STRING COHOMOLOGY OF CALABI-YAU HYPERSURFACES VIA MIRROR SYMMETRY

LEV A. BORISOV AND ANVAR R. MAVLYUTOV

Abstract. We propose a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in Batyrev’s mirror symmetry construction. The spaces are defined explicitly in terms of the corresponding reflexive polyhedra in a mirror-symmetric manner. We draw connections with other approaches to the string cohomology, in particular with the work of Chen and Ruan.

Contents

1. Introduction 1 2. String-theoretic and stringy Hodge numbers 3 3. Mirror symmetry construction of Batyrev 6 4. String cohomology construction for Calabi-Yau hypersurfaces 7 5. Hodge-Deligne numbers of affine hypersurfaces 12 6. “Deformed” rings and modules 14 7. Decomposition of stringy Hodge numbers for hypersurfaces 17 8. String cohomology construction via intersection cohomology 19 9. String cohomology vs. Chen-Ruan orbifold cohomology 20 10. Description of string cohomology inspired by vertex algebras 24 11. Appendix. G-polynomials 27 References 28

1. Introduction

The notion of orbifold cohomology has appeared in physics as a result of studying the string theory on orbifold global quotients, (see [DiHVW]). In addition to the usual cohomology of the quotient, this space was supposed to include the so-called twisted sectors, whose existence was predicted by the modular invariance condition on the partition function of the theory. Since then, there have been several attempts to give a rigorous mathematical formulation of this cohomology theory. The first two, due to [BDa] and [B3], tried to define the topological invariants of certain algebraic varieties (including orbifold global quotients) that should correspond to the dimensions of the Hodge components of a conjectural string cohomology space. These invariants should have the property arising naturally from physics: they are preserved by partial crepant resolutions; moreover, they coincide with the usual Hodge numbers for smooth varieties. Also, these invariants must be the same as those defined by physicists for orbifold global quotients. In [B3, B4], Batyrev has successfully solved this problem for a large class of singular algebraic varieties. The first mathematical definition of the orbifold cohomology space was given in [ChR] for arbitrary orbifolds. Moreover, this orbifold cohomology possesses a product structure arising as a limit of a natural quantum product. It is still not entirely clear if the dimensions of the Chen-Ruan cohomology coincide with the prescription of Batyrev whenever both are defined, but they do give the same result for reduced global orbifolds.

1991 Mathematics Subject Classification. Primary: 14M25.

Key words and phrases. Toric varieties, intersection cohomology, mirror symmetry.

In this paper, we propose a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in the Batyrev mirror symmetry construction (see [B2]), with the spaces defined rather explicitly in terms of the corresponding reflexive polyhedra. A peculiar feature of our construction is that instead of a single string cohomology space we construct a finite-dimensional family of such spaces, which is consistent with the physicists’ picture (see [G]). We verify that this construction is consistent with the previous definitions in [BDa], [B3] and [ChR], in the following sense. The (bigraded) dimension of our space coincides with the definitions of [BDa] and [B3]. In the case of hypersurfaces that have only orbifold singularities, we recover Chen-Ruan’s orbifold cohomology as one special element of this family of string cohomology spaces. We also conjecture a partial natural ring structure on our string cohomology space, which is in correspondence with the cohomology ring of crepant resolutions. This may be used as a real test of the Chen-Ruan orbifold cohomology ring. We go further, and conjecture the B-model chiral ring on the string cohomology space. This is again consistent with the description of the B-model chiral ring of smooth Calabi-Yau hypersurfaces in [M2].

Our construction of the string cohomology space for Calabi-Yau hypersurfaces is motivated by Mirror Symmetry. Namely, the description in [M3] of the cohomology of semiample hypersurfaces in toric varieties applies to the smooth Calabi-Yau hypersurfaces in [B2]. Analysis of Mirror Symmetry on this cohomology leads to a natural construction of the string cohomology space for all semiample Calabi-Yau hypersurfaces. As already mentioned, our string cohomology space depends not only on the complex structure (the defining polynomial f), but also on some extra parameter we call ω. For special values of this parameter of an orbifold Calabi-Yau hypersurface, we get the orbifold Dolbeault cohomology of [ChR]. However, for nonorbifold Calabi-Yau hypersurfaces, there is no natural special choice of ω, which means that the general definition of the string cohomology space should depend on some mysterious extra parameter. In the situation of Calabi-Yau hypersurfaces, the parameter ω corresponds to the defining polynomial of the mirror Calabi-Yau hypersurface. In general, we expect that this parameter should be related to the “stringy complexified Kähler class”, which is yet to be defined.

In an attempt to extend our definitions beyond the Calabi-Yau hypersurface case, we give a conjectural definition of string cohomology vector spaces for stratified varieties with ℚ-Gorenstein toroidal singularities that satisfy certain restrictions on the types of singular strata. This definition involves intersection cohomology of the closures of strata, and we check that it produces spaces of correct bigraded dimension. It also reproduces orbifold cohomology of a ℚ-Gorenstein toric variety as a special case.

Here is an outline of our paper. In Section 2, we examine the connection between the original definition of the string-theoretic Hodge numbers in [BDa] and the stringy Hodge numbers in [B3]. We point out that these do not always give the same result and argue that the latter definition is the more useful one. In Section 3, we briefly review the mirror symmetry construction of Batyrev, mainly to fix our notations and to describe the properties we will use in the derivation of the string cohomology. Section 4 describes the cohomology of semiample hypersurfaces in toric varieties and explains how mirror symmetry provides a conjectural definition of the string cohomology of Calabi-Yau hypersurfaces. It culminates in Conjecture 4.8, where we define the stringy Hodge spaces of semiample Calabi-Yau hypersurfaces in complete toric varieties. We spend most of the remainder of the paper establishing the expected properties of the string cohomology space. Sections 5 and 6 calculate the dimensions of the building blocks of our cohomology spaces. In Section 6, we develop a theory of deformed semigroup rings which may be of independent interest. This allows us to show in Section 7 that Conjecture 4.8 is compatible with the definition of the stringy Hodge numbers from [B3]. In the non-simplicial case, this requires the use of G-polynomials of Eulerian posets, whose relevant properties are collected in the Appendix. Having established that the dimension is correct, we try to extend our construction to the non-hypersurface case. Section 8 gives another conjectural definition of the string cohomology vector space in a somewhat more general situation. It hints that the intersection cohomology and the perverse sheaves should play a prominent role in future definitions of string cohomology. In Section 9, we connect our work with that of Chen-Ruan [ChR] and Poddar [P]. Finally, in Section 10, we provide yet another description of the string cohomology of Calabi-Yau hypersurfaces, which was inspired by the vertex algebra approach to Mirror Symmetry.

Acknowledgments. We thank Victor Batyrev, Robert Friedman, Mainak Poddar, Yongbin Ruan and Richard Stanley for helpful conversations and useful references. The second author also thanks the Max-Planck Institut für Mathematik in Bonn for its hospitality and support.

2. String-theoretic and stringy Hodge numbers

The string-theoretic Hodge numbers were first defined in the paper of Batyrev and Dais (see [BDa]) for varieties with Gorenstein toroidal or quotient singularities. In subsequent papers [B3, B4] Batyrev defined stringy Hodge numbers for arbitrary varieties with log-terminal singularities. To our knowledge, the relationship between these two concepts has never been clarified in the literature. The goal of this section is to show that the string-theoretic Hodge numbers coincide with the stringy ones under some conditions on the singular strata.

We begin with the definition of the string-theoretic Hodge numbers.

Definition 2.1. [BDa] Let X = ⋃_{i∈I} X_{i} be a stratified algebraic variety over ℂ with at most Gorenstein toroidal singularities such that for each i ∈ I the singularities of X along the stratum X_{i} of codimension k_{i} are defined by a k_{i}-dimensional finite rational polyhedral cone σ_{i}; i.e., X is locally isomorphic to

ℂ^{k−k_{i}} × U_{σ_{i}}

at each point x ∈ X_{i}, where U_{σ_{i}} is a k_{i}-dimensional affine toric variety which is associated with the cone σ_{i} (see [D]), and k = dimX. Then the polynomial

E_{st}^{BD}(X; u, v) := ∑_{i∈I} E(X_{i}; u, v) · S(σ_{i}, uv)

is called the string-theoretic E-polynomial of X. Here,

S(σ_{i}, t) := (1 − t)^{dimσ_{i}} ∑_{n∈σ_{i}} t^{degn} = (t − 1)^{dimσ_{i}} ∑_{n∈intσ_{i}} t^{−degn}

where deg is the linear function on σ_{i} that takes value 1 on the generators of one-dimensional faces of σ_{i}, and intσ_{i} is the relative interior of σ_{i}. If we write E_{st}(X; u, v) in the form

E_{st}^{BD}(X; u, v) = ∑_{p,q} a_{p,q}u^{p}v^{q},

then the numbers h_{st}^{p,q(BD)}(X) := (−1)^{p+q}a_{p,q} are called the string-theoretic Hodge numbers of X.

Remark 2.2. The E-polynomial in the above definition is defined for an arbitrary algebraic variety X as

E(X; u, v) = ∑_{p,q} e^{p,q}u^{p}v^{q},

where e^{p,q} = ∑_{k≥0}(−1)^{k}h^{p,q}(H_{c}^{k}(X)).

Stringy Hodge numbers of X are defined in terms of the resolutions of its singularities. In general, one can only define the E-function in this case, which may or not be a polynomial. We refer to [KoMo] for the definitions of log-terminal singularities and related issues.

Definition 2.3. [B3] Let X be a normal irreducible algebraic variety with at worst log-terminal singularities, ρ : Y → X a resolution of singularities such that the irreducible components D_{1}, …, D_{r} of the exceptional locus is a divisor with simple normal crossings. Let α_{j} > −1 be the discrepancy of D_{j}, see [KoMo]. Set I := {1, …, r}. For any subset J ⊂ I we consider

D_{J} := \{ \begin{matrix}⋂_{j∈J} D_{j} & if J ≠ ∅ \\ Y & if J = ∅ \end{matrix}  and  D_{J}^{∘} := D_{J} \ ⋃_{j∈ I\J} D_{j}.

We define an algebraic function E_{st}(X; u, v) in two variables u and v as follows:

E_{st}(X; u, v) := ∑_{J⊂I} E(D_{J}^{∘}; u, v) ∏_{j∈J} \frac{uv − 1}{(uv)^{a_{j}+1} − 1}

(it is assumed ∏_{j∈J} to be 1, if J = ∅). We call E_{st}(X; u, v) the stringy E-function of X. If E_{st}(X; u, v) is a polynomial, define the stringy Hodge numbers the same way as Definition 2.1 does.

It is not obvious at all that the above definition is independent of the choice of the resolution. The original proof of Batyrev uses a motivic integration over the spaces of arcs to relate the E-functions obtained via different resolutions. Since the work of D. Abramovich, K. Karu, K. Matsuki, J. Włodarsczyk [AKaMW], it is now possible to check the independence from the resolution by looking at the case of a single blowup with a smooth center compatible with the normal crossing condition.

Lemma 2.4. Let X be a disjoint union of strata X_{i}, which are locally closed in Zariski topology, and let ρ be a resolution as in Definition 2.3. For each X_{i} consider

E_{st}(X_{i} ⊆ X; u, v) := ∑_{J⊂I} E(D_{J}^{∘} ∩ ρ^{−1}(X_{i}); u, v) ∏_{j∈J} \frac{uv − 1}{(uv)^{a_{j}+1} − 1}.

Then this E-function is independent of the choice of the resolution Y. The Efunction of X decomposes as

E_{st}(X; u, v) = ∑_{i} E_{st}(X_{i} ⊆ X; u, v).

