplied by q l ,i.e., a univalued function in q , . . . , q r .3. As a first step, we show that there is a constant L > x = ( x , . . . , x m ) ∈ C m ,the expression X l ∈ L eff q l · z − l Q mi =1 Γ( x i + l i + 1) = X l ∈ L eff z − l Q ra =1 q p a ( l ) a Q mi =1 Γ( x i + l i + 1)is convergent on { ( z, q , . . . , q r ) | | z | ≥ , | q a | ≤ L } ∩ C ∗ τ × S . Using [BH06, Lemma A.4] we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − l Q ra =1 q p a ( l ) a Q mi =1 Γ( x i + l i + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A ( x )(4 m ) || l || · e − l · log | z | + P ra =1 p a ( l ) · log | q a | Let ǫ >
0, the series is absolutely and uniformly convergent if k l k · log(4 m ) − l · log | z | + r X a =1 p a ( l ) · log | q a | ≤ − ǫ || l || (16)for all l ∈ L eff . This gives the condition l · log | z | + r X a =1 p a ( l ) · ( − log | q a | ) ≥ ( ǫ + log(4 m )) · || l || || M || be the norm of the matrix ( m ia ). For | z | ≥ q ∈ S we have l · log | z | + r X a =1 p a ( l ) · ( − log | q a | ) ≥ r X a =1 p a ( l ) · ( − log | q a | ) ≥ r X a =1 p a ( l ) · min a =1 ,...,r ( − log | q a | ) ≥ || M || · || l || · min a =1 ,...,r ( − log | q a | )where we have used P ra =1 m ia p a ( l ) = l i and p a ( l ) ≥ l ∈ L eff . Thus condition (16) is satisfiedfor max a =1 ,...r | q a | ≤ e −|| M || ( ǫ + log (4 m )) =: L This shows convergence of P l ∈ L eff q l · z − l Q mi =1 Γ( x i + l i +1) on e S ∗ := { ( z, q , . . . , q r ) | | z | ≥ , | q a | ≤ L } ∩ C ∗ τ × S . From the nilpotency of the operators D i ∪ ∈ End ( H ∗ ( X Σ A , C ) we see that X l ∈ L eff q l · z − l Q mi =1 Γ( D i + l i + 1) ∈ H ∗ ( X Σ A , C ) ⊗ O an e S ∗ . For the readers convenience, we recall next how to derive the identities e (cid:3) l ( e I ) = 0 ∀ l ∈ L ( z∂ z + P ra =1 ρ ( p ∨ a ) q a ∂ q a ) ( e I ) = 0 . (17)Write e (cid:3) l := e (cid:3) − l − e (cid:3) + l where e (cid:3) − l := Q a : p a ( l ) > q p a ( l ) a Q i : l i < − l i − Q ν =0 ( P ra =1 m ia zq a ∂ q a − νz ) e (cid:3) + l := Q a : p a ( l ) < q − p a ( l ) a Q i : l i > l i − Q ν =0 ( P ra =1 m ia zq a ∂ q a − νz )Using the fact that zq a ∂ q a e I = z ( p a + p a ( l )) · e I we get e (cid:3) − l e I = Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L Y a : p a ( l ) > q p a ( l ) a Y i : l i < z − l i Y ν = l i +1 ( D i + l i + ν ) Q ra =1 q p a ( l ) a · z − l Q i Γ( D i + l i + 1)= Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L Q a : p a ( l ) > q p a ( l + l ) a · Q a : p a ( l ) < q p a ( l ) a · z − l − P i : l i < l i Q i : l i < Γ( D i + l i + l i + 1) · Q i : l i ≥ Γ( D i + l i + 1)= Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L Q a : p a ( l ) > q p a ( l ) a · Q a : p a ( l ) < q p a ( l − l ) a · z − l + P i : l i > l i Q i : l i < Γ( D i + l i + 1) · Q i : l i ≥ Γ( D i + l i − l i + 1)= Γ( T X Σ A ) · e δ · z − ρ · X l ∈ L Y a : p a ( l ) < q p a ( l ) a Y i : l i > z l i Y ν =1 − l i ( D i + l i + ν ) Q ra =1 q p a ( l ) a · z − l Q i Γ( D i + l i + 1)= e (cid:3) + l e I (18)which shows e (cid:3) l ( e I ) = 0. The second one of the equations (17) follows from z∂ z + r X a =1 ρ ( p ∨ a ) q a ∂ a ! e I = ( − ρ − l ) + r X a =1 ρ ( p ∨ a )( p a + p a ( l )) ! e I = 036ow we conclude by a classical argument from the theory of ordinary differential equations (see, e.g.,[CL55, Theorem 3.1]): Fix q ∈ S with | q a | < L , then e I ( z − , q ) satisfies a system of differentialequations in z − with a regular singularity at z − = 0. Hence e I ( z − , q ) is a multivalued analyticfunction on all of C ∗ τ ×{ q } , that is, e I is (multivalued) analytic in C ∗ τ × S ∗ , with S = { q ∈ C r | | q a |
For any homogeneous basis T , T , . . . , T s of H ∗ ( X Σ A , C ) , write e I = P st =0 e I t · T t , sothat e I t ∈ H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ S ol • ( QM loc e A ) (cid:17) by proposition 3.12, 3. Moreover, ( e I , . . . , e I s ) is a basis of H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ S ol • ( QM loc e A ) (cid:17) by proposition 3.12, 4. Using the natural duality H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ S ol • ( QM loc e A ) (cid:17) ! = (cid:16) H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ DR • ( QM loc e A ) (cid:17)(cid:17) ∨ = H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ H om D C ∗ τ × S ∗ ( O C ∗ τ × S ∗ , QM loc e A ) (cid:17) ∨ , let ( f , . . . , f s ) ∈ (cid:16) H (cid:16) ( ^ C ∗ τ × S ∗ ) an , π ∗ DR • ( QM loc e A ) (cid:17)(cid:17) s +1 be the dual basis, then we have id = s X t =0 f t ◦ e I t ∈ H (cid:16) ( C ∗ τ × S ∗ ) an , E nd D ( C ∗ τ × S ∗ ) an ( QM loc e A ) (cid:17) . In particular, seeing e I t (or, more precisely e I t (1) ) as a multivalued function in O C ∗ τ × S ∗ , we obtain arepresentation s X t =0 e I t ( z − , q , . . . , q r ) · f t (19) of the element ∈ QM loc e A , where f t are multivalued sections of the local system (( QM loc e A ) an | C ∗ τ × S ∗ ) ∇ . .3 Logarithmic Frobenius structures We derive in this subsection the existence of a Frobenius manifold with logarithmic poles associated tothe Landau-Ginzburg model of X Σ A . This extends, for the given class of functions, the constructionfrom [DS03], in the sense that we obtain a family of germs of Frobenius manifolds along the space U from the last subsection, with a logarithmic degeneration behavior at the divisor Z . For the readersconvenience, we first recall briefly the notion of a Frobenius structure with logarithmic poles, and oneof the main result from [Rei09], which produces such structures starting from a set of initial data withspecific properties. In contrast to the earlier parts of the paper, all objects in this subsection are analytic,unless otherwise stated. Definition-Lemma 3.14.
