On Calabi-Yau compactifications of Landau-Ginzburg models for coverings of projective spaces
aa r X i v : . [ m a t h . AG ] F e b ON CALABI–YAU COMPACTIFICATIONS OF LANDAU–GINZBURGMODELS FOR COVERINGS OF PROJECTIVE SPACES
VICTOR PRZYJALKOWSKI
Abstract.
We suggest the procedure that constructs a log Calabi–Yau compactificationof weak Landau–Ginzburg model of a Fano variety. We apply the suggestion for del Pezzosurfaces and coverings of projective spaces of index one. We also prove in these casesconjectures relating numbers of components of the fiber over infinity and of finite fiberswith the dimension of an anticanonical system of the Fano variety and its first nontrivialmiddle Hodge number, respectively. Introduction and setup
The mirror dual object to a smooth Fano variety X is a so called Landau–Ginzburgmodel , that is a quasi-projective variety Y with complex-valued regular function w : Y → C with certain properties called superpotential . The dual objects for all versions of mirrorsymmetry are expected to be the same (with different enhancements specific to the certainversions). Homological Mirror Symmetry [Kon94] deals with singularities of fibers, thus itneeds the fibers to be compact (or, at least, to contain all singularities). So it is importantto have w proper. However in practice it is more convenient to have Y as simple as possibleand then to construct its relative compactification. Following Givental [Gi97] and Hori–Vafa [HV00], let Y = ( C ∗ ) n , where n = dim( X ). In this case the superpotential, theregular function on the algebraic torus, can be represented by a Laurent polynomial f .We associate with it a family { f = λ, λ ∈ C } of fibers of the map ( C ∗ ) n → C given by f .We are interested in periods of this family.Strong version of Mirror Symmetry conjecture of variations of Hodge structures asso-ciate a toric Landau–Ginzburg with X . The base for this conjecture is a classical MirrorSymmetry of variations of Hodge structures, see, for instance, [Gi97]. Let be the fun-damental class of X . The series e I X ( t ) = 1 + X d> , a ∈ Z ≥ d ! h τ a i d · t d , where h τ a i d is a one-pointed genus Gromov–Witten invariant with descendants foranticanonical degree d curves on X , see [Ma99, VI-2.1], is called a constant term ofregularized I -series for X .Denote a constant term of a Laurent polynomial f by [ f ]. Define I f = P [ f i ] t i . Theorem 1.1 ([Prz08, Proposition 2.3]) . Consider a Laurent polynomial f ∈ C [ x ± , . . . , x ± n ] . The author was supported by RNF grant № hen there exist a (punctured) neighborhood U ⊂ C of the point ∞ = P \ C , a localcoordinate at ∞ on U ∪ {∞} ⊂ P , a fiberwise ( n − ω t ∈ Ω n − C ∗ ) n / C ( U ) , and a fiberwise ( n − t over U such that the Taylor expansion of the analyticfunction Z ∆ t ω t is given by the series I f . Definition 1.2.
Let X be a smooth n -dimensional Fano variety. A Laurent polynomial f X in n variables is called a weak Landau–Ginzburg model for X if e I X = I f X . Remark . Mirror symmetry associates Landau–Ginzburg model not just to a Fanovariety, but to a Fano variety together with a fixed divisor (or symplectic form) on it. Inour definition the divisor is the anticanonical one.As we mentioned above, our goal is to get a proper family. Moreover, general mirrorsymmetry expectation is that fibers of Landau–Ginzburg models are Calabi–Yau varietiesmirror dual to anticanonical sections of the initial Fano variety. This justifies the following.
Definition 1.4.