Proof. Each resolution of X induces a resolution of the complement of X̄_{i}. This shows that for each X_{i} the sum

∑_{j,X_{j}⊆X̄_{i}} E_{st}(X_{j} ⊆ X; u, v)

is independent from the choice of the resolution and is thus well-defined. Then one uses the induction on dimension of X_{i}. The last statement is clear. □

Remark 2.5. It is a delicate question what data are really necessary to calculate E_{st}(X_{i} ⊆ X; u, v). It is clear that the knowledge of a Zariski open set of X containing X_{i} is enough. However, it is not clear whether it is enough to know an analytic neighborhood of X_{i}.

We will use the above lemma to show that the string-theoretic Hodge numbers and the stringy Hodge numbers coincide in a wide class of examples.

Proposition 2.6. Let X = ⋃_{i} X_{i} be a stratified algebraic variety with at worst Gorenstein toroidal singularities as in Definition 2.1. Assume in addition that for each i there is a desingularization Y of X so that its restriction to the preimage of X_{i} is a locally trivial fibration in Zariski topology. Moreover, for a point x ∈ X_{i} the preimage in Y of an analytic neighborhood of x is complex-analytically isomorphic to a preimage of a neighborhood of {0} in U_{σ_{i}} under some resolution of singularities of U_{σ_{i}}, times a complex disc, so that the isomorphism is compatible with the resolution morphisms. Then

E_{st}^{BD}(X; u, v) = E_{st}(X; u, v).

Proof. Since E-polynomials are multiplicative for Zariski locally trivial fibrations (see [DKh]), the above assumptions on the singularities show that

E_{st}(X_{i} ⊆ X; u, v) = E(X_{i}; u, v)E_{st}({0} ⊆ U_{σ_{i}}; u, v).

We have also used here the fact that since the fibers are projective, the analytic isomorphism implies the algebraic one, by GAGA. By the second statement of Lemma 2.4, it is enough to show that

E_{st}({0} ⊆ U_{σ_{i}}; u, v) = S(σ_{i}, uv).

This result follows from the proof of [B3], Theorem 4.3 where the products

∏_{j∈J} \frac{uv − 1}{(uv)^{a_{j}+1} − 1}

are interpreted as a geometric series and then as sums of t^{deg(n)} over points n of σ_{i}. □

Corollary 2.7. String-theoretic and stringy Hodge numbers coincide for nondegenerate hypersurfaces (complete intersections) in Gorenstein toric varieties.

Proof. Indeed, in this case, the toric desingularizations of the ambient toric variety induce the desingularizations with the required properties. □

We will keep this corollary in mind and from now on will silently transfer all the results on string-theoretic Hodge numbers of hypersurfaces and complete intersections in toric varieties in [BBo], [BDa] to their stringy counterparts.

Remark 2.8. An example of the variety where string-theoretic and stringy Hodge numbers differ is provided by the quotient of ℂ^{2} × E by the finite group of order six generated by

r_{1} : (x, y; z) ↦ (xe^{2πi/3}, ye^{−2πi/3}; z), r_{2} : (x, y; z) ↦ (y, x; z + p)

where (x, y) are coordinates on ℂ^{2}, z is the uniformizing coordinate on the elliptic curve E and p is a point of order two on E. In its natural stratification, the quotient has a stratum of A_{2} singularities, so that going around a loop in the stratum results in the non-trivial automorphism of the singularity.

Remark 2.9. We expect that the stringy Hodge numbers of algebraic varieties with abelian quotient singularities coincide with the dimensions of their orbifold cohomology, [ChR]. This is not going to be true for the string-theoretic Hodge numbers. Also, the latter numbers are not preserved by the partial crepant resolutions as required by physics, see the above example. As a result, we believe that the stringy Hodge numbers are the truly interesting invariant, and that the string-theoretic numbers is a now obsolete first attempt to define them.

3. Mirror symmetry construction of Batyrev

In this section, we review the mirror symmetry construction from [B2]. We can describe it starting with a semiample nondegenerate (transversal to the torus orbits) anticanonical hypersurface X in a complete simplicial toric variety ℙ_{Σ}. Such a hypersurface is Calabi-Yau. The semiampleness property produces a contraction map, the unique properties of which are characterized by the following statement.

Proposition 3.1. [M1] Let ℙ_{Σ} be a complete toric variety with a big and nef divisor class [X] ∈ A_{d−1}(ℙ_{Σ}). Then, there exists a unique complete toric variety ℙ_{Σ_{X}} with a toric birational map π : ℙ_{Σ} ⟶ ℙ_{Σ_{X}}, such that Σ is a subdivision of Σ_{X}, π_{∗}[X] is ample and π^{∗}π_{∗}[X] = [X]. Moreover, if X = ∑_{ρ} a_{ρ}D_{ρ} is torus-invariant, then Σ_{X} is the normal fan of the associated polytope

Δ_{X} = {m ∈ M : ⟨m, e_{ρ}⟩ ≥ −a_{ρ} for all ρ} ⊂ M_{ℝ}.

Remark 3.2. Our notation is a standard one taken from [BC, C2]: M is a lattice of rank d; N = Hom(M, ℤ) is the dual lattice; M_{ℝ} and N_{ℝ} are the ℝ-scalar extensions of M and N; Σ is a finite rational polyhedral fan in N_{ℝ}; ℙ_{Σ} is a d-dimensional toric variety associated with Σ; Σ(k) is the set of all k-dimensional cones in Σ; e_{ρ} is the minimal integral generator of the 1-dimensional cone ρ ∈ Σ corresponding to a torus invariant irreducible divisor D_{ρ}.

Applying Proposition 3.1 to the semiample Calabi-Yau hypersurface, we get that the push-forward π_{∗}[X] is anticanonical and ample, whence, by Lemma 3.5.2 in [CKat], the toric variety ℙ_{Σ_{X}} is Fano, associated with the polytope Δ ⊂ M_{ℝ} of the anticanonical divisor ∑_{ρ} D_{ρ} on ℙ_{Σ}. Then, [M1, Proposition 2.4] shows that the image Y := π(X) is an ample nondegenerate hypersurface in ℙ_{Σ_{X}} = ℙ_{Δ}. The fact that ℙ_{Δ} is Fano means by Proposition 3.5.5 in [CKat] that the polytope Δ is reflexive, i.e., its dual

Δ^{∗} = {n ∈ N_{ℝ} : ⟨m, n⟩ ≥ −1 for m ∈ Δ}

has all its vertices at lattice points in N, and the only lattice point in the interior of Δ^{∗} is the origin 0. Now, consider the toric variety ℙ_{Δ^{∗}} associated to the polytope Δ^{∗} (the minimal integral generators of its fan are precisely the vertices of Δ). Theorem 4.1.9 in [B2] says that an anticanonical nondegenerate hypersurface Y^{∗} ⊂ ℙ_{Δ^{∗}} is a Calabi-Yau variety with canonical singularities. The Calabi-Yau hypersurface Y^{∗} is expected to be a mirror of Y. In particular, they pass the topological mirror symmetry test for the stringy Hodge numbers:

h_{st}^{p,q}(Y) = h_{st}^{d−1−p,q}(Y^{∗}), 0 ≤ p, q ≤ d − 1,

by [BBo, Theorem 4.15]. Moreover, all crepant partial resolutions X of Y have the same stringy Hodge numbers:

h_{st}^{p,q}(X) = h_{st}^{p,q}(Y).

Physicists predict that such resolutions of Calabi-Yau varieties have indistinguishable physical theories. Hence, all crepant partial resolutions of Y may be called the mirrors of crepant partial resolutions of Y^{∗}. To connect this to the classical formulation of mirror symmetry, one needs to note that if there exist crepant smooth resolutions X and X^{∗} of Y and Y^{∗}, respectively, then

h^{p,q}(X) = h^{d−1−p,q}(X^{∗}), 0 ≤ p, q ≤ d − 1,

since the stringy Hodge numbers coincide with the usual ones for smooth Calabi-Yau varieties. The equality of Hodge numbers is expected to extend to an isomorphism (mirror map) of the corresponding Hodge spaces, which is compatible with the chiral ring products of A and B models (see [CKat] for more details).

4. String cohomology construction for Calabi-Yau hypersurfaces

In this section, we show how the description of cohomology of semiample hypersurfaces in [M3] leads to a construction of the string cohomology space of Calabi-Yau hypersurfaces. We first review the building blocks participating in the description of the cohomology in [M3], and then explain how these building blocks should interchange under mirror symmetry for a pair of smooth Calabi-Yau hypersurfaces in Batyrev’s mirror symmetry construction. Mirror symmetry and the fact that the dimension of the string cohomology is the same for all partial crepant resolutions of ample Calabi-Yau hypersurfaces leads us to a conjectural description of string cohomology for all semiample Calabi-Yau hypersurfaces. In the next three sections, we will prove that this space has the dimension prescribed by [BDa].

The cohomology of a semiample nondegenerate hypersurface X in a complete simplicial toric variety ℙ_{Σ} splits into the toric and residue parts:

H^{∗}(X) = H_{toric}^{∗}(X) ⊕ H_{res}^{∗}(X),

where the first part is the image of the cohomology of the ambient space, while the second is the residue map image of the cohomology of the complement to the hypersurface. By [M2, Theorem 5.1],

H_{toric}^{∗}(X) ≅ H^{∗}(ℙ_{Σ})/Ann(X)  (1)

where Ann(X) is the annihilator of the class [X] ∈ H^{2}(ℙ_{Σ}). The cohomology of ℙ_{Σ} is isomorphic to

ℂ[D_{ρ} : ρ ∈ Σ(1)]/(P (Σ) + SR(Σ)),

where

P (Σ) = ⟨ ∑_{ρ∈Σ(1)}⟨m, e_{ρ}⟩D_{ρ} : m ∈ M⟩

is the ideal of linear relations among the divisors, and

SR(Σ) = ⟨D_{ρ_{1}} ⋯ D_{ρ_{k}} : {e_{ρ_{1}}, …, e_{ρ_{k}}} ⊄ σ for all σ ∈ Σ⟩

is the Stanley-Reisner ideal. Hence, H_{toric}^{∗}(X) is isomorphic to the bigraded ring

T (X)_{∗,∗} := ℂ[D_{ρ} : ρ ∈ Σ(1)]/I,

where I = (P (Σ) + SR(Σ)) : [X] is the ideal quotient, and D_{ρ} have the degree (1, 1).

The following modules over the ring T (X) have appeared in the description of cohomology of semiample hypersurfaces:

Definition 4.1. Given a big and nef class [X] ∈ A_{d−1}(ℙ_{Σ}) and σ ∈ Σ_{X}, let

U^{σ}(X) = ⟨ ∏_{ρ⊂γ∈Σ} D_{ρ} : intγ ⊂ intσ⟩

be the bigraded ideal in ℂ[D_{ρ} : ρ ∈ Σ(1)], where D_{ρ} have the degree (1,1). Define the bigraded space

T^{σ}(X)_{∗,∗} = U^{σ}(X)_{∗,∗}/I^{σ},

where

I^{σ} = {u ∈ U^{σ}(X)_{∗,∗} : uvX^{d−dimσ} ∈ (P (Σ)+SR(Σ)) for v ∈ U^{σ}(X)_{dimσ−∗,dimσ−∗}}.

Next, recall from [C1] that any toric variety ℙ_{Σ} has a homogeneous coordinate ring

S(ℙ_{Σ}) = ℂ[x_{ρ} : ρ ∈ Σ(1)]

with variables x_{ρ} corresponding to the irreducible torus invariant divisors D_{ρ}. This ring is graded by the Chow group A_{d−1}(ℙ_{Σ}), assigning [∑_{ρ} a_{ρ}D_{ρ}] to deg(∏_{ρ} x_{ρ}^{a_{ρ}}). For a Weil divisor D on ℙ_{Σ}, there is an isomorphism H^{0}(ℙ_{Σ}, O_{ℙ_{Σ}}(D)) ≅ S(ℙ_{Σ})_{α}, where α = [D] ∈ A_{d−1}(ℙ_{Σ}). If D is torus invariant, the monomials in S(ℙ_{Σ})_{α} correspond to the lattice points of the associated polyhedron Δ_{D}.