Let M be a complex manifold of dimension bigger or equal to one, and Z ⊂ M be a simple normal crossing divisor.1. Suppose that ( M \ Z, ◦ , g, e, E ) is a Frobenius manifold. Then we say that it has a logarithmic polealong Z (or that ( M, Z, ◦ , g, e, E ) is a logarithmic Frobenius manifold for short) if ◦ ∈ Ω M (log Z ) ⊗ ⊗ Θ M (log Z ) , g ∈ Ω M (log Z ) ⊗ , E, e ∈ Θ(log Z ) and if g is non-degenerate on Θ M (log Z ) .2. A log-trTLEP(n) -structure on M is a holomorphic vector bundle H → P z × M such that p ∗ p ∗ H = H (where p : P z × M ։ M is the projection) which is equipped with an integrable connection ∇ witha pole of type along { } × M and a logarithmic pole along ( P z × Z ) ∪ ( {∞} × M ) and a flat, ( − n -symmetric, non-degenerate pairing P : H ⊗ ι ∗ H → O P z × M ( − n, n ) .3. Any logarithmic Frobenius manifold gives rise to a log-trTLEP(n) -structures on M , basically bysetting H := p ∗ Θ(log Z ) , ∇ := ∇ LC − z ◦ + (cid:0) U z − V (cid:1) dzz , where ∇ LC is the Levi-Civita connectionof g on Θ(log Z ) , U := E ◦ ∈ E nd (Θ(log Z )) and V := ∇ LC • E − Id ∈ E nd (Θ(log Z )) (see [Rei09,proposition 1.7 and proposition 1.10] for more details). Under certain conditions, a given log-trTLEP(n)-structure can be unfolded to a logarithmic Frobeniusmanifold. This is summarized in the following theorem which we extract from [Rei09, theorem 1.12],notice that a non-logarithmic version of it was shown in [HM04], and goes back to earlier work ofDubrovin and Malgrange (see the references in [HM04]).
Theorem 3.15.
Let ( N, be a germ of a complex manifold and ( Z, ⊂ ( N, a normal crossingdivisor. Let ( H , be a germ of a log-trTLEP(n) -structure on N . Suppose moreover that there is aglobal section ξ ∈ H ( P × N, H ) whose restriction to {∞} × N is horizontal for the residue connection ∇ res ,τ : H /τ H → H /τ H ⊗ Ω {∞}× N (log ( {∞} × Z )) and which satisfies the following three conditions1. The map from Θ(log Z ) | → p ∗ H | induced by the Higgs field [ z ∇ • ]( ξ ) : Θ(log Z ) → p ∗ H isinjective (injectivity condition (IC)) ,2. The vector space p ∗ H | is generated by ξ | (0 , and its images under iteration of the elements ofEnd ( p ∗ H | ) induced by U and by [ z ∇ X ] ∈ for any X ∈ Θ(log Z ) (generation condition (GC)) ,3. ξ is an eigenvector for the residue endomorphism V ∈ E nd O {∞}× N ( H /z − H ) (eigenvector condition(EC)) .Then there exists a germ of a logarithmic Frobenius manifold ( M, e Z ) , which is unique up to canonicalisomorphism, a unique embedding i : N ֒ → M with i ( M ) ∩ e Z = i ( Z ) and a unique isomorphism H → (id P z × i ) ∗ p ∗ Θ M (log e Z ) of log-trTLEP(n) -structures. Using proposition 3.10, we show now how to associate a logarithmic Frobenius manifold to the Landau-Ginzburg model (
W, q ) of the toric manifold X Σ A . Theorem 3.16.
1. Let X Σ A be a smooth toric weak Fano manifold, defined by a fan Σ A . Let ( W, q ) : S → C t × S be the Landau-Ginzburg model of X Σ A and let q , . . . , q r be the coordinates on S defined by the choice of a nef basis p , . . . , p r of L ∨ . Consider the tuple ( [ QM e A , ∇ , P ) associ-ated to ( W, q ) by proposition 3.10. Then the corresponding analytic object ( [ QM e A , ∇ , P ) an is a log-trTLEP(n) -structure on U ,an . . There is a canonical Frobenius structure on ( U ,an × C µ − r , with a logarithmic pole along ( Z × C µ − r , , where, as before, Z = S ra =1 { q a = 0 } ⊂ U ,an ⊂ C r .Proof.