Let X be a smooth n -dimensional Fano variety and let f X be its weakLandau–Ginzburg model. Consider the diagram( C ∗ ) n (cid:31) (cid:127) / / f X (cid:15) (cid:15) Y w (cid:15) (cid:15) (cid:31) (cid:127) / / Z u (cid:15) (cid:15) C C (cid:31) (cid:127) / / P , where Y is a smooth open Calabi–Yau variety, Z is smooth, − K Z ∼ u − ( pt ), and w and u are proper. The pair ( Y, w ) is called
Calabi–Yau compactification of f X , while thepair ( Z, u ) is called log Calabi–Yau compactification .The notion of weak Landau–Ginzburg model is motivated by the following result,which can be considered as a cornerstone of the field. Let X be a smooth toric vari-ety, let N ≃ Z N , N R = N ⊗ R , let Σ in N R be its fan, and let { v i ∈ N | i ∈ I } be a set ofintegral generators of rays of Σ, where I is a finite set indexing rays. We use the standardnotation x v = x v · . . . · x v n n for formal variables x i and v = ( v , . . . , v n ) ∈ N . Theorem 1.5 ([Gi97], see also [HV00]) . The Laurent polynomial f X = P i ∈ I x v i is a weakLandau–Ginzburg model for X .More generally, Givental proved an analog of Theorem 1.5 for complete intersectionsin smooth toric varieties provided they admit so called nef-partitions . (Below we give thedefinition applied to the case of weighted complete intersections we are interested in.) Thetotal space of Landau–Ginzburg model is not an algebraic torus in this case, but a completeintersection therein. This approach was extended to the case for complete intersectionsin Grassmannians [BCFKS97] (see also [EHX97]) or partial flag manifolds [BCFKS98]using “good” (so called small) degenerations of the ambient varieties to toric varieties,replacing the ambient variety by the toric one. n [Ba97] it was suggested that Theorem 1.5 holds for a Fano variety admitting smalltoric degeneration. In a lot of cases (see the references after Definition 1.6 below) Given-tal’s Landau–Ginzburg models for complete intersections are birational to an algebraictorus, and superpotentials for these Landau–Ginzburg models supported in the torus cor-respond to toric degenerations in a way similar to the one from Theorem 1.5. Howeverthe toric varieties appearing in this construction can be not “good”: they can be verysingular. We claim that this approach, after some modification, can be applied for any toric degeneration, see, for instance, [Prz13].Recall that for a toric variety T with fan Σ in N ≃ Z n the fan polytope F ( T ) is aconvex hull in N R ≃ N ⊗ R of integral generators of rays of Σ, and for a Laurent polyno-mial f ∈ C [ N ], its Newton polytope N ( f ) is a convex hull in N R of non-zero monomialsof f (considered as elements of N ). Definition 1.6 ([Prz18a, Definition 3.5]) . Let X be a smooth Fano variety of dimension n .Then a Laurent polynomial f in n variables is called toric Landau–Ginzburg model for X if f is a weak Landau–Ginzburg model for X , if f admits a Calabi–Yau compactification,and if there exists a degeneration X T of X to a toric variety T such that N ( f ) = F ( T ).(The latter condition is called the toric condition .)Toric Landau–Ginzburg models exist for del Pezzo surfaces and Fanothreefolds ([Prz13], [ILP13], [CCGK16], [KKPS19]) complete intersections([CCG + ], [ILP13], [PSh15a], [Prz18b]); some partial results are known for Grassmannians([PSh14], [PSh15b]) and complete intersections in toric varieties ([Gi97], [DH15]).In [Prz13, Conjecture 36] the existence of toric Landau–Ginzburg model for any smoothFano variety is declared.Let us discuss what is known for smooth Fano complete intersections in weighted pro-jective spaces (weighted complete intersections). For a background of weighted completeintersections one can consult [Do82], [IF00], or [PSh], see also [PSh20].Let X be a smooth Fano weighted complete intersection of multidegree ( d , . . . , d k ) ina weighted projective space P ( a , . . . , a N ). Let d = i X = P a i − P d j be its Fano index. Definition 1.7. A nef-partition ( I , . . . , I k ) for X is a splitting { , . . . , N } = I ⊔ I ⊔ . . . ⊔ I k such that P j ∈ I i a j = d i for every i = 0 , . . . , k . The nef-partition is called nice if thereexists an index r ∈ I such that a r = 1.Conjecturally, nice nef-partitions exist for all smooth Fano weighted complete intersec-tions. The evidence for this is the following. Theorem 1.8 ([Prz11, Theorem 9], [PSh19a, Theorem 1.3], [PSh, Proposition 11.4.5]) . Nice nef partitions exist for smooth Fano weighted complete intersections of Cartier divi-sors, or of codimension 2, or of dimension at most 5.Given a nef-partition, in a spirit of Givental’s suggestion for complete intersectionsin smooth toric varieties, one can write down Landau–Ginzburg model for the weightedcomplete intersection. Moreover, if the nef-partition is nice, the total space of this modelis birational to an algebraic torus. efinition 1.9. Let X be a well formed Fano weighted complete intersection. Supposethat there exists a nice nef-partition ( I , . . . , I k ) for X . Let a i , . . . , a im i denote the el-ements of I i for 0 ≤ i ≤ k , and suppose that a = 1. The Landau–Ginzburg model ofGivental’s type is the Laurent polynomial( J ) f X = (1 + x , + . . . + x ,m − ) d · . . . · (1 + x k, + . . . + x k,m k − ) d k k Q i =0 m i − Q j =1 x a i,j i,j + x , + . . . + x ,m − in variables x i,j , 1 ≤ i ≤ k , 1 ≤ j ≤ m i − Proposition 1.10.