In [BC], the following rings have been used to describe the residue part of cohomology of ample hypersurfaces in complete simplicial toric varieties:

Definition 4.2. [BC] Given f ∈ S(ℙ_{Σ})_{β}, set J_{0}(f) := ⟨x_{ρ}∂f/∂x_{ρ} : ρ ∈ Σ(1)⟩ and J_{1}(f) := J_{0}(f) : x_{1} ⋯ x_{n}. Then define the rings R_{0}(f) = S(ℙ_{Σ})/J_{0}(f) and R_{1}(f) = S(ℙ_{Σ})/J_{1}(f), which are graded by the Chow group A_{d−1}(ℙ_{Σ}).

In [M3, Definition 6.5], similar rings were introduced to describe the residue part of cohomology of semiample hypersurfaces:

Definition 4.3. [M3] Given f ∈ S(ℙ_{Σ})_{β} of big and nef degree β = [D] ∈ A_{d−1}(ℙ_{Σ}) and σ ∈ Σ_{D}, let J_{0}^{σ}(f) be the ideal in S(ℙ_{Σ}) generated by x_{ρ}∂f/∂x_{ρ}, ρ ∈ Σ(1) and all x_{ρ′} such that ρ′ ⊂ σ, and let J_{1}^{σ}(f) be the ideal quotient J_{0}^{σ}(f) : (∏_{ρ⊄σ} x_{ρ}). Then we get the quotient rings R_{0}^{σ}(f) = S(ℙ_{Σ})/J_{0}^{σ}(f) and R_{1}^{σ}(f) = S(ℙ_{Σ})/J_{1}^{σ}(f) graded by the Chow group A_{d−1}(ℙ_{Σ}).

As a special case of [M3, Theorem 2.11], we have:

Theorem 4.4. Let X be an anticanonical semiample nondegenerate hypersurface defined by f ∈ S_{β} in a complete simplicial toric variety ℙ_{Σ}. Then there is a natural isomorphism

⨁_{p,q} H^{p,q}(X) ≅ ⨁_{p,q} T (X)_{p,q} ⊕ ( ⨁_{σ∈Σ_{X}} T^{σ}(X)_{s,s} ⊗ R_{1}^{σ}(f)_{(q−s)β+β_{1}^{σ}}),

where s = (p + q − d + dimσ + 1)/2 and β_{1}^{σ} = deg(∏_{ρ_{k}⊂σ} x_{k}).

By the next statement, we can immediately see that all the building blocks R_{1}^{σ}(f)_{(q−s)β+β_{1}^{σ}} of the cohomology of partial resolutions in Theorem 4.4 are independent of the resolution and intrinsic to an ample Calabi-Yau hypersurface:

Proposition 4.5. [M3] Let X be a big and nef nondegenerate hypersurface defined by f ∈ S_{β} in a complete toric variety ℙ_{Σ} with the associated contraction map π : ℙ_{Σ} ⟶ ℙ_{Σ_{X}}. If f_{σ} ∈ S(V (σ))_{β^{σ}} denotes the polynomial defining the hypersurface π(X) ∩ V (σ) in the toric variety V (σ) ⊂ ℙ_{Σ_{X}} corresponding to σ ∈ Σ_{X}, then, there is a natural isomorphism induced by the pull-back:

H^{d(σ)−∗,∗−1}H^{d(σ)−1}(π(X) ∩ 𝕋_{σ}) ≅ R_{1}(f_{σ})_{∗β^{σ}−β_{0}^{σ}} ≅R_{1}^{σ}(f)_{∗β−β_{0}+β_{1}^{σ}},

where d(σ) = d − dimσ, 𝕋_{σ} ⊂ V (σ) is the maximal torus, and β_{0} and β_{0}^{σ} denote the anticanonical degrees on ℙ_{Σ} and V (σ), respectively.

Given a mirror pair (X, X^{∗}) of smooth Calabi-Yau hypersurfaces in Batyrev’s construction, we expect that, for a pair of cones σ and σ^{∗} over the dual faces of the reflexive polytopes Δ^{∗} and Δ, T^{σ}(X)_{s,s} with s = (p+q−d+dimσ+1)/2, in H^{p,q}(X) interchanges, by the mirror map (the isomorphism which maps the quantum cohomology of one Calabi-Yau hypersurface to the B-model chiral ring of the other one), with R_{1}^{σ^{∗}}(g)_{(p+q−dimσ^{∗})β^{∗}/2+β_{1}^{σ^{∗}}} in H^{d−1−p,q}(X^{∗}) (note that dimσ^{∗} = d−dimσ+1), where g ∈ S(ℙ_{Σ^{∗}})_{β^{∗}} determines X^{∗}. For the 0-dimensional cones σ and σ^{∗}, the interchange goes between the polynomial part R_{1}(g)_{∗β^{∗}} of one smooth Calabi-Yau hypersurface and the toric part of the cohomology of the other one. This correspondence was already confirmed by the construction of the generalized monomialdivisor mirror map in [M3]. On the other hand, one can deduce that the dimensions of these spaces coincide for the pair of 3-dimensional smooth Calabi-Yau hypersurfaces, by using Remark 5.3 in [M1]. The correspondence between the toric and polynomial parts was discussed in [CKat].

Now, let us turn our attention to a mirror pair of semiample singular Calabi-Yau hypersurfaces Y and Y^{∗}. We know that their string cohomology should have the same dimension as the usual cohomology of possible crepant smooth resolutions X and X^{∗}, respectively. Moreover, the A-model and B-model chiral rings on the string cohomology should be isomorphic for X and X^{∗}, respectively. We also know that the polynomial g represents the complex structure of the hypersurface Y^{∗} and its resolution X^{∗}, and, by mirror symmetry, g should correspond to the complexified Kähler class of the mirror Calabi-Yau hypersurface. Therefore, based on the mirror correspondence of smooth Calabi-Yau hypersurfaces, we make the following prediction for the small quantum ring presentation on the string cohomology space:

QH_{st}^{p,q}(Y) ≅ ⨁_{(σ,σ^{∗})} R_{1}(ω_{σ^{∗}})_{(p+q−dimσ^{∗}+2)β^{σ^{∗}} /2−β_{0}^{σ^{∗}}} ⊗ R_{1}(f_{σ})_{(q−p+d−dimσ+1)β^{σ}/2−β_{0}^{σ}},

(2)

where the sum is by all pairs of the cones σ and σ^{∗} (including 0-dimensional cones) over the dual faces of the reflexive polytopes, and where ω_{σ^{∗}} ∈ S(V (σ^{∗}))_{β^{σ^{∗}}} is a formal restriction of ω ∈ S(ℙ_{Δ^{∗}})_{β^{∗}}, which should be related to the complexified Kähler class of the mirror (we will discuss this in Section 9). This construction can be rewritten in simpler terms, which will help us to give a conjectural description of the usual string cohomology space for all semiample Calabi-Yau hypersurfaces.

First, recall Batyrev’s presentation of the toric variety ℙ_{Δ} for an arbitrary polytope Δ in M (see [B1], [C2]). Consider the Gorenstein cone K over Δ×{1} ⊂ M⊕ℤ. Let S_{Δ} be the subring of ℂ[t_{0}, t_{1}^{±1}, …, t_{d}^{±1}] spanned over ℂ by all monomials of the form t_{0}^{k}t^{m} = t_{0}^{k}t_{1}^{m_{1}} ⋯ t_{d}^{m_{d}} where k ≥ 0 and m ∈ kΔ. This ring is graded by the assignment deg(t_{0}^{k}t^{m}) = k. Since the vector (m, k) ∈ K if and only if m ∈ kΔ, the ring S_{Δ} is isomorphic to the semigroup algebra ℂ[K]. The toric variety ℙ_{Δ} can be represented as

Proj(S_{Δ}) = Proj(ℂ[K]).

The ring S_{Δ} has a nice connection to the homogeneous coordinate ring S(ℙ_{Δ}) = ℂ[x_{ρ} : ρ ∈ Σ_{Δ}(1)] of the toric variety ℙ_{Δ}, corresponding to a fan Σ_{Δ}. If β ∈ A_{d−1}(ℙ_{Δ}) is the class of the ample divisor ∑_{ρ∈Σ_{Δ}(1)} b_{ρ}D_{ρ} giving rise to the polytope Δ, then there is a natural isomorphism of graded rings

ℂ[K] ≅ S_{Δ} ≅ ⨁_{k=0}^{∞} S(ℙ_{Δ})_{kβ},  (3)

sending (m, k) ∈ ℂ[K]_{k} to t_{0}^{k}t^{m} and ∏_{ρ} x_{ρ}^{kb_{ρ}+⟨m,e_{ρ}⟩}, where e_{ρ} is the minimal integral generator of the ray ρ. Now, given f ∈ S(ℙ_{Δ})_{β}, we get the ring R_{1}(f). The polynomial f = ∑_{m∈Δ} f(m)x_{ρ}^{b_{ρ}+⟨m,e_{ρ}⟩}, where f(m) are the coefficients, corresponds by the isomorphisms (3) to ∑_{m∈Δ} f(m)t_{0}t^{m} ∈ (S_{Δ})_{1} and ∑_{m∈Δ} f(m)[m, 1] ∈ ℂ[K]_{1} (the brackets [ ] are used to distinguish the lattice points from the vectors over ℂ), which we also denote by f. By the proof of [BC, Theorem 11.5], we have that

(S(ℙ_{Δ})/J_{0}(f))_{kβ} ≅ (S_{Δ}/⟨t_{i}∂f/∂t_{i} : i = 0, …, d⟩)_{k} ≅ R_{0}(f, K)_{k},

where R_{0}(f, K) is the quotient of ℂ[K] by the ideal generated by all “logarithmic derivatives” of f:

∑_{m∈Δ}((m, 1) · n)f(m)[m, 1]

for n ∈ N ⊕ ℤ. The isomorphisms (3) induce the bijections

S(ℙ_{Δ})_{kβ−β_{0}} \underrightarrow{∏_{ρ} x_{ρ}} ⟨∏_{ρ} x_{ρ}⟩_{kβ} ≅ (I_{Δ}^{(1)})_{k} ≅ ℂ[K^{∘}]_{k}

(β_{0} = deg(∏_{ρ} x_{ρ})), where I_{Δ}^{(1)} ⊂ S_{Δ} is the ideal spanned by all monomials t_{0}^{k}t^{m} such that m is in the interior of kΔ, and ℂ[K^{∘}] ⊂ ℂ[K] is the ideal spanned by all lattice points in the relative interior of K. Since the space R_{1}(f)_{kβ−β_{0}} is isomorphic to the image of ⟨∏_{ρ} x_{ρ}⟩_{kβ} in (S(ℙ_{Δ})/J_{0}(f))_{kβ},

R_{1}(f)_{kβ−β_{0}} ≅ R_{1}(f, K)_{k},

where R_{1}(f, K) is the image of ℂ[K^{∘}] in the graded ring R_{0}(f, K).

The above discussion applies well to all faces Γ in Δ. In particular, if the toric variety V (σ) ⊂ ℙ_{Δ} corresponds to Γ, and β^{σ} ∈ A_{d−dimσ−1}(V (σ)) is the restriction of the ample class β, then

S(V (σ))_{∗β^{σ}} ≅ ℂ[C],

where C is the Gorenstein cone over the polytope Γ × {1}. This induces an isomorphism

R_{1}(f_{σ})_{∗β^{σ}−β_{0}^{σ}} ≅ R_{1}(f_{C}, C),

where f_{C} = ∑_{m∈Γ} f(m)[m, 1] in ℂ[C]_{1} is the projection of f to the cone C. Now, we can restate our conjecture (2) in terms of Gorenstein cones:

⨁_{p,q} QH_{st}^{p,q}(Y) ≅ ⨁_{\begin{matrix}p,q \\ (C,C^{∗}) \end{matrix}} R_{1}(ω_{C^{∗}}, C^{∗})_{(p+q−d+dimC^{∗}+1)/2} ⊗ R_{1}(f_{C}, C)_{(q−p+dimC)/2},

where the sum is by all dual faces of the reflexive Gorenstein cones K and K^{∗}. This formula is already supported by Theorem 8.2 in [BDa], which for ample Calabi-Yau hypersurfaces in weighted projective spaces gives a corresponding decomposition of the stringy Hodge numbers (see Remark 5.2 in the next section). A generalization of [BDa, Theorem 8.2] will be proved in Section 7, justifying the above conjecture in the case of ample Calabi-Yau hypersurfaces in Fano toric varieties.