1. This follows directly from the properties of [ QM e A , ∇ and P as described in proposition3.10.2. We apply theorem 3.15 to the germ ( N,
0) := ( U ,an ,
0) and the germ of the log-trTLEP(n)-structure ( [ QM e A , ∇ , P ) an . Define the section ξ to be the class of 1 in F ⊂ H ( P z × U , [ QM e A ),recall that F ∼ = H ( P × { } , ( d QM e A ) | P z ×{ } ) was defined as the subspace of E ∼ = ( QM e A ) | C z ×{ } generated by monomials in ( zq a ∂ q a ) a =1 ,...,r . The ∇ res ,τ -flatness of ξ follows from proposition3.10, 6. Conditions (IC) and (GC) are a consequence of the identification of ( QM e A ) | (0 , with( H ∗ ( X Σ A , C ) , ∪ ) (lemma 3.8, 1.) and the fact that the latter algebra is “ H -generated”, i.e., fromthe description given by formula (10). More precisely, the action of the logarithmic Higgs fields[ zq a ∂ q a ] on H ( P z , b E ) ∼ = F ∼ = ( QM e A ) (0 , correspond, under the isomorphism α from lemma3.8 exactly to the multiplication with the divisors classes D a ∈ H ( X Σ A , C ) on H ∗ ( X Σ A , C ), and H -generation implies that the images under iteration of these multiplications generate the wholevector space ( QM e A ) | (0 , . Finally, condition (EC) follows from proposition 3.9, 2. Hence theconditions of theorem 3.15 are satisfied and yield the existence of a Frobenius structure on a germ( N × C µ − r , ξ by the universality property of theorem 3.15. Remark:
It follows from conditions (GC) and (EC) that ξ is a primitive and homogeneous section inthe sense of [DS03] (this notion goes back to the theory of “primitive forms” of K. Saito). In particular,for a representative U ,an of the germ ( U ,an ,
0) and any point q ∈ U ,an \ Z , the Frobenius structurefrom theorem 3.15, 2., is one of those constructed in loc.cit. It is a natural to ask the following Question 3.17.
Is the (restriction of the) Frobenius structure from above to a small neighborhood of q ∈ U ,an \ Z the canonical Frobenius structure of the map f W ( − , q ) : S → C t from [DS03] (see also[Dou09])? Notice that for X Σ A = P n , it follows from the computations done in [DS04] (which concern the moregeneral case of weighted projective spaces), that this question can be answered in the affirmative. D -module and the mirror correspondence We start this section by recalling for the readers convenience some well-known constructions from quan-tum cohomology of smooth projective varieties, mainly in order to fix the notations. In particular, weexplain the so-called quantum D -module (resp. the Givental connection) and the J -function. We nextshow that the quantum D -module can be identified with the object [ QM e A constructed in the last section.This identification uses the famous I = J -theorem of Givental and can be seen as the essence of the mirrorcorrespondence for smooth toric weak Fano varieties. As a consequence, using the results of subsection3.3, we obtain a mirror correspondence as an isomorphism of logarithmic Frobenius manifolds. We review very briefly some well known constructions from quantum cohomology of smooth projectivecomplex varieties and explain the the so-called quantum D -module, also called Givental connection. Definition-Lemma 4.1.
Let X be smooth and projective over C with dim C ( X ) = n . Choose onceand for all a homogeneous basis T , T , . . . , T r , T r +1 , . . . , T s of H ∗ ( X, C ) , where T = 1 ∈ H ( X, C ) , T , . . . , T r are nef classes in H ( X, Z ) (here and in what follows, we consider without mentioning onlythe torsion free parts of the integer cohomology groups) and T i ∈ H k ( X, C ) with k > for all i > r . If X = X Σ A is toric and weak Fano, then we suppose moreover that T i = p i , i.e, that the basis T , . . . , T s extends the basis of L ∨ ∼ = H ( X Σ A , C ) chosen at the beginning of section 3.1. We write t , . . . , t s for thecoordinates induced on the space H ∗ ( X, C ) . We denote by ( − , − ) the Poincar´e pairing on H ∗ ( X, C ) and by ( T k ) k =0 ,...,s the dual basis with respect to ( − , − ) . . For any effective class β ∈ H ( X, Z ) / Tors denote by M ,n,β ( X ) the Deligne-Mumford stack ofstable maps f : C → X from rational nodal pointed curves C to X such that f ∗ ([ C ]) = [ β ] . For any i = 1 , . . . , n , let ω π be the relative dualizing sheaf of the “forgetful” morphism π : M ,n +1 ,β ( X ) →M ,n,β ( X ) (i.e., the morphism forgetting the i -th point and stabilizing if necessary) which representsthe universal family. Define a Cartier divisor L i := x ∗ i ( ω π ) on M ,n,β ( X ) , where x i : M ,n,β ( X ) →M ,n +1 ,β ( X ) is the i -th marked point, and put ψ i = c ( L i ) .2. For any tuple α , . . . , α n ∈ H ∗ ( X, C ) , let h ψ i α , . . . , ψ i n n α n i ,n,β := Z [ M ,n,β ( X )] virt ψ i ev ( α ) ∪ . . . ∪ ψ i n n ev n ( α n ) and put h α , . . . , α n i ,n,β := h ψ α , . . . , ψ n α n i ,n,β . Here ev i : M ,n,β ( X ) → X is the i -th evalua-tion morphism ev i ([ C, f, ( x , . . . , x n )]) := f ( x i ) and [ M ,n,β ( X )] virt is the so-called virtual funda-mental class of M ,n,β ( X ) , which has dimension dim C ( X ) + R β c ( X ) + n − . h α , . . . , α n i ,n,β iscalled a Gromov-Witten invariant and h ψ i α , . . . , ψ i n n α n i ,n,β is a Gromov-Witten invariant withgravitational descendent.3. Let α, γ, τ ∈ H ∗ ( X, C ) be given, write τ = τ ′ + δ where δ ∈ H ( X, C ) and τ ′ ∈ H ( X, C ) ⊕ H > ( X, C ) . Define the big quantum product to be α ◦ τ γ : = X β ∈ Eff X X n,k ≥ n ! h α, γ, τ, . . . , τ | {z } n − times , T k i ,n +3 ,β T k Q β = X β ∈ Eff X X n,k ≥ e δ ( β ) n ! h α, γ, τ ′ , . . . , τ ′ | {z } n − times , T k i ,n +3 ,β T k Q β ∈ H ∗ ( X, C ) ⊗ C [[ t ]][[Eff X ]] (20) where Eff X is the semigroup of effective classes in H ( X, Z ) , i.e., the intersection of H ( X, Z ) with the Mori cone in H ( X, R ) . Notice that in order to obtain the last equality, we have used thedivisor axiom for Gromov-Witten invariants, see, e.g., [CK99, section 7.3.1].The Novikov ring C [[Eff X ]] was introduced to split the contribution of the different β ∈ Eff X ,as otherwise the formula above would not be a formal power series. However, if one knows theconvergence of this power series, one can set Q = 1 .4. Suppose that as before α, γ ∈ H ∗ ( X, C ) and that δ ∈ H ( X, C ) . Define the small quantum product by α ⋆ δ γ := s X k =0 X β ∈ Eff X e δ ( β ) h α, γ, T k i , ,β T k Q β ∈ H ∗ ( X, C ) ⊗ O anH ( X, C ) [[Eff X ]] . As we have seen, the quantum product exists as defined only formally near the origin in H ∗ ( X, C ).However, we will need to consider the asymptotic behavior of the quantum product in another limitdirection inside this cohomology space. For that purpose we will use the following Theorem 4.2 ([Iri07, theorem 1.3]) . The quantum product for a projective smooth toric variety isconvergent on a simply connected neighborhood W of (cid:8) τ = τ ′ + δ ∈ H ∗ ( X, C ) | ℜ ( δ ( β )) < − M ∀ β ∈ Eff X \{ } , k τ ′ k < e − M (cid:9) for some M ≫ , here k · k can be taken to be the standard hermitian norm on H ∗ ( X, C ) induced by thebasis T , . . . , T s . If α and γ are seen as sections of the tangent bundle of the cohomology space, we also write α ◦ γ forthe quantum product, which is also a section of T H ∗ ( X, C ).The next step is to define the Givental connection, also known as the quantum D -module. For a smoothtoric weak Fano manifold, this is the object that we will compare to the various hypergeometric differentialsystems constructed in the last section from the Landau-Ginzburg model of this variety.40 efinition-Lemma 4.3.
1. Write p : P z × W ։ W for the projection, and let F big := p ∗ T W be thepull-back of the tangent bundle of W . Define a connection with a logarithmic pole along {∞} × W and with pole of type along { } × W on F big by putting for any s ∈ H ( P z × W, F big ) ∇ Giv ∂ tk s := ∇ res ,z − ∂ tk ( s ) − z · T k ◦ T l ∇ Giv ∂ z s := z (cid:0) E ◦ sz + µ ( s ) (cid:1) (21) where µ ∈ End C ( H ∗ ( X, C )) is the grading operator already used in definition 3.11, E := s X i =0 (cid:18) − deg( T i )2 (cid:19) + r X a =1 k a ∂ T a is the so-called Euler field which is defined by P ra =1 k a T a = c ( X ) and where ∇ res ,z − is theconnection on T W defined by the affine structure on H ∗ ( X, C ) . Notice that by its very definition,the residue connection of ∇ Giv along z − = 0 is ∇ res ,z − , whence its name. We have that ( ∇ Giv ) =0 , and this integrability condition encodes many of the properties of the quantum product (mostnotably its associativity, which is expressed by a system of partial differential equations, known asWitten-Dijkgraaf-Verlinde-Verlinde equations). We sometimes use the dual Givental connection,which is defined by ˆ ∇ Giv := ι ∗ ∇ Giv , recall that ι ( z, t ) = ( − z, t ) .2. Define the pairing P : F big ⊗ ι ∗ F big −→ O P z × W ( − n, n )( a, b ) z n ( a ( z ) , b ( − z )) (22)
3. The tuple ( F big , ∇ Giv , P ) is a trTLEP(n) -structure on W in the sense of [HM04, definition 4.1](i.e., the non-logarithmic version of definition-lemma 3.14, 2.). We call it the quantum D -moduleor Givental connection of H ∗ ( X, C ) .4. Write W ′ := { τ ∈ W | τ ′ = 0 } and let F := p ∗ ( T H ∗ ( X, C ) | W ′ ) . We equip F with a con-nection and a pairing defined by formulas (21) and (22) . Then ( F , ∇ Giv , P ) is a trTLEP(n) -structure on W ′ ⊂ H ( X, C ) , which we call the small quantum D -module. We have ( F , ∇ Giv , P ) =( F big , ∇ Giv , P ) | P z × W ′ . Next we show that the small quantum D -module can be considered in a natural way as a bundle overthe partial compactification of the K¨ahler moduli space that we already encountered in the last section. Lemma 4.4.