Let X be a well formed Fano weighted complete intersection. Supposethat there exists a nice nef-partition for X , and let f X be defined by ( J ). Then X [ f uX ] t u = X j Q kr =0 ( d r v )! Q ≤ j ≤ m i ≤ i ≤ k ( a ij v )! t d v . One can deduce the following from [Gi97, Theorem 0.1].
Theorem 1.11 (see [Prz07, Theorem 1.1]) . Let X be a smooth well formed Fano weightedcomplete intersection. Suppose that there exists a nice nef-partition for X , and let f X bedefined by ( J ). Then e I X = X j Q kr =0 ( d r j )! Q Ns =0 ( a s j )! t d j . Corollary 1.12.
Let X be a smooth well formed Fano weighted complete intersection.Suppose that there exists a nice nef-partition for X . Then the polynomial f X givenby ( J ) is a weak Landau–Ginzburg model for X . Theorem 1.13 ([ILP13, Theorem 3.1]) . Let X be a well formed Fano weighted completeintersection. Suppose that there exists a nice nef-partition for X . Then the polynomial f X given by ( J ) satisfies the toric condition.The existence of Calabi–Yau compactification is proven only for complete intersectionsin usual projective spaces. Theorem 1.14 ([PSh15a, Theorem 5.1]) . Let X ⊂ P N be a smooth Fano complete inter-section. Then the polynomial f X given by ( J ) admits a Calabi–Yau compactification.The proof of Theorem 1.14 is not so easy to generalize to weighted complete intersec-tions. More promising approach is the log Calabi–Yau compactification procedure sug-gested in [Prz17]. First, it does not use the specificity of complete intersections. Second, itgives a precise description (up to codimension one) of the fiber over infinity as a boundarydivisor of some toric variety. These fibers play a key role in Katzarkov–Kontsevich–Pantevconjectures [KKP17] and in P = W conjecture [HKP20]. This approach was applied forFano threefolds [Prz17] and complete intersections in usual projective spaces [Prz18b].Briefly, the approach is the following. ompactification Construction . Let us have a smooth Fano variety X of dimension n and its weak Landau–Ginzburg model f X satisfying the toric condition. In particular, X has a degeneration to a toric variety T such that ∆ = N ( f X ) = F ( T ) ⊂ N R . Let ∇ = ∆ ∨ = { x ∈ M R = N ∨ ⊗ R | h x, y i ≤ − , y ∈ ∆ } be the polytope dual to ∆. Assume that ∇ is integral (that is, ∆ is reflexive) and admitsa unimodular triangulation; in other words, there exists a smooth toric variety e T ∨ suchthat F ( e T ∨ ) = ∇ . Thus { f X = λ, λ ∈ C } can be compactified to an anticanonical pencilin e T ∨ . If f X has a particular shape, then the base locus of this pencil is a union of smoothcodimension 2 subvarieties (possibly with multiplicities). Moreover, the base locus doesnot contain codimension 2 toric strata of e T ∨ . If we blow up one of the components ofthe base locus, we get an anticanonical pencil whose base locus is a union of smoothcodimension 2 subvarieties again, and the anticanonical class of the blown up family isstill linearly equivalent to a fiber. Repeating such blow ups one by one, we arrive tothe required log Calabi–Yau compactification u : Z → P . As a bonus of this procedurewe get a description of the fiber over infinity u − ( ∞ ) as the boundary divisor (up tocodimension 2) for e T ∨ . In particular, it consists of k components, where k is the numberof integral boundary points of ∇ .This log Calabi–Yau compactification procedure suggests the following. Conjecture 1.16 ([ChP20, Conjecture 1.6]) . Let X be a smooth Fano variety, andlet ( Z, u ) be a log Calabi–Yau compactification of its toric Landau–Ginzburg model.Then u − ( ∞ ) consists of χ (cid:0) O ( − K X ) (cid:1) − h (cid:0) O X ( − K X ) (cid:1) − Remark . Conjecture 1.16 is motivated by the following. Let ∆ be a Newton polytopefor f X . Consider the degeneration of X to the toric variety T ∆ . Since this degenerationis flat, one has χ ( X ) = χ ( T ∆ ) . On the other hand, the log pair ( T ∆ , − K T ∆ ) is Kawamata log terminal, see [Ko95, Propo-sition 