It is known that the string cohomology, which should be the limit of the quantum cohomology ring, of smooth Calabi-Yau hypersurfaces should be the same as the usual cohomology. We also know the property that the quantum cohomology spaces should be isomorphic for the ample Calabi-Yau hypersurface Y and its crepant resolution X. Therefore, it makes sense to compare the above description of QH_{st}^{p,q}(Y) with the description of the cohomology of semiample Calabi-Yau hypersurfaces X in Theorem 4.4. We can see that the right components in the tensor products coincide, by Proposition 4.5 and the definition of R_{1}(f_{C}, C). On the other hand, the left components in QH_{st}^{p,q}(Y) for the ample Calabi-Yau hypersurface Y do not depend on a resolution, while the left components T^{σ}(X) in H^{p,q}(X) for the resolution X depend on the Stanley-Reisner ideal SR(Σ). This hints us to the following definitions:

Definition 4.6. Let C be a Gorenstein cone in a lattice L, subdivided by a fan Σ, and let ℂ[C] and ℂ[C^{∘}], where C^{∘} is the relative interior of C, be the semigroup rings. Define “deformed” ring structures ℂ[C]^{Σ} and ℂ[C]^{Σ} on ℂ[C] and ℂ[C^{∘}], respectively, by the rule: [m_{1}][m_{2}] = [m_{1} + m_{2}] if m_{1}, m_{2} ⊂ σ ∈ Σ, and [m_{1}][m_{2}] = 0, otherwise.

Given g = ∑_{m∈C,degm=1} g(m)[m], where g(m) are the coefficients, let

R_{0}(g, C)^{Σ} = ℂ[C]^{Σ}/Z · ℂ[C]^{Σ}

be the graded ring over the graded module

R_{0}(g, C^{∘})^{Σ} = ℂ[C^{∘}]^{Σ}/Z · ℂ[C^{∘}]^{Σ},

where Z = {∑_{m∈C,degm=1}(m·n)g(m)[m] : n ∈ Hom(L, ℤ)}. Then define R_{1}(g, C)^{Σ} as the image of the natural homomorphism R_{0}(g, C^{∘})^{Σ} ⟶ R_{0}(g, C)^{Σ}.

Remark 4.7. In the above definition, note that if Σ is a trivial subdivision, we recover the spaces R_{0}(g, C) and R_{1}(g, C) introduced earlier. Also, we should mention that the Stanley-Reisner ring of the fan Σ can be naturally embedded into the “deformed” ring ℂ[C]^{Σ}, and this map is an isomorphism when the fan Σ is smooth.

Here is our conjecture about the string cohomology space of semiample Calabi-Yau hypersurfaces in a complete toric variety.

Conjecture 4.8. Let X ⊂ ℙ_{Σ} be a semiample anticanonical nondegenerate hypersurface defined by f ∈ H^{0}(ℙ_{Σ}, O_{ℙ_{Σ}}(X)) ≅ ℂ[K]_{1}, and let ω be a generic element in ℂ[K^{∗}]_{1}, where K^{∗} is the reflexive Gorenstein cone dual to the cone K over the reflexive polytope Δ associated to X. Then there is a natural isomorphism:

H_{st}^{p,q}(X) ≅ ⨁_{C⊆K} R_{1}(ω_{C^{∗}}, C^{∗})_{(p+q−d+dimC^{∗}+1)/2}^{Σ} ⊗ R_{1}(f_{C}, C)_{(q−p+dimC)/2},

where C^{∗} ⊆ K^{∗} is a face dual to C, and where f_{C}, ω_{C^{∗}} denote the projections of f and ω to the respective cones C and C^{∗}. (Here, the superscript Σ denotes the subdivision of K^{∗} induced by the fan Σ.)

Since the dimension of the string cohomology for all crepant partial resolutions should remain the same and should coincide with the dimension of the quantum string cohomology space, we expect that

dimR_{1}(ω_{C^{∗}}, C^{∗})^{Σ} = dimR_{1}(ω_{C^{∗}}, C^{∗}) ,  (4)

which will be shown in Section 6 for a projective subdivision Σ. Conjecture 4.8 will be confirmed by the corresponding decomposition of the stringy Hodge numbers in Section 7. Moreover, in Section 9, we will derive the Chen-Ruan orbifold cohomology as a special case of Conjecture 4.8 for ample Calabi-Yau hypersurfaces in complete simplicial toric varieties.

5. Hodge-Deligne numbers of affine hypersurfaces

Here, we compute the dimensions of the spaces R_{1}(g, C) from the previous section. It follows from Proposition 4.5 that these dimensions are exactly the Hodge-Deligne numbers of the minimal weight space on the middle cohomology of a hypersurface in a torus. An explicit formula in [DKh] and [BDa] for the E-polynomial of a nondegenerate affine hypersurface whose Newton polyhedra is a simplex leads us to the answer for the graded dimension of R_{1}(g, C) when C is a simplicial Gorenstein cone. However, it was very difficult to compute the Hodge-Deligne numbers of an arbitrary nondegenerate affine hypersurface. This was a major technical problem in the proof of mirror symmetry of the stringy Hodge numbers for Calabi-Yau complete intersections in [BBo]. Here, we will present a simple formula for the Hodge-Deligne numbers of a nondegenerate affine hypersurface.

Before we start computing gr.dim.R_{1}(g, C), let us note that for a nondegenerate g ∈ ℂ[C]_{1} (i.e., the corresponding hypersurface in Proj(ℂ[C]) is nondegenerate):

gr.dim.R_{0}(g, C) = S(C, t),

where the polynomial S is the same as in Definition 2.1 of the stringy Hodge numbers. This was shown in [B1, Theorem 4.8 and 2.11] (see also [Bo1]).

When the cone C is simplicial, we already know the formula for the graded dimension of R_{1}(g, C):

Proposition 5.1. Let C be a simplicial Gorenstein cone, and let g ∈ ℂ[C]_{1} be nondegenerate. Then

gr.dim.R_{1}(g, C) = S̃(C, t)

where S̃(C, t) = ∑_{C_{1}⊆C} S(C_{1}, t)(−1)^{dimC−dimC_{1}}.

Proof. The polynomial S̃(C, t) was introduced with a slightly different notation in [BDa, Definition 8.1] for a lattice simplex. One can check that S̃(C, t) in this proposition is equivalent to the one in [BDa, Corollary 6.6]. From the previous section and [B1, Proposition 9.2], we know that

R_{1}(g, C) ≅ Gr_{F} W_{dimZ_{g}}H^{dimZ_{g}}(Z_{g}),

where Z_{g} is the nondegenerate affine hypersurface determined by g in the maximal torus of Proj(ℂ[C]). By [BDa, Proposition 8.3],

E(Z_{g}; u, v) = \frac{(uv − 1)^{dimC−1} + (−1)^{dimC}}{uv} + (−1)^{dimC} ∑_{\begin{matrix}C_{1}⊆C \\ dimC_{1}>1 \end{matrix}} \frac{u^{dimC_{1}}}{uv} S̃(C_{1}, u^{−1}v).

Now, note that the coefficients e^{p,q}(Z_{g}) at the monomials u^{p}v^{q} with p + q = dimZ_{g} are related to the Hodge-Deligne numbers by the calculations in [DKh]:

e^{p,q}(Z_{g}) = (−1)^{dimC}h^{p,q}(H^{dimZ_{g}}(Z_{g})) + (−1)^{p}δ_{pq}C_{dimC−1}^{p},

where δ_{pq} is the Kronecker symbol and C_{dimC−1}^{p} is the binomial coefficient. Comparing this with the above formula for E(Z_{g}; u, v), we deduce the result. □

Remark 5.2. By the above proposition, we can see that [BDa, Theorem 8.2] gives a decomposition of the stringy Hodge numbers of ample Calabi-Yau hypersurfaces in weighted projective spaces in correspondence with Conjecture 4.8.

Next, we generalize the polynomials S̃(C, t) from Proposition 5.1 to nonsimplicial Gorenstein cones in such a way that they would count the graded dimension of R_{1}(g, C).

Definition 5.3. Let C be a Gorenstein cone in a lattice L. Then set

S̃(C, t) := ∑_{C_{1}⊆C} S(C_{1}, t)(−1)^{dimC−dimC_{1}}G([C_{1}, C], t),

where G is a polynomial (from Definition 11.1 in the Appendix) for the partially ordered set [C_{1}, C] of the faces of C that contain C_{1}.

Remark 5.4. It is not hard to show that the polynomial S̃(C, t) satisfies the duality

S̃(C, t) = t^{dimC} S̃(C, t^{−1})

based on the duality properties of S and the definition of G-polynomials. However, the next result and Proposition 4.5 imply this fact.

Proposition 5.5. Let C be a Gorenstein cone, and let g ∈ ℂ[C]_{1} be nondegenerate. Then

gr.dim.R_{1}(g, C) = S̃(C, t).

Proof. As in the proof of Proposition 5.1, we consider a nondegenerate affine hypersurface Z_{g} determined by g in the maximal torus of Proj(ℂ[C]). Then [BBo, Theorem 3.18] together with the definition of S gives

E(Z_{g}; u, v) = \frac{(uv − 1)^{dimC−1}}{uv} + \frac{(−1)^{dimC}}{uv} ∑_{C_{2}⊆C} B([C_{2}, C]^{∗}; u, v)S(C_{2}, vu^{−1})u^{dimC_{2}},

where the polynomials B are from Definition 11.3. We use Lemma 11.4 and Definition 5.3 to rewrite this as

E(Z_{g}; u, v) = \frac{(uv − 1)^{dimC−1}}{uv} + \frac{(−1)^{dimC}}{uv} ×

× ∑_{C_{2}⊆C_{1}⊆C} u^{dimC_{2}}S(C_{2}, u^{−1}v)G([C_{2}, C_{1}], u^{−1}v)(−u)^{dimC_{1}−dimC_{2}}G([C_{1}, C]^{∗}, uv)

= \frac{(uv − 1)^{dimC−1}}{uv} + \frac{(−1)^{dimC}}{uv} ∑_{C_{1}⊆C} u^{dimC_{1}} S̃(C_{1}, u^{−1}v)G([C_{1}, C]^{∗}, uv).

The definition of G-polynomials assures that the degree of u^{dimC_{1}}G([C_{1}, C]^{∗}, uv) is at most dimC with the equality only when C_{1} = C. Therefore, the graded dimension of R_{1}(g, C) can be read off the same way as in the proof of Proposition 5.1 from the coefficients at total degree dimC − 2 in the above sum. □

6. “Deformed” rings and modules

While this section may serve as an invitation to a new theory of “deformed” rings and modules, the goal here is to prove the equality (4), by showing that the graded dimension formula of Proposition 5.5 holds for the spaces R_{1}(g, C)^{Σ} from Definition 4.6. To prove the formula we use the recent work of Bressler and Lunts (see [BreL], and also [BaBrFK]). This requires us to first study Cohen-Macaulay modules over the deformed semigroup rings ℂ[C]^{Σ}.

First, we want to generalize the nondegeneracy notion:

Definition 6.1. Let C be a Gorenstein cone in a lattice L, subdivided by a fan Σ. Given g = ∑_{m∈C,degm=1} g(m)[m], get

g_{j} = ∑_{m∈C,degm=1}(m · n_{j})g(m)[m], for j = 1, …, dimC

where {n_{1}, …, n_{dimC}} ⊂ Hom(L, ℤ) descends to a basis of Hom(L, ℤ)/C^{⊥}. The element g is called Σ-regular (nondegenerate) if {g_{1}, …, g_{dimC}} forms a regular sequence in the deformed semigroup ring ℂ[C]^{Σ}.