1. Consider the natural action of πiH ( X, Z ) on H ∗ ( X, C ) by translation. Thenthe set W is invariant under this action. Write V for the quotient space, and π : W ։ V for the projection map. Then there is a trTLEP(n) -structure ( G big , ∇ Giv , P ) on V such that π ∗ ( G big , ∇ Giv , P ) = ( F big , ∇ Giv , P ) . ( G big , ∇ Giv , P ) is also called quantum D -module of X .2. The algebraic quotient of H ( X, C ) by πiH ( X, Z ) is the torus Spec C [ H ( X, Z )] , which we call S to be consistent with the notation of the previous section in case that X is toric weak Fano.Then the small quantum D -module descends to V ′ = S an ∩ V , i.e, there is a vector bundle G on P z × V ′ , a connection ∇ Giv and a pairing P such that ( G , ∇ Giv , P ) is a trTLEP(n) -structureon V ′ and such that π ∗ ( G , ∇ Giv , P ) = ( F , ∇ Giv , P ) , where π : W ′ ։ V ′ is again the projectionmap. We also call ( G , ∇ Giv , P ) the small quantum D -module. Obviously, we have again that ( G , ∇ Giv , P ) = ( G big , ∇ Giv , P ) | P z × V ′ .If X is Fano, then ( G , ∇ Giv , P ) has an algebraic structure, i.e., it is defined as an algebraic bundleover P z × S .Proof. The first statement and the first part of the second one are immediate consequences of the divisoraxiom already mentioned above. If X is Fano, then as R β c ( X ) > β ∈ Eff X , for fixed n onlyfinitely many Gromov-Witten invariants can be non-zero, this implies the algebraicity of G .41 orollary 4.5. Using the choice of the nef basis T , . . . , T r of H ( X, Z ) (consisting of the classes p , . . . , p r ∈ L ∨ if X = X Σ A is toric weak Fano), we obtain an embedding H ( X, C ) / πiH ( X, Z ) ֒ → C r ,with complement a normal crossing divisor Z = S ra =1 { q a = 0 } , if q a = e t a for a = 1 , . . . , r . Denoteby V ′ the closure of the image of V ′ under this embedding. Then there is an extension ( G , ˆ ∇ Giv , P ) of ( G , ˆ ∇ Giv , P ) to a log-trTLEP(n) -structure on V ′ . Moreover, consider the partial compactification V := (cid:8) ( t , q , . . . , q r , t r +1 , . . . , t s } | k q k < e − M , k ( t , t r +1 , . . . , t s ) k < e − M (cid:9) ⊂ H ( X, C ) ⊕ C r ⊕ L k> H k ( X, C ) of V , then there is a structure of a logarithmic Frobenius manifold on V restricting to the germ of aFrobenius manifold defined by the quantum product at any point ( t , q , . . . , q r , t r +1 , . . . , t s ) ∈ H ( X, C ) ⊕ H ( X, C ) / πiH ( X, Z ) ⊕ L k> H k ( X, C ) .Proof. Both statements follow from [Rei09, section 2.2, proposition 1.7 and proposition 1.10]. J -function, Givental’s theorem and mirror correspondence In order to compare the quantum D -module G to the hypergeometric system [ QM e A from the last section,we will use a particular multivalued section of G , called the J -function. Givental has shown in [Giv98]that I = J for Fano varieties and that equality holds after a change of coordinates in the weak Fanocase. We use this equality to identify the two log-trTLEP(n)-structures and deduce an isomorphism ofFrobenius manifolds with logarithmic poles.Actually, Givental’s theorem is broader as it also treats the case of nef complete intersections in toricvarieties, however, the B-model has a different shape for those varieties, the most prominent examplebeing the quintic hypersurface in P . In this case (this is true whenever the complete intersection isCalabi-Yau) the mirror is an ordinary variation of pure polarized Hodge structures, whereas in oursituation the Landau-Ginzburg model gives rise to a non-commutative Hodge structure as discussed insection 5. We plan to discuss the relation between the B-model of a (weak) Fano variety and that of itssubvarieties in a subsequent paper.We start with the definition of the J -function. It is convenient to introduce at the same time anendomorphism valued series which is closely related J . We suppose from now on that X = X Σ A is asmooth toric weak Fano variety. Definition 4.6.
1. Define a
End ( H ∗ ( X Σ A , C )) -valued power series in z − , t , . . . , t r by L ( δ, z − )( T a ) := e − δ/z T a − X β ∈ Eff X Σ A \{ } j =0 ,...,s e δ ( β ) (cid:28) e − δ/z T a z + ψ , T j (cid:29) , ,β T j , here the gravitational descendent GW-invariant h T j z + ψ , i , ,β has to be understood as the formalsum − P k ≥ ( − z ) − k − h ψ k T j , i , ,β .2. Define the H ∗ ( X Σ A , C ) -valued power series J by J ( δ, z − ) := e δz · X β ∈ Eff X Σ A \{ } j =0 ,...,s e δ ( β ) (cid:28) T j z − ψ , (cid:29) , ,β T j . Notice that any product of cohomology classes appearing in the definition of L and J is the cup product. Observe that L has the factorization L = S ◦ ( e − δ/z ) where S is the following End ( H ∗ ( X Σ A , C ))-valuedpower series S ( δ, z − )( T a ) := T a − X β ∈ Eff X Σ A \{ } j =0 ,...,s e δ ( β ) (cid:28) T a z + ψ , T j (cid:29) , ,β T j , D -module with a hypergeometric systemfrom the last chapter is the following famous result of Givental. Theorem 4.7 ([Giv98, theorem 0.1]) . The coordinate change κ from 3.12, 3., transforms the I -functioninto the J -function, i.e., we have I = (id C τ × κ ) ∗ J . In particular, it follows from proposition 3.12, 3.that J defines a (multivalued) holomorphic mapping from C τ × S ∗ to H ∗ ( X Σ A , C ) . If X Σ A is Fano, then I = J . Denote by S the matrix-valued function which represents the endomorphism function S with respect tothe basis T , . . . , T s . Similarly, K i is the constant matrix representing the cup product with T i , Ω i is theconnection matrix of ˆ ∇ Giv zq i ∂q i and V the matrix diag(deg( T ) , . . . , deg( T s )). We have the following Lemma 4.8 ([Iri06, lemma 2.1,2.2]) .
1. The matrix-valued function S satisfies the following differ-ential equations: zq i ∂ S ∂q i − S · K i + Ω i · S = 0 , z ∂∂z + r X i =1 (deg q i ) q i ∂∂q i ! S + [ V , S ] = 0 .
2. The
End ( H ∗ ( X Σ A , C )) -valued power series S satisfies S ∗ ( δ, z − ) · S ( δ, − z − ) = id , where ( − ) ∗ denotes the adjoint with respect to the Poincar´e pairing. In particular S is invertible. The main properties of the J -function and of the endomorphism function L are summarized in thefollowing proposition. Proposition 4.9.