3.7] for the terminology. Thus, by Kodaira vanishing (see, for example, [KM98,Theorem 2.70]), one has h ( − K X ) = h ( − K T ∆ ) . The anticanonical linear system of T ∆ can be described as a linear system of Laurentpolynomials supported on its dual polytope ∇ . Suppose that the log Calabi–Yau com-pactification procedure described in Compactification Construction 1.15 is applicable for f X . In particular, ∇ is integral and T ∇ admits a toric crepant resolution e T ∇ → T ∇ . Thedimension of the anticanonical linear system of T ∆ is the number of integral points onthe boundary of ∇ . Since these boundary points are in one-to-one correspondence withboundary divisors of e T ∇ and, thus, with irreducible components of the fiber u − ( ∞ ), theconjecture follows. he fiber over infinity is not the only one that can be reducible. Another reducible fibersof log Calabi–Yau compactification also predict invariants of the initial Fano variety. Conjecture 1.18 ([PSh15a, Conjecture 1.1]) . Let X be a smooth Fano variety of di-mension n and let ( Y, w ) be a Calabi–Yau compactification of a toric Landau–Ginzburgmodel for X . Put κ Y = ♯ (cid:2) irreducible components of all reducible fibers of ( Y, w ) (cid:3) − ♯ (cid:2) reducible fibers (cid:3) . Then one has h ,n − pr ( X ) = κ Y , where h , pr ( X ) = h , ( X ) + 1 and h ,n − pr ( X ) = h ,n − ( X )for n > ∇ admits an unimodular triangulation, sothat there is a smooth toric variety whose fan polytope is ∇ . Finally, we need to have“good” base locus of the pencil compactified in the smooth toric variety.In fact toric Landau–Ginzburg models usually have very specific Newton polytopes andcoefficients, so that the second and third problems are easy to solve. However one can notlaunch the compactification procedure as it is for the non-reflexive case, which is commonalready for weighted complete intersections.We claim that Compactification Construction 1.15 after some modification can be ap-plied in more general cases. Proposition 1.19 (Proposition 2.1) . Let S be a smooth del Pezzo surface. Then stan-dard (see Section 2.1) weak Landau–Ginzburg model f S for S admits a log Calabi–Yaucompactification. In particular, f S is a toric Landau–Ginzburg model. Moreover, Conjec-tures 1.16 and 1.18 hold for it. Theorem 1.20 (Theorem 3.1) . Let X be a smooth index one covering of a projectivespace. Then weak Landau–Ginzburg model of Givental’s type is a toric Landau–Ginzburgmodel for X . It admits a log Calabi–Yau compactification. Conjecture 1.16 holds for it.In Section 4 we outline the suggested generalization of the compactification procedure.The author is grateful to I. Cheltsov, A. Corti, and C. Shramov for useful discussions.2. Del Pezzo surfaces
Recall that del Pezzo surfaces can be described as either a quadric surface or an anti-canonical degree d ≤ S d which is a blow up of P in r = 9 − d general enoughpoints. For d ≥ S d is very ample. Landau–Ginzburg modelsfor smooth del Pezzo surfaces and general divisors on them were suggested in [AKO06].They are pencils of elliptic curves with nodal singular fibers and one fiber (“over infinity”)which is a wheel of 12 − d smooth rational curves. Construction of toric Landau–Ginzburgmodel for any divisor on S d , d ≥
3, one can find in [Prz17]. For general divisors its logCalabi–Yau compactification is the pencil constructed in [AKO06]. The construction oftoric Landau–Ginzburg models uses Gorenstein toric degenerations obtained in [Ba85].Del Pezzo surfaces of degrees 1 and 2 are weighted complete intersections, so they have eak Landau–Ginzburg models of Givental’s type. We call the weak Landau–Ginzburgmodels mentioned above (for all del Pezzo surfaces) standard . Proposition 2.1.
Let S be a smooth del Pezzo surface. Then standard weak Landau–Ginzburg model f S for S admits a log Calabi–Yau compactification. In particular, f S isa toric Landau–Ginzburg model. Moreover, Conjectures 1.16 and 1.18 hold for it. Proof.