Remark 6.2. When Σ is a trivial subdivision, [B1, Theorem 4.8] shows that the above definition is consistent with the previous notion of nondegeneracy corresponding to the transversality of a hypersurface to torus orbits.

Theorem 6.3. (i) The ring ℂ[C]^{Σ} and its module ℂ[C^{∘}]^{Σ} are Cohen-Macaulay. (ii) A generic element g ∈ ℂ[C]_{1} is Σ-regular. Moreover, for a generic g the sequence {g_{1}, …, g_{dimC}} from Definition 6.1 is ℂ[C^{∘}]^{Σ}-regular. (iii) If g ∈ ℂ[C]_{1} is Σ-regular, then the sequence {g_{1}, …, g_{dimC}} is ℂ[C^{∘}]^{Σ}-regular.

Proof. Part (ii) follows from the proofs of Propositions 3.1 and 3.2 in [Bo1]. The reader should notice that the proofs use degenerations defined by projective simplicial subdivisions, and any fan admits such a subdivision.

Then, part (ii) implies (i), by the definition of Cohen-Macaulay, while part (iii) follows from (i) and Proposition 21.9 in [E]. □

As a corollary of Theorem 6.3, we get the following simple description of Σregular elements:

Lemma 6.4. An element g ∈ ℂ[C]_{1} is Σ-regular, if and only if its restriction to all maximum-dimensional cones C′ ∈ Σ(dimC) is nondegenerate in ℂ[C′].

Proof. Since ℂ[C]^{Σ} is Cohen-Macaulay, the regularity of a sequence is equivalent to the quotient by the sequence having a finite dimension, by [Mat, Theorem 17.4].

One can check that ℂ[C]^{Σ} is filtered by the modules R_{k} defined as the span of [m] such that the minimum cone that contains m has dimension at least k. The kth graded quotient of this filtration is the direct sum of ℂ[C_{1}^{∘}] by all k-dimensional cones C_{1} of Σ. If g is nondegenerate for every cone of maximum dimension, then its projection to any cone C_{1} is nondegenerate, and Theorem 6.3 shows that it is nondegenerate for each ℂ[C_{1}^{∘}]. Then by decreasing induction on k one shows that R_{k}/{g_{1}, …, g_{dimC}}R_{k} is finite-dimensional.

In the other direction, it is easy to see that for every C′ ∈ Σ the ℂ[C]^{Σ}-module ℂ[C′] is a quotient of ℂ[C]^{Σ}, which gives a surjection

ℂ[C]^{Σ}/{g_{1}, …, g_{dimC}}ℂ[C]^{Σ} ⟶ ℂ[C′]/{g_{1}|_{C′}, …, g_{dimC}|_{C′}}ℂ[C′] ⟶ 0.

□

The above lemma implies that the property of Σ-regularity is preserved by the restrictions:

Lemma 6.5. Let C be a Gorenstein cone in a lattice L, subdivided by a fan Σ. If g ∈ ℂ[C]_{1} is Σ-regular, then g ∈ ℂ[C_{1}]_{1} is Σ-regular for all faces C_{1} ⊆ C.

Proof. Let g ∈ ℂ[C]_{1} be Σ-regular. By Lemma 6.4, the restriction g_{C′} is nondegenerate in ℂ[C′] for all C′ ∈ Σ(dimC). Since the property of nondegeneracy associated with a hypersurface is preserved by the restrictions, g_{C′_{1}} is nondegenerate in ℂ[C′_{1}] for all C′_{1} ∈ Σ(dimC_{1}) contained in C_{1}. Applying Lemma 6.4 again, we deduce the result. □

The next result generalizes [B1, Proposition 9.4] and [Bo1, Proposition 3.6].

Proposition 6.6. Let g ∈ ℂ[C]_{1} be Σ-regular, then R_{0}(g, C)^{Σ} and R_{0}(g, C^{∘})^{Σ} have graded dimensions S(C, t) and t^{k}S(C, t^{−1}), respectively, and there exists a nondegenerate pairing

⟨ , ⟩ : R_{0}(g, C)_{k}^{Σ} × R_{0}(g, C^{∘})_{dimC−k}^{Σ} → R_{0}(g, C^{∘})_{dimC}^{Σ} ≅ ℂ,

induced by the multiplicative R_{0}(g, C)^{Σ}-module structure.

Proof. It is easy to see that the above statement is equivalent to saying that ℂ[C^{∘}]^{Σ} is the canonical module for ℂ[C]^{Σ}. When Σ consists of the faces of C only, this is well-known (cf. [D]). To deal with the general case, we will heavily use the results of [E], Chapter 21.

We denote A = ℂ[C]^{Σ}. For every cone C_{1} of Σ the vector spaces ℂ[C_{1}] and ℂ[C_{1}^{∘}] are equipped with the natural A-module structures. By Proposition 21.10 of [E], modified for the graded case, we get

Ext_{A}^{i}(ℂ[C_{1}], w_{A})≅\{ \begin{matrix}ℂ[C_{1}^{∘}], & i = codim(C_{1}) \\ 0, & i ≠ codim(C_{1}) \end{matrix}

where w_{A} is the canonical module of A.

Consider now the complex F of A-modules

0 ⟶ F^{0} ⟶ F^{1} ⟶ ⋯ ⟶ F^{d} ⟶ 0

where

F^{n} = ⨁_{C_{1}∈Σ,codim(C_{1})=n} ℂ[C_{1}]

and the differential is a sum of the restriction maps with signs according to the orientations. The nontrivial cohomology of F is located at F^{0} and equals ℂ[C^{∘}]^{Σ}. Indeed, by looking at each graded piece separately, we see that the cohomology occurs only at F^{0}, and then the kernel of the map to F^{1} is easy to describe. We can now use the complex F and the description of Ext_{A}^{i}(ℂ[C_{1}], w_{A}) to try to calculate Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A}). The resulting spectral sequence degenerates immediately, and we conclude that Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A}) has a filtration such that the associated graded module is naturally isomorphic to

⨁_{C_{1}∈Σ} ℂ[C_{1}^{∘}].

By duality of maximal Cohen-Macaulay modules (see [E]), it suffices to show that Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A})≅A, but the above filtration only establishes that it has the correct graded pieces, so extra arguments are required. Let C′ be a cone of Σ of maximum dimension. We observe that F contains a subcomplex F′ such that

F^{′n} = ⨁_{C_{1}⊆C′} ℂ[C_{1}].

Similar to the case of F, the cohomology of F′ occurs only at F^{′0} and equals ℂ[C^{′∘}]. By snake lemma, the cohomology of F/F′ also occurs at the zeroth spot and equals ℂ[C^{∘}]^{Σ}/ℂ[C^{′∘}]. By looking at the spectral sequences again, we see that

Ext^{>0}(ℂ[C^{∘}]^{Σ}/ℂ[C^{′∘}], w_{A}) = 0

and we have a grading preserving surjection

Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A}) ⟶ Hom_{A}(ℂ[C^{′∘}], w_{A}) ⟶ 0.

Since Hom_{A}(ℂ[C′], w_{A})≅ℂ[C^{′∘}], duality of maximal Cohen-Macaulay modules over A shows that

Hom_{A}(ℂ[C^{′∘}], w_{A})≅ℂ[C′]

so for every m ∈ C′ the element [m] of A does not annihilate the degree zero element of Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A}). By looking at all C′ together, this shows that

Hom_{A}(ℂ[C^{∘}]^{Σ}, w_{A})≅A

which finishes the proof. □

Proposition 6.7. Let g ∈ ℂ[C]_{1} be Σ-regular, then the pairing ⟨ , ⟩ induces a symmetric nondegenerate pairing { , } on R_{1}(g, C)^{Σ}, defined by

{x, y} = ⟨x, y′⟩

where y′ is an element of R_{0}(g, C^{∘})^{Σ} that maps to y.

Proof. The nondegeneracy of the pairing { , } follows from that of ⟨ , ⟩. The pairing is symmetric, because it comes from the commutative product on ℂ[C^{∘}]^{Σ}. □

Theorem 6.8. Let C be a Gorenstein cone subdivided by a projective fan Σ. If g ∈ ℂ[C]_{1} is Σ-regular, then the graded dimension of R_{1}(g, C)^{Σ} is S̃(C, t).

Proof. We will use the description of Bressler and Lunts [BreL] of locally free flabby sheaves on the finite ringed topological space associated to the cone C. We recall here the basic definitions. Consider the set P of all faces of the cone C. It is equipped with the topology in which open sets are subfans, i.e. the sets of faces closed under the operation of taking a face. Bressler and Lunts define a sheaf A of graded commutative rings on P whose sections over each open set is the ring of continuous piecewise polynomial functions on the union of all strata of this set. The grading of linear functions will be set to 1, contrary to the convention of [BreL].

They further restrict their attention to the sheaves F of A-modules on P that satisfy the following conditions.

• For every face C_{1} of C, sections of F over the open set that corresponds to the union of all faces of C_{1} is a free module over the ring of polynomial functions on C_{1}.

• F is flabby, i.e. all restriction maps are surjective.

We will use the following crucial result.

Theorem 6.9. [BreL] Every sheaf F that satisfies the above two properties is isomorphic to a direct sum of indecomposable graded sheaves L_{C_{1}}t^{i}, where C_{1} is a face of C and t^{i} indicates a shift in grading. For each indecomposable sheaf L_{C_{1}} the space of global sections Γ(P, L_{C_{1}}) is a module over the polynomial functions on C of the graded rank G([C_{1}, C]^{∗}, t) where [C_{1}, C] denotes the Eulerian subposet of P that consists of all faces of C that contain C_{1}.

Now let us define a sheaf B(g) on P whose sections over the open subset I ∈ P are ℂ[∩_{i∈I}C_{i}]^{Σ}. It is clearly a flabby sheaf, which can be given a grading by deg( ). Moreover, B(g) can be given a structure of a sheaf of A modules as follows. Every linear function φ on a face C_{1} defines a logarithmic derivative

∂_{φ}g := ∑_{m∈C_{1},degm=1} ϕ(m)g(m)[m]

of g, which is an element of the degree 1 in ℂ[C_{1}]^{Σ}. Then the action of φ is given by the multiplication by ∂_{φ}g, and this action is extended to all polynomial functions on the cone C_{1}. Similar construction clearly applies to continuous piecewise polynomial functions for any open set of P.

Proposition 6.6 assures that B(g) satisfies the second condition of Bressler and Lunts, and can therefore be decomposed into a direct sum of L_{C_{1}}t^{i} for various C_{1} and i. The definition of R_{1}(g, C)^{Σ} implies that its graded dimension is equal to the graded rank of the stalk of B(g) at the point C ∈ P. Since the graded rank of B is S(C, t), we conclude that

S(C, t) = ∑_{C_{1}⊆C} gr.dim.R_{1}(g_{C_{1}}, C_{1})^{Σ}G([C_{1}, C]^{∗}, t).

To finish the proof of Theorem 6.8, it remains to apply Lemma 11.2. □

7. Decomposition of stringy Hodge numbers for hypersurfaces

In this section, we prove a generalization of [BDa, Theorem 8.2] for all Calabi-Yau hypersurfaces, which gives a decomposition of the stringy Hodge numbers of the hypersurfaces. First, we recall a formula for the stringy Hodge numbers of Calabi-Yau hypersurfaces obtained in [BBo]. Then using a bit of combinatorics, we rewrite this formula precisely to the form of [BDa, Theorem 8.2] with S̃ defined in the previous section.