1. For any α ∈ H ∗ ( X Σ A , C ) , we have ˆ ∇ Giv ∂ tk L · α = ˆ ∇ Giv q k ∂ qk L · α = 0ˆ ∇ Giv z ∂ z L · α = L · ( zµ − c ( X Σ A ) ∪ ) · α
2. The endomorphism-valued function L is invertible.3. We have J = L − ( T ) = P st =0 ( s t , T ) T t , with s t = L ( T t ) :4. Both L and J are convergent on P z \ { } × ( S ∗ ∩ V ′ ) .Proof.
1. The first formula can be found in [Pan98, equation (25)] and the second follows from lemma4.8 by a straightforward calculation.2. This follows from the second point of 4.8.3. See, e.g. [CK99, lemma 10.3.3].4. The multivalued functions ( s t , T ) are holomorphic in C τ × S ∗ as this is true for J by theorem 4.7and proposition 3.12, 3. Using the formula ˆ ∇ Giv q a ∂ qa ( s t , T l ) = ( s t , T a ◦ T l ) we conclude that s t is amultivalued section of G which is holomorphic in C τ × ( S ∗ ∩ V ′ ), because monomials of the form T n ◦ . . . ◦ T n r r are a basis of G in this domain.Next we will define a twist of the endomorphism-valued function L to produce truly flat sections of theGivental connection. Define e L = L ◦ z − µ ◦ z ρ = S ◦ e − δ/z ◦ z − µ ◦ z ρ . If we set e s t = e L ( T t ), where as before ρ = c ( X Σ A ) ∈ H ( X Σ A , C ) = L ∨ , then it is a straightforward computation to see that ˆ ∇ Giv e s t = 0 for t = 0 , . . . , s . As L resp. e L is invertible, we obtain that e s t is a basis of multivalued flat sections.We also need to define a twisted J -function, namely e J := P tt =0 e J t T t := P st =0 ( e s t , T ) T t = e L − ( T ). Thisyields, similarly to equation (19), the following formula1 = T = s X t =0 e J t e s t ∈ H ( C ∗ τ × V ′ , G ) (23)43he following proposition uses all the previous results to identify the differential systems defined on bothsides of the mirror correspondence. Proposition 4.10.
Let W be a sufficiently small open neighborhood of ∈ C r,an which is contained in S ∩ V ′ ∩ U ,an and such that κ induces an automorphism of W . There is an isomorphism φ : ( [ QM e A ) an | P z × W −→ (id P z × κ ) ∗ G | P z × W of log-trTLEP(n) -structures on W .Proof. Define a morphism of vector bundles with connection ϕ : (cid:16) ( QM e A ) an | C z × W , ∇ (cid:17) −→ (id C z × κ ) ∗ (cid:16) G | C z × W , ˆ ∇ Giv (cid:17) T , where the connection operator ∇ on the left hand side is the one from theorem 3.7. The first taskis to show that ϕ is well-defined, i.e., that the following holds: Put e (cid:3) ′ l := (id C z × κ ) ∗ e (cid:3) l and e E ′ :=(id C z × κ ) ∗ ( z ∂ z + P ra =1 ρ ( p ∨ a ) zq a ∂ q a ), then we have to show that e (cid:3) ′ l ( q , . . . , q r , z, ˆ ∇ Giv zq ∂ q , . . . , ˆ ∇ Giv zq r ∂ qr )(1) = 0 ∀ l ∈ L e E ′ (cid:16) q , . . . , q r , z, ˆ ∇ Giv z ∂ z , ˆ ∇ Giv zq ∂ q , . . . , ˆ ∇ Giv zq r ∂ qr (cid:17) (1) = 0 . Obviously, the objects on the left hand side of these equations are sections of (id C z × κ ) ∗ G | C z × W , i.e., theycannot have support on a proper subvariety, hence, it suffices to show that they are zero on C ∗ τ × ( W ∩ S ).On that subspace we can use the presentation 1 = P st =0 e J t · e s t from equation (23). As the multivaluedsections e s t are flat for ˆ ∇ Giv it follows that we have to show that e (cid:3) ′ l ( e J t ) = e (cid:3) l ((id C ∗ τ × κ ) ∗ e J t ) = 0 e E ′ ( e J t ) = (cid:0) z ∂ z + P ra =1 ρ ( p ∨ a ) zq a ∂ q a (cid:1) ((id C ∗ τ × κ ) ∗ e J t ) = 0 . This is obvious by theorem 4.7 and by the equations (17) in the proof of proposition 3.12. Hence weobtain that ϕ is a well-defined morphism of locally free sheaves compatible with the connection operatorson both sides.Next we show the the surjectivity of ϕ : As we are allowed to replace W by a smaller open neighborhoodof 0 ∈ C r , one easily sees that it suffices to show that ϕ is surjectiv on the germs at (0 ,
0) of bothmodules. Namely, we have flat structures on C ∗ τ × ( W ∩ S ) and on C ∗ τ × Z a for all a = 1 , . . . , r , so thatif ϕ is surjective at some point in C ∗ τ × ( W ∩ S ) resp. at some point in C ∗ τ × Z a , it will be surjectiveon all of C ∗ τ × ( W ∩ S ) resp. C ∗ τ × Z a . By Nakayama’s lemma, surjectivity on the germs at (0 ,
0) isguaranteed once we have surjectivity at the fibre at (0 , H ∗ ( X Σ A , C ) (for G | (0 , , this isomorphism holds by definition, and for ( QM e A ) an | (0 , , this islemma 3.8, 1.). Now by comparison of ranks, we obtain that ϕ is an isomorphism.It remains to show that ϕ can be extended to an isomorphism of log-trTLEP(n)-structures on W .First notice that ϕ yields an identification of the local systems ( QM loc e A ) ∇| C ∗ τ × ( W ∩ S ) and G ˆ ∇ Giv | C ∗ τ × ( W ∩ S ) .In particular, it follows then from lemma 3.8, 3. that the monodromy of G ˆ ∇ Giv | C ∗ τ × ( W ∩ S ) around Z a = { q a = 0 } is unipotent (this can also be shown by a direct calculation). Hence by using the the samearguments as in proposition 3.10 it suffices to identify the punctual trTLEP(n)-structures ( d QM e A ) | P ×{ } and G | P ×{ } . We already have such an identification on C z × { } by restricting the above isomorphism ϕ to C z × { } . Moreover, consider a basis w , . . . , w µ of ( QM e A ) | C z ×{ } as in the proof of proposition3.9, 1., which extends the basis T , T , . . . , T r , T r +1 , . . . , T s of H ∗ ( X Σ A , C ) = ( QM e A ) | (0 , . Then by thedefinition of the Givental connection and of the morphism ϕ , the restriction ϕ | C z ×{ } maps this basis isto T , . . . , T s ∈ G | C z ×{ } ∼ = ⊕ st =0 O C z T t . Remark also that the connection matrices in these bases of ∇ on( QM e A ) | C z ×{ } resp. ˆ ∇ Giv on G | (0 , are equal, this follows from formula (13) resp. formula (21). Hence44 extends to an isomorphism of P z -bundles e φ : ( d QM e A ) | P z ×{ } → G | P z ×{ } = (cid:0) (id P z × κ ) ∗ G (cid:1) | P z ×{ } ,compatible with the connections. By the same argument, this isomorphism also respects the pairings P on both sides, as it restricts to the identity at z = 0.As discussed above, we obtain from ϕ and e φ an isomorphism φ : ( [ QM e A ) an | P z × W −→ (id P z × κ ) ∗ G | P z × W of log-trTLEP(n)-structures on W , as required.As a consequence, we can now deduce an isomorphism of logarithmic Frobenius structures defined bythe quantum product resp. by the Landau-Ginzburg model (through the construction from subsection3.3) of X Σ A . Theorem 4.11.