Let deg( S ) ≥
3. Then f S admits the log Calabi–Yau compactification givenby [Prz17, Compactification Construction 16]. Conjecture 1.16 holds for S by [Prz17,Remark 17]. One can check Conjecture 1.18 compactifying f S as a pencil of elliptic curvesin P and resolve the base locus of the pencil. (In fact this is just another description ofCompactification Construction 1.15.) For instance, if d = 3, then f S = ( x + y + 1) xy . Using the embedding T ( x, y ) ⊂ P ( x : y : z ) where T ( x, y ) ∼ = ( C ∗ ) denotes the algebraictorus with coordinates x and y , one can compactify the family of fibers for f S to theelliptic pencil F = {F ( λ : µ ) | ( λ : µ ) ∈ P } , where F ( λ : µ ) = { µ ( x + y + z ) = λxyz } . This pencil is generated by the triple line l = { x + y + z = 0 } and the union of lines l = { x = 0 } , l = { y = 0 } , l = { z = 0 } . The base locus of F consists of three points p i = l ∩ l i , i = 1 , ,
3. At each of them, thefiber over the point (0 : 1) ∈ P has multiplicity 3, and any other fiber has multiplicity 1.Let us resolve the base locus of F . First, resolve it in a neighborhood of the point p . Forthis blow it up. Since the multiplicity of F (0:1) = 3 l at p is 3, the exceptional curve liesin the fiber over (0 : 1) of the proper transform ¯ F of the pencil F , and the multiplicityof this fiber along the curve is 2. So the intersection of the exceptional curve with thefiber over infinity is a base point of multiplicity 2 of ¯ F . Proceeding like this, make twomore blow ups to resolve the base locus of F in the neighborhood of p (note that thefirst three exceptional divisors lie in the central fibers of the corresponding pencils, whilethe last one is horizontal). Similarly, resolve the base locus of F in a neighborhood of thepoints p and p .Note that after this resolution the anticanonical class is still the fiber, so we arrive toa log Calabi–Yau compactification of f X . Finally note that we get 3 exceptional curvesover each point. Two of them lie in the central fiber (the one containing the triple line),and one is horizontal. Thus we get 1 + 3 · e E . The fiber over infinity consists of 3lines — proper transforms of l , l , and l . Thus, Conjectures 1.16 and 1.18 hold in thiscase. ow let deg( S ) = 2. Then S can be described as a quartic hypersurface in P (1 , , , S is the polynomial f = ( x + y + 1) xy . Its Newton polytope ∆ is the convex hull of vertices (3 , −
1) , ( − ,
3) , ( − , − ∇ = ∆ ∨ is the convex hull of vertices (1 , , − , − ). Thus ∇ isnot integral, so that we can not apply Compactification Construction 1.15. However westill can construct a toric variety whose rays are generated by the vertices of ∇ . Its fanis generated by integral points (1 , , − , − P . The pencilof fibers for f is now generated by the fiber over infinity, which is a union of three lines l , l , l , one of which has multiplicity 2, and the central fiber, which is a line l in generalposition taken with multiplicity 4. In other words, as above we can compactify the pencilgiven by f to the pencil F = {F ( λ : µ ) , ( λ : µ ) ∈ P } , where F ( λ : µ ) = { µ ( x + y + z ) = λxyz } , so that l = { x + y + z = 0 } , l = { x = 0 } , l = { y = 0 } , l = { z = 0 } . In particular, F = F (0:1) = 4 l and F ∞ = F (1:0) = l + l + 2 l . We also have F t ∼ − K P + l .Denote p i = l ∩ l i . Let us resolve the base locus of F . First, resolve it in a neighborhoodof the point p . For this blow it up. Since the multiplicity of F (0:1) = 4 l at p is 4, theexceptional curve lies in the fiber over (0 : 1) of the blown up pencil, and this curve hasmultiplicity 3 at this fiber. So the intersection of the exceptional curve with the fiber overinfinity is a base point of multiplicity 3 of the blown up pencil. Proceeding like this, make3 more blow ups to resolve the base locus of F in the neighborhood of p (note that thefirst 3 exceptional divisors lie in the fiber over (0 : 1), while the last one is horizontal).Similarly, resolve the base locus of F in a neighborhood of the point p . Now blow up p .Let e be the exceptional curve for this blow up. Since mult p F = 4, while mult p F ∞ is 2,then the multiplicity of e at the fiber over (0 : 1) of blown up family is 4 − e ∩ l . After this blow up we arriveto a base point free family e Z = { e Z t , t ∈ P } . One has e Z t ∼ − K e Z + l . Moreover, since l = 1 on P and we blow up two smooth points lying on l , we have l = − e Z . Thus,contracting l on e Z , we get the pencil Z = { Z t , t ∈ P } with smooth total space Z and − K Z ∼ Z t . This means that { Z t } is the required log Calabi–Yau compactification for f .Conjectures 1.16 and 1.18 easily follow from this construction. Indeed, Z ∞ = l + l is awheel of 2 rational curves, and the fiber Z consists of l , e , and three exceptional divisorslying over each of p and p .Similarly one can get the log Calabi–Yau compactification for deg( S ) = 1. In thiscase S can be described as a sextic hypersurface in P (1 , , , S is the polynomial f = ( x + y + 1) xy . he polytope dual to N ( f ) is a convex hull of points (1 , , ), ( − , − ). We com-pactify our family to the pencil F = {F ( λ : µ ) , ( λ : µ ) ∈ P } , where F ( λ : µ ) = { µ ( x + y + z ) = λxy z } , so that for l = { x + y + z = 0 } , l = { x = 0 } , l = { y = 0 } , l = { z = 0 } we have F = F (0:1) = 6 l and F ∞ = F (1:0) = l + 2 l + 3 l . We also have F t ∼ − K P + l + 2 l .Denote p i = l ∩ p i . As above, making six blow ups for p , three blow ups for p ,and two blow ups for p , we get the family e F = { e F t , t ∈ P } with total space e Z and e F t ∼ − K e Z + l + 2 l . Note that l = − l = − e Z . Let ϕ : e Z → Z ′ be thecontraction of l . Then Z ′ is smooth and ϕ ( l ) = −
1, so that we can contract ϕ ( l ) andget the log Calabi–Yau compactification Z of f X . Conjectures 1.16 and 1.18 easily followfrom this construction.Finally note that toric condition holds for standard weak Landau–Ginzburg models ofdel Pezzo surfaces by definition. (cid:3) Index one coverings of projective spaces
In this section we generalize the log Calabi–Yau compactification procedure for delPezzo surface of degree 2 constructed in the proof of 2.1 to index one smooth Fanocoverings of projective spaces. Let X be index 1 smooth Fano variety which is a a -to-1-covering of a projective space branched in a divisor of degree d . Let α = ( a − d =dim( X ). Then X can be described as a hypersurface of degree ad in P ( 1 , . . . , | {z } α +1 times , d ) . Thus its Givental’s Landau–Ginzburg model is f X = ( x + . . . + x α + 1) ad x · . . . · x α . Theorem 3.1.