The stringy Hodge numbers of a Calabi-Yau complete intersection have been calculated in [BBo] in terms of the numbers of integer points inside multiples of various faces of the reflexive polytopes Δ and Δ^{∗} as well as some polynomial invariants of partially ordered sets. A special case of the main result in [BBo] is the following description of the stringy E-polynomials of Calabi-Yau hypersurfaces.

Theorem 7.1. [BBo] Let K ⊂ M ⊕ ℤ be the Gorenstein cone over a reflexive polytope Δ ⊂ M. For every (m, n) ∈ (K, K^{∗}) with m · n = 0 denote by x(m) the minimum face of K that contains m and by x^{∗}(n) the dual of the minimum face of K^{∗} that contains n. Also, set A_{(m,n)}(u, v) be

\frac{(−1)^{dim(x^{∗}(n))}}{uv} (v − u)^{dim(x(m))}(uv − 1)^{d+1−dim(x^{∗}(n))}B([x(m), x^{∗}(n)]^{∗}; u, v)

where the function B is defined in Definition 11.3 in the Appendix. Then

E_{st}(Y; u, v) = ∑_{(m,n)∈(K,K^{∗}),m·n=0} (\frac{u}{v})^{deg m} A_{(m,n)}(u, v) (\frac{1}{uv} )^{deg n}

for an ample nondegenerate Calabi-Yau hypersurface Y in ℙ_{Δ} = Proj(ℂ[K]).

The mirror duality E_{st}(Y; u, v) = (−u)^{d−1}E_{st}(Y^{∗}; u^{−1}, v) was proved in [BBo] as the immediate corollary of the above formula and the duality property B(P; u, v) = (−u)^{rkP} B(P^{∗}; u^{−1}, v). It was not noticed there that Lemma 11.2 allows one to rewrite the B-polynomials in terms of G-polynomials, which we will now use to give a formula for the E_{st}(Y; u, v), explicitly obeying the mirror duality. The next result is a generalization of Theorem 8.2 in [BDa] with S̃ from Definition 5.3.

Theorem 7.2. Let Y be an ample nondegenerate Calabi-Yau hypersurface in ℙ_{Δ} = Proj(ℂ[K]). Then

E_{st}(Y; u, v) = ∑_{C⊆K}(uv)^{−1}(−u)^{dimC} S̃(C, u^{−1}v)S̃(C^{∗}, uv).

Proof. First, observe that the formula for E_{st}(Y; u, v) from Theorem 7.1 can be written as

∑_{m,n,C_{1},C_{2}} \frac{(−1)^{dimC_{2}^{∗}}}{uv} (v−u)^{dim(C_{1})}B([C_{2}, C_{1}^{∗}]; u, v)(uv−1)^{dimC_{2}} (\frac{u}{v})^{degm} (\frac{1}{uv} )^{degn}

where the sum is taken over all pairs of cones C_{1} ⊆ K, C_{2} ⊆ K^{∗} that satisfy C_{1} · C_{2} = 0 and all m and n in the relative interiors of C_{1} and C_{2}, respectively. We use the standard duality result (see Definition 2.1)

∑_{n∈int(C)} t^{−deg(n)} = (t − 1)^{−dimC}S(C, t)

to rewrite the above formula as

\frac{1}{uv} ∑_{C_{1}·C_{2}=0}(−1)^{dim(C_{2}^{∗})}u^{dimC_{1}}B([C_{2}, C_{1}^{∗}]; u, v)S(C_{1}, u^{−1}v)S(C_{2}, uv).

Then apply Lemma 11.4 to get

E_{st}(Y; u, v) = \frac{1}{uv} ∑_{C∈K} ∑_{C_{1}⊆C,C_{2}⊆C^{∗}}(−1)^{dim(C_{2}^{∗})}u^{dimC_{1}} ×

× G([C_{1}, C], u^{−1}v)(−u)^{dimC_{1}^{∗}−dimC^{∗}}G([C_{2}, C^{∗}], uv)S(C_{1}, u^{−1}v)S(C_{2}, uv).

It remains to use Definition 5.3. □

8. String cohomology construction via intersection cohomology

Here, we construct the string cohomology space for ℚ-Gorenstein toroidal varieties, satisfying the assumption of Proposition 2.6. The motivation for this construction comes from the conjectural description of the string cohomology space for ample Calabi-Yau hypersurfaces and a look at the formula in [BDa, Theorem 6.10] for the stringy E-polynomial of a Gorenstein variety with abelian quotient singularities. This immediately leads to a decomposition of the string cohomology space as a direct sum of tensor products of the usual cohomology of a closure of a strata with the spaces R_{1}(g, C) from Proposition 5.1. Then the property that the intersection cohomology of an orbifold is naturally isomorphic to the usual cohomology leads us to the construction of the string cohomology space for ℚ-Gorenstein toroidal varieties. We show that this space has the dimension prescribed by Definition 2.1 for Gorenstein complete toric varieties and the nondegenerate complete intersections in them.

Conjectural Definition 8.1. Let X = ⋃_{i∈I} X_{i} be a Gorenstein complete variety with quotient abelian singularities, satisfying the assumption of Proposition 2.6. The stringy Hodge spaces of X are naturally isomorphic to

H_{st}^{p,q}(X) ≅ ⨁_{\begin{matrix}i∈I \\ k≥0 \end{matrix}} H^{p−k,q−k}(\overline{X}_{i}) ⊗ R_{1}(ω_{σ_{i}}, σ_{i})_{k},

where σ_{i} is the Gorenstein simplicial cone of the singularity along the strata X_{i}, and ω_{σ_{i}} ∈ ℂ[σ_{i}]_{1} are nondegenerate such that, for σ_{j} ⊂ σ_{i}, ω_{σ_{i}} maps to ω_{σ_{j}} by the natural projection ℂ[σ_{i}] ⟶ ℂ[σ_{j}].

Remark 8.2. Since \overline{X}_{i} is a compact orbifold, the coefficient e^{p,q}(\overline{X}_{i}) at the monomial u^{p}v^{q} in the polynomial E(\overline{X}_{i}; u, v) is equal to (−1)^{p+q}h^{p,q}(\overline{X}_{i}), by Remark 2.2. Therefore, Proposition 5.1 shows that the above decomposition of H_{st}^{p,q}(X) is in correspondence with [BDa, Theorem 6.10], and the dimensions h_{st}^{p,q}(X) coincide with those from Definition 2.3.

Since we expect that the usual cohomology must be replaced in Definition 8.1 by the intersection cohomology for Gorenstein toroidal varieties, the next result is a natural generalization of Theorem 6.10 in [BDa].

Theorem 8.3. Let X = ⋃_{i∈I} X_{i} be a Gorenstein complete toric variety or a nondegenerate complete intersection of Cartier hypersurfaces in the toric variety, where the stratification is induced by the torus orbits. Then

E_{st}(X; u, v) = ∑_{i∈I} E_{int}(\overline{X}_{i}; u, v) · S̃(σ_{i}, uv),

where σ_{i} is the Gorenstein cone of the singularity along the strata X_{i}.

Proof. Similarly to Corollary 3.17 in [BBo], we have

E_{int}(\overline{X}_{i}; u, v) = ∑_{X_{j}⊆\overline{X}_{i}} E(X_{i}; u, v) · G([σ_{i} ⊆ σ_{j}]^{∗}, uv).

Hence, we get

∑_{i∈I} E_{int}(\overline{X}_{i}; u, v) · S̃(σ_{i}, uv) = ∑_{i∈I} ∑_{X_{j}⊆\overline{X}_{i}} E(X_{j}; u, v)G([σ_{i} ⊆ σ_{j}]^{∗}, uv)S̃(σ_{i}, uv)

= ∑_{j∈I} E(X_{j}; u, v)( ∑_{σ_{i}⊆σ_{j}} G([σ_{i} ⊆ σ_{j}]^{∗}, uv)S̃(σ_{i}, uv)) = ∑_{j∈I} E(X_{j}; u, v)S(σ_{j}, uv),

where at the last step we have used the formula for S̃ and Lemma 11.2. □

Based on the above theorem, we propose the following conjectural description of the stringy Hodge spaces for ℚ-Gorenstein toroidal varieties.

Conjectural Definition 8.4. Let X = ⋃_{i∈I} X_{i} be a ℚ-Gorenstein d-dimensional complete toroidal variety, satisfying the assumption of Proposition 2.6. The stringy Hodge spaces of X are defined by:

H_{st}^{p,q}(X) := ⨁_{\begin{matrix}i∈I \\ k≥0 \end{matrix}} H_{int}^{p−k,q−k}(\overline{X}_{i}) ⊗ R_{1}(ω_{σ_{i}}, σ_{i})_{k},

where σ_{i} is the Gorenstein cone of the singularity along the strata X_{i}, and ω_{σ_{i}} ∈ ℂ[σ_{i}]_{1} are nondegenerate such that, for σ_{j} ⊂ σ_{i}, ω_{σ_{i}} maps to ω_{σ_{j}} by the natural projection ℂ[σ_{i}] ⟶ ℂ[σ_{j}]. Here, p, q are rational numbers from [0, d], and we assume that H_{int}^{p−k,q−k}(\overline{X}_{i}) = 0 if p − k or q − k is not a non-negative integer.

Remark 8.5. Toric varieties and nondegenerate complete intersections of Cartier hypersurfaces have the stratification induced by the torus orbits which satisfies the assumptions in the above definition.

9. String cohomology vs. Chen-Ruan orbifold cohomology

Our next goal is to compare the two descriptions of string cohomology for Calabi-Yau hypersurfaces to the Chen-Ruan orbifold cohomology. Using the work of [P], we will show that in the case of ample orbifold Calabi-Yau hypersurfaces the three descriptions coincide. We refer the reader to [ChR] for the orbifold cohomology theory and only use [P] in order to describe the orbifold cohomology for complete simplicial toric varieties and Calabi-Yau hypersurfaces in Fano simplicial toric varieties.

From Theorem 1 in [P, Section 4] and the definition of the orbifold Dolbeault cohomology space we deduce:

Proposition 9.1. Let ℙ_{Σ} be a d-dimensional complete simplicial toric variety. Then the orbifold Dolbeault cohomology space of ℙ_{Σ} is

H_{orb}^{p,q}(ℙ_{Σ}; ℂ) ≅ ⨁_{\begin{matrix}σ∈Σ \\ l∈ℚ \end{matrix}} H^{p−l,q−l}(V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt,

where T (σ)_{l} = {∑_{ρ⊂σ} a_{ρ}[e_{ρ}] ∈ N : a_{ρ} ∈ (0, 1), ∑_{ρ⊂σ} a_{ρ} = l} (when σ = 0, set l = 0 and T (σ)_{l} = ℂ), and V (σ) is the closure of the torus orbit corresponding to σ ∈ Σ. Here, p and q are rational numbers in [0, d], and H^{p−l,q−l}(V (σ)) = 0 if p−l or q − l is not integral. (The elements of ⊕_{0≠σ∈Σ,l}T (σ)_{l} correspond to the twisted sectors.)

In order to compare this result to the description in Definition 8.4, we need to specify the ω_{σ_{i}} for the toric variety ℙ_{Σ}. The stratification of ℙ_{Σ} is given by the torus orbits: ℙ_{Σ} = ∪_{σ∈Σ}𝕋_{σ}. The singularity of the variety ℙ_{Σ} along the strata 𝕋_{σ} is given by the cone σ, so we need to specify a nondegenerate ω_{σ} ∈ ℂ[σ]_{1} for each σ ∈ Σ. If ω_{σ} = ∑_{ρ⊂σ} ω_{ρ}[e_{ρ}] with ω_{ρ} ≠ 0, then one can deduce that ω_{σ} is nondegenerate using Remark 6.2 and the fact that the nondegeneracy of a hypersurface in a complete simplicial toric variety (in this case, it corresponds to a simplex) is equivalent to the nonvanishing of the logarithmic derivatives simultaneously. So, picking any nonzero coefficients ω_{ρ} for each ρ ∈ Σ(1) gives a nondegenerate ω_{σ} ∈ ℂ[σ]_{1} satisfying the condition of Definition 8.4. For such ω_{σ}, note that the set Z = {∑_{ρ⊂σ}(e_{ρ} ·m)ω_{ρ}e_{ρ} : m ∈ Hom(N, ℤ)} is a linear span of e_{ρ} for ρ ⊂ σ. Hence,

R_{0}(ω_{σ}, σ)_{l} = (ℂ[σ]/Z · ℂ[σ])_{l} ≅ ⨁_{t∈T̃(σ)_{l}} ℂt,

where T̃(σ)_{l} = {∑_{ρ⊂σ} a_{ρ}e_{ρ} ∈ N : a_{ρ} ∈ [0, 1), ∑_{ρ⊂σ} a_{ρ} = l}, and

R_{1}(ω_{σ}, σ)_{l} ≅ ⨁_{t∈T (σ)_{l}} ℂt.