There is a unique isomorphism germ
M ir : ( W × C µ − r , → ( V, which mapsthe logarithmic Frobenius manifold from corollary 4.5 (A-side) to that of theorem 3.16 (B-side) andwhose restriction to W corresponds to the isomorphism φ of log-trTLEP(n) -structures from above. Inparticular, it induces the identity on the tangent spaces at the origin, i.e., on ( H ∗ ( X Σ A , C ) , ∪ ) .Proof. This is a direct consequence of the uniqueness statement in theorem 3.15, using the last proposi-tion.
In this section we will use the results from the previous parts of the paper to show, via the fundamentaltheorem [Sab08, theorem 4.10], that the quantum D -module on the K¨ahler moduli space underlies avariation of pure polarized non-commutative Hodge structures. Moreover, we study the asymptoticbehavior near the large radius limit point and show that the associated harmonic bundle is tame in thesense of Mochizuki and Simpson (see, e.g., [Moc02, definition 4.4]) along the boundary divisor. We startby recalling briefly the necessary definitions. Definition 5.1 ([Her03, definition 2.12],[HS10, definition 2.1],[KKP08, definition 2.7]) . Let M be acomplex manifold and n ∈ Z be an integer. A variation of TERP-structures on M of weight n consistsof the following set of data.1. A holomorphic vector bundle H on C z × M with an algebraic structure in z -direction, i.e., a locallyfree O M [ z ] -module.2. A R -local system L on C ∗ z × M , together with an isomorphism iso : L ⊗ R O an C ∗ z × M → H an | C ∗ z × M such that the connection ∇ induced by iso has a pole of type along { }× M and a regular singularityalong {∞} × M .3. A polarizing form P : L ⊗ ι ∗ L → i n R C ∗ z × M , which is ( − n -symmetric and which induces a non-degenerate pairing P : H ⊗ O C z × M ι ∗ H → z n O C z × M , here non-degenerate means that we obtain a non-degenerate symmetric pairing [ z − n P ] : H /z H ×H /z H → O M . We also recall the notions of pure and pure polarized TERP-structures.
Definition 5.2.
Let ( H , L , P, n ) be a variation of TERP-structures on M . Write γ : P × M → P × M for the involution ( z, x ) ( z − , x ) and consider γ ∗ H , which is a holomorphic vector bundleover ( P \{ } ) × M . Define a locally free O P C anM -module b H , where O P C anM is the subsheaf of C an P × M consisting of functions annihilated by ∂ z by gluing H and γ ∗ H via the following identification on C ∗ z × M .Let x ∈ M and z ∈ C ∗ z and define c : H | ( z,x ) −→ ( γ ∗ H ) | ( z,x ) a -parallel transport of z − n · a. hen c is an anti-linear involution and identifies H | C ∗ z × M with γ ∗ H | C ∗ z × M . Notice that c restricts to thecomplex conjugation (with respect to L ) in the fibres over S × M .1. ( H , L , P, n ) is called pure iff b H = p ∗ p ∗ b H , where p : P × M ։ M . A variation of pure TERP-structures is also called variation of (pure) non-commutative Hodge structures ( ncHodge struc-ture for short).2. Let ( H , L , P, n ) be pure, then by putting h : p ∗ b H ⊗ C anM p ∗ b H −→ C anM ( s, t ) z − n P ( s, c ( t )) we obtain a hermitian form on p ∗ b H . We call ( H , L , P, n ) a pure polarized TERP resp. ncHodgestructure if this form is positive definite (at each point x ∈ M ). Remarks:
We comment on the differences between this definition and those in [HS10] resp. [KKP08].1. One may want, depending on the actual geometric situation to be considered, the local system L to be defined over Q (as in [KKP08]) or even over Z . This corresponds to the notion of real resp.rational Hodge structures and to the choice of a lattice for them in ordinary Hodge theory.2. The reason for considering TERP-structures, and not only ncHodge structures, which are pureby definition (this condition is called opposedness condition in [KKP08]) is that there are naturalexamples of TERP-structures which are not pure (see, e.g., [HS10, section 9]).3. A ncHodge structure in the sense of [KKP08] does not contain any polarization data. However, thestructures we are considering, i.e., those defined by (families of) algebraic functions are polarizablein a natural way, so that it seems reasonable to include these data in the definition.4. We did not put the Q -structure axiom from [KKP08] in the definition of an ncHodge structure.This property, roughly stating that the Stokes structure defined by the pole of ∇ along z = 0 (incase it is irregular) is already defined on the local system L , and not only on its complexification L ⊗ R C was part of the definition of a mixed TERP-structure in [HS07]. It turns out that in somesituations (see, e.g., [Moc08a, section 8]), this property is actually something to be proved, whichis why we exclude this condition from the definition of a ncHodge structure. Notice however thatin the geometric situations we are studying, this condition will always be satisfied.The following theorem is the first result of this section. Theorem 5.3.