The Laurent polynomial f X is a toric Landau–Ginzburg model for X . Itadmits a log Calabi–Yau compactification. Conjecture 1.16 holds for it. Proof.
The polynomial f X is a weak Landau–Ginzburg model by Corollary 1.12. It satis-fies the toric condition by Theorem 1.13. Now construct a log Calabi–Yau compactifica-tion. (This proves the first and the second assertions of the theorem.)The Newton polytope ∆ for f X is a convex hull of rows of the matrix ad − − . . . − − ad − . . . − . . . . . . . . . . . . − − . . . ad − − − . . . − . he dual polytope ∆ ∨ is a convex hull of rows of the matrix . . .
00 1 . . . . . . . . . . . . . . . . . . − d − d . . . − d . The toric variety whose rays are generated by rows of this matrix is P = P α , and integralgenerators of these rays are the rows of the latter matrix with the last one scaled by d .This means that the compactified (in P ) pencil F corresponding to f X is generated bythe fiber over infinity F ∞ , which is is a union of coordinate hyperplanes for P with oneof them taken with multiplicity d , and the central (that is, lying over (0 : 1) fiber F ,which is a hyperplane (general with respect to the coordinate hyperplanes) taken withmultiplicity ad . In other words, we compactify the family via the embedding T ( x , . . . , x γ ) ⊂ P ( x : . . . : x α ) , where T ( x , . . . , x α ) ∼ = ( C ∗ ) α denotes the algebraic torus with coordinates x , . . . , x α , toget the pencil F = {F ( λ : µ ) , ( λ : µ ) ∈ P } , where F ( λ : µ ) = { µ · ( x + . . . + x α ) ad = λ · x d · x · . . . · x α } , so that for H = { x + . . . + x α = 0 } and H i = { x i = 0 } , i = 1 , . . . , α , we have F = F (0:1) = adH, F ∞ = F (1:0) = dH + H + . . . + H α . Since − K P ∼ H + . . . + H α , we have F t ∼ − K P + ( d − H . If we denote B i = H ∩ H i ,then the base locus of F is the union of B i , i = 0 , . . . , α . We have mult B i F = ad , whilemult B F ∞ = d andmult B i F ∞ = 1 , i = 1 , . . . , α. Blow up P along B . We get an exceptional divisor E and the pencil F with F = E + adH and F ∞ = dH + H + . . . + H α . (We again denote divisors and their stricttransforms of divisors by the same symbols.) Denoting B = E ∩ H and B i = H ∩ H i , B ′ i = H ∩ H i for i = 1 , . . . , α , we havemult B F = ( a − d, mult B F ∞ = d, mult B i F = ad, mult B i F ∞ = 1 , i = 1 , . . . , α. The base locus of F is a union of several smooth codimension 2 strata: its irreduciblecomponents are B i , i = 0 , . . . , α , and B i = E ∩ H i , i = 1 , . . . , α . We also have F t ∼ − K Z + ( d − H , where Z is the total space of F .Repeating this procedure a − B of thebase locus and get the pencil F ′ = {F ′ t , t ∈ P } with total space Z ′ . The base locus for F ′ is a union of smooth codimension 2 subvarieties of Z ′ that intersect each other and F ′∞ transversally. Moreover, F ′ t ∼ − K Z ′ + ( d − H .Choose a toric structure on the original P α with boundary divisors H , H, H , . . . , H α .Then the blow ups we consider above are toric and thus Z ′ is also toric. The rays of the an of Z ′ are generated by the vectors v = (1 , , , . . . , ,v = (0 , , , . . . , ,. . .v α − = (0 , , , . . . , ,v α = ( − , − , − , . . . , − ,u = (1 , , , . . . , ,u = (2 , , , . . . , ,. . .u a = ( a, , , . . . , , where v corresponds to the strict transform of H , v i , i = 0 , , . . . , v α correspond to stricttransforms of H i and v corresponds to the strict transform of H . Since v = u + v + v + . . . + v α , we can blow down H on Z ′ to a smooth point and get a smooth pencil F ′′ = {F ′′ t , t ∈ P } with total space Z ′′ . Moreover, F ′′ t ∼ − K Z ′′ . Now we can proceed as in Compactifica-tion Construction 1.15, cf. [Prz17, Proposition 26], and get the required log Calabi–Yaucompactification Z → P .Finally, the fiber of Z → P over (1 : 0) is a normal crossing divisor, and its its dualintersection complex is a boundary of a ( d − a ( d − − α = ( a − d components. By [Do82, Theorem 3.3.4] and [PSh19b, Corollary 3.3] one has h ( O X ( − K X )) = ( a − d + 1 , so Conjecture 1.16 holds. (cid:3) Corollary 3.2.