This shows that the orbifold Dolbeault cohomology for complete simplicial toric varieties can be obtained as a special case of the description of string cohomology in Definition 8.4.

We will now explain how the parameter ω should be related to the complexified Kähler class. We do not have the definition of the ”orbifold” Kähler cone even for simplicial toric varieties. However, we know the Kähler classes in H^{2}(ℙ_{Σ}, ℝ).

Proposition 9.2. Let ℙ_{Σ} be a projective simplicial toric variety, then H^{2}(ℙ_{Σ}, ℝ) ≅ P L(Σ)/M_{ℝ}, where P L(Σ) is the set of Σ-piecewise linear functions φ : N_{ℝ} ⟶ ℝ, which are linear on each σ ∈ Σ. The Kähler cone K(Σ) ⊂ H^{2}(ℙ_{Σ}, ℝ) of ℙ_{Σ} consists of the classes of the upper strictly convex Σ-piecewise linear functions.

One may call K(Σ) the “untwisted” part of the orbifold Kähler cone. So, we can introduce the untwisted complexified Kähler space of the complete simplicial toric variety:

K_{ℂ}^{untwist}(ℙ_{Σ}) = {ω ∈ H^{2}(ℙ_{Σ}, ℂ) : Im(ω) ∈ K(Σ)}/imH^{2}(ℙ_{Σ}, ℤ).

Its elements may be called the untwisted complexified Kähler classes. We can find a generic enough ω ∈ K_{ℂ}^{untwist}(ℙ_{Σ}) represented by a complex valued Σ-piecewise linear function φ_{ω} : N_{ℂ} ⟶ ℂ such that φ_{ω}(e_{ρ}) ≠ 0 for ρ ∈ Σ(1). Setting ω_{ρ} = exp(φ_{ω}(e_{ρ})) produces our previous parameters ω_{σ} for σ ∈ Σ. This is how we believe ω_{σ} should relate to the complexified Kähler classes, up to perhaps some instanton corrections.

We next turn our attention to the case of an ample Calabi-Yau hypersurface Y in a complete simplicial toric variety ℙ_{Σ}. Section 4.2 in [P] works with a generic nondegenerate anticanonical hypersurface. However, one can avoid the use of Bertini’s theorem and state the result without “generic”. It is shown that the nondegenerate anticanonical hypersurface X is a suborbifold of ℙ_{Σ}, the twisted sectors of Y are obtained by intersecting with the closures of the torus orbits and the degree shifting numbers are the same as for the toric variety ℙ_{Σ}. Therefore, we conclude:

Proposition 9.3. Let Y ⊂ ℙ_{Σ} be an ample Calabi-Yau hypersurface in a complete simplicial toric variety. Then

H_{orb}^{p,q}(Y; ℂ) ≅ ⨁_{\begin{matrix}σ∈Σ \\ l∈ℤ \end{matrix}} H^{p−l,q−l}(Y ∩ V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt,

where T (σ)_{l} = {∑_{ρ⊂σ} a_{ρ}[e_{ρ}] ∈ N : a_{ρ} ∈ (0, 1), ∑_{ρ⊂σ} a_{ρ} = l} (when σ = 0, set l = 0 and T (σ)_{l} = ℂ).

As in the case of the toric variety, we pick ω_{σ} = ∑_{ρ⊂σ} ω_{ρ}e_{ρ} with ω_{ρ} ≠ 0. Then, by the above proposition,

H_{st}^{p,q}(Y) ≅ H_{orb}^{p,q}(Y; ℂ).

We now want to show that the description in Proposition 9.3 is equivalent to the one in Conjecture 4.8. First, note that the proper faces C^{∗} of the Gorenstein cone K^{∗} in Conjecture 4.8 one to one correspond to the cones σ ∈ Σ. Moreover, the rings ℂ[C^{∗}] ≅ ℂ[σ] are isomorphic in this correspondence. If we take ω ∈ ℂ[K^{∗}]_{1}^{Σ} to be [0, 1] + ∑_{ρ∈Σ(1)} ω_{ρ}[e_{ρ}, 1], then ω is Σ-regular and

R_{1}(C^{∗}, ω_{C^{∗}})_{l} ≅ R_{1}(ω_{σ}, σ)_{l} ≅ ⊕_{t∈T (σ)_{l}}ℂt.

On the other hand, the Hodge component H^{p−l,q−l}(Y ∩ V (σ)) decomposes into the direct sum

H_{toric}^{p−l,q−l}(Y ∩ V (σ)) ⊕ H_{res}^{p−l,q−l}(Y ∩ V (σ))

of the toric and residue parts. Since Y ∩ V (σ) is an ample hypersurface, from [BC, Theorem 11.8] and Section 4 it follows that

H_{res}^{p−l,q−l}(Y ∩ V (σ)) ≅ R_{1}(f_{C}, C)_{q−l+1},

where C ⊂ K is the face dual to C^{∗} which corresponds to σ, p + q − 2l = dimY ∩ V (σ) = d − dimσ − 1 = d − dimC^{∗} − 1. If p + q − 2l ≠ d − dimC^{∗} − 1, then H_{res}^{p−l,q−l}(Y ∩ V (σ)) = 0. Hence, we get

⨁_{\begin{matrix}σ∈Σ \\ l∈ℤ \end{matrix}} H_{res}^{p−l,q−l}(Y ∩ V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt ≅ ⨁_{0≠C⊆K} R_{1}(ω_{C^{∗}}, C^{∗})_{a}^{Σ} ⊗ R_{1}(f_{C}, C)_{b},

where a = (p + q − d + dimC^{∗} + 1)/2 and b = (q − p + dimC)/2. We are left to show that

⨁_{\begin{matrix}σ∈Σ \\ l∈ℤ \end{matrix}} H_{toric}^{p−l,p−l}(Y ∩ V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt ≅ R_{1}(ω, K^{∗})_{p+1}^{Σ}.  (5)

Notice that the dimensions of the spaces on both sides coincide, so it suffices to construct a surjective map between them. This will follow from the following proposition.

Proposition 9.4. Let ℙ_{Σ} = Proj(ℂ[K]) be the Gorenstein Fano simplicial toric variety, where K as above. Then there is a natural isomorphism:

H_{st}^{p,p}(ℙ_{Σ}) ≅ ⨁_{\begin{matrix}σ∈Σ \\ l∈ℤ \end{matrix}} H^{p−l,p−l}(V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt ≅ R_{0}(ω, K^{∗})_{p}^{Σ},

where ω = [0, 1] + ∑_{ρ∈Σ(1)} ω_{ρ}[e_{ρ}, 1] with ω_{ρ} ≠ 0.

Proof. First, observe that the dimensions of the spaces in the isomorphisms coincide by our definition of string cohomology, Proposition 6.6 and [BDa, Theorem 7.2]. So, it suffices to construct a surjective map between them.

We know the cohomology ring of the toric variety:

H^{∗}(V (σ)) ≅ ℂ[D_{ρ} : ρ ∈ Σ(1), ρ + σ ∈ Σ(dimσ + 1)]/(P (V (σ)) + SR(V (σ))),

where

SR(V (σ)) = ⟨D_{ρ_{1}} ⋯ D_{ρ_{k}} : {e_{ρ_{1}}, …, e_{ρ_{k}}} ⊄ τ for all σ ⊂ τ ∈ Σ(dimσ + 1)⟩

is the Stanley-Reisner ideal, and

P (V (σ)) = ⟨ ∑_{ρ∈Σ(1),ρ+σ∈Σ(dimσ+1)}⟨m, e_{ρ}⟩D_{ρ} : m ∈ M ∩ σ^{⊥}⟩.

Define the maps from H^{p−l,p−l}(V (σ)) ⊗ ⨁_{t∈T (σ)_{l}} ℂt to R_{0}(ω, K^{∗})^{Σ} by sending D_{ρ_{1}} ⋯ D_{ρ_{p−l}} ⊗ t to ω_{ρ_{1}}[e_{ρ_{1}}] ⋯ ω_{ρ_{p−l}}[e_{ρ_{p−l}}] · t ∈ ℂ[N]^{Σ}. One can easily see that these maps are well defined. To finish the proof we need to show that the images cover R_{0}(ω, K^{∗})^{Σ}. Every lattice point [n] in the boundary of K^{∗} lies in the relative interior of a face C ⊂ K^{∗}, and can be written as a linear combination of the minimal integral generators of C:

[n] = ∑_{[e_{ρ},1]∈C}(a_{ρ} + b_{ρ})[e_{ρ}, 1],

where a_{ρ} ∈ (0, 1) and b_{ρ} are nonnegative integers. Let C′ ⊆ C be the cone spanned by those [e_{ρ}, 1] for which a_{ρ} ≠ 0. The lattice point ∑_{[e_{ρ},1]∈C′}(a_{ρ})[e_{ρ}, 1] projects to one of the elements t from T (σ)_{l} for some l and σ corresponding to C′. Using the relations ∑_{ρ∈Σ(1)} ω_{ρ}⟨m, e_{ρ}⟩[e_{ρ}, 1] in the ring R_{0}(ω, K^{∗})^{Σ}, we get that

[n] = ∑_{[e_{ρ},1]∈C′}(a_{ρ})[e_{ρ}, 1] + ∑_{ρ+σ∈Σ(dimσ+1)} b′_{ρ}[e_{ρ}, 1],

which comes from H^{p−l,p−l}(V (σ)) ⊗ ℂt for an appropriate p. The surjectivity now follows from the fact that the boundary points of K^{∗} generate the ring C[K^{∗}]^{Σ}/⟨ω⟩. □

The isomorphism (5) follows from the above proposition and the presentation:

H_{toric}^{∗}(Y ∩ V (σ)) ≅ H^{∗}(V (σ))/Ann([Y ∩ V (σ)])

(see (1)). Indeed, the map constructed in the proof of Proposition 9.4 produces a well defined map between the right hand side in (5) and R_{0}(ω, K^{∗})^{Σ}/Ann([0, 1]) because the annihilator of [Y ∩V (σ)] maps to the annihilator of [0, 1]. On the other hand,

(R_{0}(ω, K^{∗})^{Σ}/Ann([0, 1]))_{p} ≅ R_{1}(ω, K^{∗})_{p+1}^{Σ},

which is induced by the multiplication by [0, 1] in R_{0}(ω, K^{∗})^{Σ}.

Conjecture 9.5. We expect that the product structure on H_{st}^{∗}(ℙ_{Σ}) is given by the ring structure R_{0}(ω, K^{∗})^{Σ}. Also, the ring structure on R_{1}(ω, K^{∗})_{∗+1}^{Σ} induced from R_{0}(ω, K^{∗})^{Σ}/Ann([0, 1]) should give a subring of H_{st}^{∗}(Y) for a generic ω in Conjecture 4.8. Moreover,

⨁_{p,q} R_{1}(ω_{C^{∗}}, C^{∗})_{(p+q−d+dimC^{∗}+1)/2}^{Σ} ⊗ R_{1}(f_{C}, C)_{(q−p+dimC)/2},

should be the module over the ring R_{0}(ω, K^{∗})^{Σ}/Ann([0, 1]):

a · (b ⊗ c) = āb ⊗ c,

for a ∈ R_{0}(ω, K^{∗})^{Σ}/Ann([0, 1]) and (b ⊗ c) from a component of the above direct sum, where ā is the image of a induced by the projection R_{0}(ω, K^{∗})^{Σ} ⟶ R_{0}(ω_{C^{∗}}, C^{∗})^{Σ}.