The restriction to C z × ( W ∩ S ) of the quantum D -module G underlies a variation of(pure) polarized ncHodge structures of weight n on W ∩ S .Proof. We will show that QM loc e A is a polarized ncHodge structure on S , then the statement followsfrom proposition 4.10. We first show that QM loc e A is equipped with structures as in definition 5.1, thatis, that it underlies a variation of TERP-structures. Then we deduce from [Sab08] that this structure ispure and polarized.It follows from corollary 3.4 that QM loc e A is a locally free O C z × S -module, equipped with a connectionoperator with a pole of type 1 along { } × S and that moreover we have a non-degenerate pairing P : QM loc e A ⊗ ι ∗ QM loc e A → z n O C z × S . Recall also from the proof of theorem 2.4 and of corollary 3.3that the D P z × S -module QM loc e A ⊗ O P z × S O P z × S equals FL τt ( H ( W, q ) + O S ). Now the Riemann-Hilbertcorrespondence gives DR • ( H ( W, q ) + O S ) = p H R • ( W, q ) ∗ C S , where p H • is the perverse cohomologyfunctor (see, e.g., [Dim04]). Hence DR • ( H ( W, q ) + O S ) carries a real (resp. rational) structure, namely, p H R • ( W, q ) ∗ R S (resp. p H R • ( W, q ) ∗ Q S ). We then deduce from [Sab97, theorem 2.2] that the thelocal system of flat sections of (( QM loc e A ) an , ∇ ) is equipped with a real or even rational structure. Onecould also invoke the recent preprint [Moc10] and show that H ( W, q ) + O S is a R -(or Q -)holonomic D -module in the sense of [Moc10, definition 7.6], which holds due to the regularity of H ( W, q ) + O S . It46hen follows from loc.cit., section 9, that this real or rational structure is preserved under the standardfunctors (direct image, inverse image, tensor product) in particular, under (partial) Fourier-Laplacetransformation (the elementary irregular rank one module has an obvious real/rational structure). HenceFL τt ( H ( W, q ) + O S ) has a real (resp. rational) structure, which shows that QM loc e A underlies a variationof TERP-structures on S .It remains to show that this structure is pure and polarized in the sense of definition 5.2. It is sufficientto do this for the restriction ( QM loc e A ) | C z ×{ q } for all q ∈ S . Write W q for the restriction W | pr − ( q ) : q − ( q ) → C t , then the restriction of the tuple ( QM loc e A , QM loc e A , P ) to C z × { q } is exactly the tuple( G, G , b P ) associated to W q which was considered in [Sab08, theorem 4.10], where one has to use thecomparison result [Sab11, lemma 5.9] to identify (possibly up to a non-zero constant, see the remarkafter the proof of lemma 3.8) the pairing P defined on QM loc e A with the pairing b P from [Sab08, theorem4.10]. Then it is shown in loc.cit. that one can associated to ( G, G , b P ) an integrable polarized twistorstructure, which means exactly that the variation of TERP-structures ( QM loc e A ) | C z ×{ q } is pure polarized,i.e., that it is a variation of (pure) polarized ncHodge structures.In order to state the second result of this section, recall the following fact (see, e.g., [HS07, lemma 3.12]). Proposition 5.4.
Let ( H , L , P ) be a variation of polarized ncHodge structures of weight n on M . Put E := p ∗ b H , which is a real-analytic bundle equipped with a holomorphic structure defined by the isomor-phism E ∼ = H /z H ⊗ O M C anM , a Higgs field θ := [ z ∇ z ] ∈ E nd O M ( H /z H ) ⊗ Ω M and the hermitian metric h from above. Then the tuple ( E, ∂, θ, h ) (where ∂ is the operator defining the holomorphic structure on E ) is a harmonic bundle in the sense of [Sim88]. Let (
E, ∂, θ, h ) be the harmonic bundle associated by the last proposition to the ncHodge structure QM loc e A on S (resp. G on W ∩ S ). The next result concerns the asymptotic behavior of E along theboundary divisor Z = S ra =1 { q a = 0 } . Theorem 5.5.
Put e U := ( U \ Z ) an ⊂ S ,an . Then the restriction of the harmonic bundle ( E, ∂, θ, h ) to e U is tame along Z in the sense of [Moc02, definition 4.4].Proof. Recall that the tameness property of a harmonic bundle defined by a variation of polarizedncHodge structures can be expressed in the chosen coordinates q , . . . , q r as follows: Write the Higgsfield θ ∈ E nd O e U ( H /z H ) ⊗ Ω e U as θ = r X a =1 θ a dq a q a with θ i ∈ E nd O e U ( H /z H ). Then ( E, ∂, θ, h ) is called tame iff the coefficients of the characteristic poly-nomials of all θ i extend to holomorphic functions on U an . Now consider the locally free O C z × U -module QM e A from theorem 3.7. The connection ∇ : QM e A −→ QM e A ⊗ z − Ω C z × U (log (( { } × U ) ∪ ( C z × Z )))induces θ ′ := [ z ∇ ] ∈ E nd O Uan (cid:16) ( QM e A ) an |{ }× U an (cid:17) ⊗ Ω U an (log Z )As θ ′ restricts to θ on e U , we see that if we write θ ′ = P ra =1 θ ′ a dq a q a , then θ ′ a is the holomorphic extensionof θ a we are looking for. References [Ado94] Alan Adolphson,
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