Let Z be any log Calabi–Yau compactification of f X . Then it is ofHodge–Tate type, that is, h p,q ( Z ) = 0 if p = q . Proof.
This follows from the proof of Theorem 3.1 and the fact that any two log Calabi–Yau compactifications differ by flops (this can be easily checked by constructing a varietydominating two different compactifications). (cid:3)
The construction used in the proof of Theorem 3.1 enables one to count components ofthe cental fiber to prove Conjecture 1.18.
Example 3.3.
Let X ⊂ P (1 , , , ,
3) be a sextic double solid threefold. Let us constructthe log Calabi–Yau compactification following the proof of Theorem 3.1 and keeping trackexceptional divisors lying over the central fiber.One has f X = ( x + x + x + 1) x x x , so that F = {F ( λ : µ ) , ( λ : µ ) ∈ P } , where F ( λ : µ ) = { µ · ( x + x + x + x ) = λx x x x . Denoting H = { x + x + x + x = 0 } and H i = { x i = 0 } for i = 1 , , , e have F = 6 H and F ∞ = 3 H + H + H + H . Denote also B i = H ∩ H i and P ij = H ∩ H i ∩ H j .Resolving the base locus of F , we blow up smooth curves (components of the baselocus) one by one. If a base curve C has multiplicity m in the fiber over infinity and am in the central one, then among a exceptional divisors lying over a general point of C one (the last) exceptional divisor is horizontal for the resolved family (that is, it projectssurjectively to the base of the family), and a − B i F = 6 , i = 0 , . . . , , and mult B F ∞ = 3 , mult B i F ∞ = 1 , i = 1 , , , we have (6 : 3 −
1) + 3 · (6 : 1 −
1) = 16 exceptional divisors in the central fiber lying overgeneral points of the components of the base locus.Now we need to count exceptional divisors lying over the points P ij . Set P = P j forsome 1 ≤ j ≤
3. Let E be the exceptional divisor of the blow up of B . Then, afterblowing up B , we get one extra base curve E ∩ H j of multiplicity 3 at the central fiberand multiplicity 1 at the fiber over infinity. The blow up of E ∩ H does not produce anynew base curves. Thus, we have 3 : 1 − P , and totally 3 · P j .Now set P = P ij , 1 ≤ i < j ≤
3. Resolving the base locus of the pencil in a neigh-borhood of general point of B i , we have five exceptional divisors E , . . . , E in the centralfiber lying over P , and the multiplicity of E s at the central fiber is 6 − s , so that we havefive new components B sj = E s ∩ H j . The multiplicity of B sj at the central fiber is 6 − s ,while its multiplicity at the fiber over infinity is 1. Thus, in total we have(5 : 1 −
1) + (4 : 1 −
1) + (3 : 1 −
1) + (2 : 1 −
1) + (1 : 1 −
1) = 10exceptional divisors in the central fiber lying over P , and 3 ·
10 = 30 divisors lying over P ij .Summing up, we have 1 + 16 + 6 + 30 = 53 components of the central fiber of logCalabi–Yau compactification for f X , so Conjecture 1.18 holds for it, since h ( X ) = 52.Conjecture 1.18 for X is proven in [Prz13], but the calculations we did to check it in factare similar to ones suggested in [ChP18].The general case can be done similarly, it just differs by more noisy combinatorics.That is, we need to count exceptional divisors lying over the strata of base locus of theinitial family which are intersections of the multiple hyperplane with some number ofhyperplanes with some multiplicities. This approach was developed in [PSh15a]. Say, inthe neighborhood of B ij = B i ∩ B j for 1 ≤ i < j ≤ α (we use the notation of the proof ofTheorem 3.1) the pencil looks like { µz ad = λxy } , where x, y, z are coordinates in A α +1 .By [PSh15a, Proposition 4.7], the number of components of the central fiber lying over B ij is (cid:18) ad − (cid:19) = ( ad − ad − . Problem 3.4.