We can also say about the product structure on the B-model chiral ring. The space R_{1}(f, K) ≅ R_{0}(f, K)/Ann([0, 1]) in Conjecture 4.8, which lies in the middle cohomology ⊕_{p+q=d−1}H_{st}^{p,q}(Y), should be a subring of the B-model chiral ring, and

⨁_{p,q} R_{1}(ω_{C^{∗}}, C^{∗})_{(p+q−d+dimC^{∗}+1)/2}^{Σ} ⊗ R_{1}(f_{C}, C)_{(q−p+dimC)/2},

should be the module over the ring R_{1}(f, K), similarly to the above description in the previous paragraph.

These ring structures are consistent with the products on the usual cohomology and the B-model chiral ring H^{∗}(X, ⋀^{∗} T_{X}) of the smooth semiample Calabi-Yau hypersurfaces X in [M3, Theorem 2.11(a,b)] and [M2, Theorem 7.3(i,ii)].

10. Description of string cohomology inspired by vertex algebras

Here we will give yet another description of the string cohomology spaces of Calabi-Yau hypersurfaces. It will appear as cohomology of a certain complex, which was inspired by the vertex algebra approach to Mirror Symmetry.

We will state the result first in the non-deformed case, and it will be clear what needs to be done in general. Let K and K^{∗} be dual reflexive cones of dimension d + 1 in the lattices M and N respectively. We consider the subspace ℂ[L] of ℂ[K] ⊗ ℂ[K^{∗}] as the span of the monomials [m, n] with m · n = 0. We also pick non-degenerate elements of degree one f = ∑_{m} f_{m}[m] and g = ∑_{n} g_{n}[n] in ℂ[K] and ℂ[K^{∗}] respectively.

Consider the space

V = Λ^{∗}(N_{ℂ}) ⊗ ℂ[L].

Lemma 10.1. The space V is equipped with a differential D given by

D := ∑_{m} f_{m}⌟m ⊗ (π_{L} ∘ [m]) + ∑_{n} g_{n}(∧n) ⊗ (π_{L} ∘ [n])

where [m] and [n] means multiplication by the corresponding monomials in ℂ[K] ⊗ ℂ[K^{∗}] and π_{L} denotes the natural projection to ℂ[L].

Proof. It is straightforward to check that D^{2} = 0. □

Theorem 10.2. Cohomology H of V with respect to D is naturally isomorphic to

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗})

where C_{ℂ}^{∗} denotes the vector subspace of N_{ℂ} generated by C^{∗}.

Proof. First observe that V contains a subspace

⨁_{C⊆K} Λ^{∗}N_{ℂ} ⊗ (ℂ[C^{∘}] ⊗ ℂ[C^{∗∘}])

which is invariant under D. It is easy to calculate the cohomology of this subspace under D, because the action commutes with the decomposition ⊕_{C}. For each C, the cohomology of D on Λ^{∗}N_{ℂ} ⊗ (ℂ[C^{∘}] ⊗ ℂ[C^{∗∘}]) is naturally isomorphic to

Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C^{∘}) ⊗ R_{0}(g, C^{∗∘}),

because Λ^{∗}N_{ℂ} ⊗ (ℂ[C^{∘}] ⊗ ℂ[C^{∗∘}]) is a tensor product of the Koszul complex for ℂ[C^{∘}] and the dual of the Koszul complex for ℂ[C^{∗∘}]. As a result, we have a map

α : H_{1} → H, H_{1} := ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C^{∘}) ⊗ R_{0}(g, C^{∗∘}).

Next, we observe that V embeds naturally into the space

⨁_{C⊆K} Λ^{∗}N_{ℂ} ⊗ (ℂ[C] ⊗ ℂ[C^{∗}])

as the subspace of the elements compatible with the restriction maps. This defines a map

β : H → ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C) ⊗ R_{0}(g, C^{∗}) =: H_{2}.

We observe that the composition β∘α is precisely the map induced by embeddings C^{∘} ⊆ C and C^{∗∘} ⊆ C^{∗}, so its image in H_{2} is

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗}).

As a result, what we need to show is that α is surjective and β is injective. We can not do this directly, instead, we will use spectral sequences associated to two natural filtrations on V.

First, consider the filtration

V = V^{0} ⊃ V^{1} ⊃ … ⊃ V^{d+1} ⊃ V^{d+2} = 0

where V^{p} is defined as Λ^{∗}N_{ℂ} tensored with the span of all monomials [m, n] for which the smallest face of K that contains m has dimension at least p. It is easy to see that the spectral sequence of this filtration starts with

H_{3} := ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C^{∘}) ⊗ R_{0}(g, C^{∗}).

Analogously, we have a spectral sequence from

H_{4} := ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C) ⊗ R_{0}(g, C^{∗∘})

to H, which gives us the following diagram.

H_{3}
↗ ⇓ ↘
H_{1} → H → H_{2}
↘ ⇑ ↗
H_{4}

We remark that the spectral sequences mean that H is a subquotient of both H_{3} and H_{4}, i.e. there are subspaces I_{3}^{+} and I_{3}^{−} of H_{3} such that H ≃ I_{3}^{+}/I_{3}^{−}, and similarly for H_{4}. Moreover, the above diagram induces commutative diagrams

0  0
↓  ↑
I_{3}^{−}  H_{1} → H → H_{2}
↓  ↘ ↑ ↗
I_{3}^{+}  I_{4}^{+}
↗ ↓ ↘  ↑
H_{1} → H → H_{2}  I_{4}^{−}
↓  ↑
0  0

with exact vertical lines. Indeed, the filtration V^{∗} induces a filtration on the subspace of V

⨁_{C⊆K} Λ^{∗} ⊗ ℂ[C^{∘}] ⊗ ℂ[C^{∗∘}].

The resulting spectral sequence degenerates immediately, and the functoriality of spectral sequences assures that there are maps from H_{1} as above. Similarly, the space

⨁_{C⊆K} Λ^{∗} ⊗ ℂ[C] ⊗ ℂ[C^{∗}]

has a natural filtration by the dimension of C that induces the filtration on V. Functoriality then gives the maps to H_{4}.

We immediately get

Im(β) ⊆ Im(H_{3} → H_{2}) ∩ Im(H_{4} → H_{2})

which implies that

Im(β) = Im(β ∘ α) = ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗}).

Analogously, Ker(α) = Ker(β ∘ α), which shows that

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗})

is a direct summand of H.

The fact that

Ker(α) ⊇ Ker(H_{1} → H_{4})

= ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ Ker(R_{0}(f, C^{∘}) → R_{0}(f, C)) ⊗ R_{0}(g, C^{∗∘})

implies that I_{3}^{−} contains the image of this space under H_{1} → H_{3}, which is equal to

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ Ker(R_{0}(f, C^{∘}) → R_{0}(f, C)) ⊗ R_{1}(g, C^{∗}).

Similarly, I_{3}^{+} is contained in the preimage of

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗})

under H_{3} → H_{2}, which is

⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{0}(f, C^{∘}) ⊗ R_{1}(g, C^{∗}).

As a result,

H = ⨁_{C⊆K} Λ^{dimC^{∗}}C_{ℂ}^{∗} ⊗ R_{1}(f, C) ⊗ R_{1}(g, C^{∗}).

□

Remark 10.3. If one replaces ℂ[K^{∗}] by ℂ[K^{∗}]^{Σ} in the definition of ℂ[L], then the statement and the proof of Theorem 10.2 remain intact. In addition, one can make a similar statement after replacing Λ^{∗}N_{ℂ} by Λ^{∗}M_{ℂ} and switching contraction and exterior multiplication in the definition of D. It is easy to see that the resulting complex is basically identical, though various gradings are switched. This should correspond to a switch between A and B models.

We will now briefly outline the connection between Theorem 10.2 and the vertex algebra approach to mirror symmetry, developed in [Bo2] and further explored in [MaS]. The vertex algebra that corresponds to the N=2 superconformal field theory is expected to be the cohomology of a lattice vertex algebra Fock_{M⊕N}, built out of M ⊕ N, by a certain differential D_{f,g} that depends on the defining equations f and g of a mirror pair. The space Λ^{∗}(N_{ℂ}) ⊗ ℂ[L] corresponds to a certain subspace of Fock_{M⊕N} such that the restriction of D_{f,g} to this subspace coincides with the differential D of Theorem 10.2. We can not yet show that this is precisely the chiral ring of the vertex algebra, so the connection to vertex algebras needs to be explored further.

11. Appendix. G-polynomials

A finite graded partially ordered set is called Eulerian if every its nontrivial interval contains equal numbers of elements of even and odd rank. We often consider the poset of faces of the Gorenstein cone K over a reflexive polytope Δ with respect to inclusions. This is an Eulerian poset with the grading given by the dimension of the face. The minimum and maximum elements of a poset are commonly denoted by ˆ0 and ˆ1.

Definition 11.1. [S1] Let P = [ˆ0, ˆ1] be an Eulerian poset of rank d. Define two polynomials G(P, t), H(P, t) ∈ ℤ[t] by the following recursive rules:

G(P, t) = H(P, t) = 1 if d = 0;

H(P, t) = ∑_{ˆ0<x≤ˆ1}(t − 1)^{ρ(x)−1}G([x, ˆ1], t) (d > 0),

G(P, t) = τ_{<d/2} ((1 − t)H(P, t)) (d > 0),

where τ_{<r} denotes the truncation operator ℤ[t] → ℤ[t] which is defined by

τ_{<r} (∑_{i} a_{i}t^{i}) = ∑_{i<r} a_{i}t^{i}.

The following lemma will be extremely useful.

Lemma 11.2. For every Eulerian poset P = [ˆ0, ˆ1] of positive rank there holds

∑_{ˆ0≤x≤ˆ1}(−1)^{rk[ˆ0,x]}G([ˆ0, x]^{∗}, t)G([x, ˆ1], t) = ∑_{ˆ0≤x≤ˆ1} G([ˆ0, x], t)G([x, ˆ1]^{∗}, t)(−1)^{rk[x,ˆ1]} = 0

where ()^{∗} denotes the dual poset. In other words, G( , t) and (−1)^{rk}G( ^{∗}, t) are inverses of each other in the algebra of functions on the posets with the convolution product.

Proof. See Corollary 8.3 of [S2]. □ The following polynomial invariants of Eulerian posets have been introduced in

[BBo].

Definition 11.3. Let P be an Eulerian poset of rank d. Define the polynomial B(P; u, v) ∈ ℤ[u, v] by the following recursive rules:

B(P; u, v) = 1 if d = 0,

∑_{ˆ0≤x≤ˆ1} B([ˆ0, x]; u, v)u^{d−ρ(x)}G([x, ˆ1], u^{−1}v) = G(P, uv).

Lemma 11.4. Let P = [ˆ0, ˆ1] be an Eulerian poset. Then

B(P; u, v) = ∑_{ˆ0≤x≤ˆ1} G([x, ˆ1]^{∗}, u^{−1}v)(−u)^{rkˆ1−rkx}G([ˆ0, x], uv).

Proof. Indeed, one can sum the recursive formulas for B([ˆ0, y]) for all ˆ0 ≤ y ≤ ˆ1 multiplied by G([y, ˆ1]^{∗}, u^{−1}v)(−u)^{rkˆ1−rky} and use Lemma 11.2. □

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Department of Mathematics, Columbia University, New York, NY 10027, USA

   E-mail address: lborisov@math.columbia.edu

Max-Planck-Institut für Mathematik, Bonn, D-53111, Germany

   E-mail address: anvar@mpim-bonn.mpg.de