Prove Conjecture 1.18 for toric Landau–Ginzburg models of Givental’stype for index one coverings of projective spaces. The most promising approach is togeneralize one suggested in [PSh15a]. . The general case
Compactifications constructed in Sections 2 and 3 suggest the following log Calabi–Yaucompactification procedure for weighted complete intersections. Consider a smooth Fanoweighted complete intersection X ⊂ P having a nice nef-partition. Let us use the notationof Definitions 1.7 and 1.9. In particular, let us have a weak Landau–Ginzburg model ofGivental’s type f X corresponding to X . One can directly prove the following. Lemma 4.1.
The polytope ∇ dual to the fan polytope F ( f X ) is generated by the rowsof the matrix M = i X a . . . . . . . . . − . . . − i X a . . . . . . . . . − . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i X a m . . . . . . − . . . − − i X a − i X a . . . − i X a . . . . . . − . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i X a k . . . − . . . −
10 0 . . . . . . i X a k . . . − . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i X a kmk − . . . −
10 0 . . . . . . − i X a k − i X a k . . . − i X a k − . . . −
10 0 . . . . . . . . . i X a , − . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . i X a ,m − . Remark . Let us call a nef-partition ( I , . . . , I k ) strong if a ,j = 1 for j = 1 , . . . , m and a i,j divides d i for all i, j . We expect that for any smooth Fano weighted completeintersection we always can find a strong nef-partition. For instance, this holds for thecases listed in Theorem 1.8. (Of course, a weighted complete intersection can have notstrong nef-partition as well; the example is a complete intersection of quartic and sexticin P (1 , , , , , , l be the number of weights of P which are equal to 1. Assume for simplicitythat i X = 1, so that m = 0 and we do not have the SE block of the matrix M . By [Do82,Theorem 3.3.4] and [Do82, Section 3.4.3], one has h ( O X ( − K X )) = h ( O X ( − l. The integral points of ∇ correspond to those rows of M that have a ij = 1 (plus the origin,of course), so there are l − T be the toric variety whose rays aregenerated by rays of M . Then T = P m × . . . × P m k . Compactify the family of fibersfor f X in T to a family F . Then F ∞ is a union of boundary divisors for T , and theirmultiplicities are equal to a ij . The fiber F is a union of components of type P m × . . . × P m i − × H i × P m i +1 × P × P m k , where H i is a hyperplane section of P m i ; multiplicity of such component is equal to d i .Let D ij be the boundary divisor that correspond to the row of M associated with a ij . ote that F λ ∼ − K T + X ( a ij − D ij . So we suggest the following compactification procedure. Choose a strong nef partiti-on I = ( I , . . . , I k ) and a weak Landau–Ginzburg model f X associated with it. Com-pactify f X and get the pencil F . Let D , . . . , D r be the boundary divisors of T , andlet B i ⊂ D i be the component of the base locus lying on D i . Resolve the base locus ina neighbourhood of B , that is, blow it up several times. Note that since I is strong,all exceptional divisors, except for the last horizontal one, lie in the central fiber. Blowdown D . Then resolve the base locus in the neighborhood of B and blow down B .Repeating this procedure for all D i we get a family Z ′ = { Z ′ t , t ∈ P } with Z ′ t ∼ − K Z ′ .Note that the base locus of { Z ′ t } is a union of smooth components of codimension 2,and these components do not coincide with intersections of components of the fiber overinfinity. Now we can proceed as in Compactification Construction 1.15, cf. [Prz17, Propo-sition 26], and get the required log Calabi–Yau compactification Z = { Z t } . Note that ifthe suggested compactification procedure gives a log Calabi–Yau compactification, thenConjecture 1.16 holds for X . Remark . If we choose a non-strong nef-partition, we get exceptional divisors at thefiber over infinity, so we need to prove that we can contract them.We do not have a proof (except for the cases considered in Proposition 2.1, Theorem 3.1,and some others) that this construction gives a log Calabi–Yau compactification, that is,that we can contract the components to get smooth varieties. In fact it can happenthat the varieties after the contraction is not smooth, but they admits a small crepantnon-projective resolution (cf. [ChP18, Remarks 2.1.5 and 10.1.4]), or, in other words, thecontractions are not projective (over the fiber over infinity).
Problem 4.4.
Prove that the suggested compactification procedure gives a log Calabi–Yau compactification.
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Victor Przyjalkowski
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina street, Moscow, Russia. [email protected], [email protected]@mi-ras.ru, [email protected]