Deformations of Calabi-Yau manifolds in Fano toric varieties

In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.


Introduction
In this paper, we focus our attention on Calabi-Yau manifolds, i.e., projective manifolds with trivial canonical bundle and without holomophic p-forms. More precisely, if we focus on dimension greater than or equal to three, Z is a Calabi-Yau manifold of dimension n if the canonical bundle K Z := Ω n Z is trivial and H 0 (Z, Ω p Z ) vanishes for p in between 0 and n. Since the canonical bundle is trivial, Z has unobstructed deformations, i.e., the moduli space of deformations of Z is smooth. This is the famous Bogomolov-Tian-Todorov Theorem [Bo78,Ti87,To89]. A more algebraic proof of this fact [Kaw92,Ra92,IM10] shows that the functor Def Z of infinitesimal deformations of Z is smooth too. In particular, the dimension of the moduli space at the point corresponding to Z is the dimension of H 1 (Z, T Z ), where T Z denotes the tangent bundle of Z. Although we know that the moduli space is smooth, we still miss a geometric understanding of it; for instance, the number of its irreducible components is unknown. A famous conjecture by M. Reid claims that the moduli space of simply connected smooth Calabi-Yau threefolds is connected via conifold transitions [Re87]. The general picture is still unknown but in some cases there has been quite a lot of progress. For example, the moduli spaces of complete intersection Calabi-Yau 3-folds in products of projective spaces are connected with each other by a sequence of conifold transitions (see [Wa06] and references therein).
If Z is contained in an ambient manifold X, we can investigate the deformation functor H X Z of deformations of Z in X (fixed) and the forgetful functor φ : H X Z → Def Z , which associates with an infinitesimal deformation of Z in X the isomorphism class of the deformation of Z. For example, if φ is smooth we can conclude that all deformations of Z lie in X and, since Def Z is smooth, the functor H X Z is also smooth [Se06, Proposition 2.2.5]. For every Calabi-Yau manifold Z of dimension at least 3 in projective space, the embedded deformations of Z in P n are unobstructed. This follows from the vanishing H 1 (Z, T P n |Z ) = 0 [Hu95, Corollary A.2] that implies that the forgetful morphism φ : H P n Z → Def Z is smooth, i.e., all deformations of Z as an abstract variety are contained in P n . Note that dimension at least 3 is fundamental, since in dimension 2 the same statement does not hold (see also Remark 2.3). Moreover, since Def Z is smooth, we can conclude that H P n Z is also smooth. Note that this does not imply that any two Calabi-Yau manifolds of the same dimension in P n are deformation equivalent: for instance, explicit examples of threefolds in P 6 that are not deformation equivalent are constructed in [Be09].
The projective space P n is a toric Fano manifold, i.e., a smooth toric variety with ample anticanonical bundle. Therefore, it is natural to investigate whether the previous results for P n can be generalised to any toric Fano variety F , not simply P n or the smooth ones. The interest in toric Fano varieties is motivated both from the mathematics and the physics viewpoint; indeed these varieties have an essential role in the Minimal Model Program and Mirror Symmetry (see, for instance, [Ro20] for a recent work on the latter topic). In [Pe19], the author investigates deformation theory of toric Fano varieties.
In [BI16], we investigated Calabi-Yau manifolds that are anticanonical divisors in toric Fano manifolds of dimension greater than or equal to 4. In particular, we proved that the forgetful morphism φ : H F Z → Def Z is smooth, i.e., all deformations of Z as abstract variety are contained in F [BI16, Proposition 1].
In this paper, we generalise these results considering as ambient space a projective simplicial toric Fano variety F and as subvariety a Calabi-Yau manifold Z embedded in the Zariski open set of regular points of F . Under this assumption we investigate the forgetful morphism φ : H F Z → Def Z . In particular, the following holds (Theorem 3.5). Theorem 1.1. Let F be a projective simplicial toric Fano variety with K F = − ρ∈Σ(1) D ρ its canonical bundle and Z ⊂ F a Calabi-Yau sumbanifold of dimension greater than or equal to 3, embedded in the Zariski open set of regular points of F . If for all ρ ∈ Σ(1) we have In particular, if Z is a Calabi-Yau manifold, of dimension greater than or equal to 3, which is a complete intersection of very ample divisors, then the previous theorem applies if the restriction of all D ρ to Z are nef divisors (Corollary 3.8). We prove Theorem 3.5 by showing the vanishing H 1 (Z, T F |Z ) = 0, which is a sufficient condition for the smoothness of the forgetful morphism φ : H F Z → Def Z . This implication is well known for the smooth case , see for example [Se06, Proposition 3.2.9]. It can be also proved via Horikawa's costability theorem for the inclusion morphism Z ֒→ F [Ho76] or [Se06, Section 3.4.5]. For the reader's convenience, we give an explicit proof of this fact under our assumptions (see Theorem 2.1). Note that the vanishing H 1 (Z, T F |Z ) = 0 is not a necessary condition for the smoothness of the forgetful morphism φ : H F Z → Def Z (Remark 2.4).
In [Be09], the author also focuses her attention on Calabi-Yau threefolds of codimension 4 in P 7 with Picard number equals to 1. Using Commutative Algebra methods, new examples are built and their Hodge numbers are investigated. Then, following this approach, we devote our attention to the computation of Hodge numbers of Calabi-Yau submanifolds Z in a toric Fano variety F . In particular, our calculations focus on the cases with H 1 (Z, T F |Z ) = 0 and dim Z = 3, 4 (Section 4). These includes some examples of Calabi-Yau threefold in weighted projective spaces (Section 4.1).
Throughout the paper, we work over the field of complex numbers. If not otherwise stated, by a toric variety F we mean a projective simplicial toric Fano variety F . We denote by Z a sumbanifold of F embedded in the Zariski open set of regular points of F ; thus, Z can be covered by smooth affine open sets.
In Section 2 we collect some results on toric Fano varieties and we prove the main theorem on the smoothness of the forgetful functor (Theorem 2.1). Section 3 is devoted to examples of Calabi-Yau submanifolds Z in toric Fano variety F , such that the forgetful functor is smooth. Finally, Section 4 contains some computations on the Hodge numbers of Calabi-Yau threefolds and fourfolds in a toric Fano variety. In particular, we describe examples of Calabi-Yau threefolds in weighted projective spaces and complete intersections fourfolds.

Embeddings in Fano varieties
In this section, we will follow the notation of the book [CLS11], we refer the reader to this book and especially to Chapter 4 for further details. Let F be a projective simplicial toric variety with no torus factors, i.e., {u ρ |ρ ∈ Σ(1)} spans N R , where Σ is the fan of F in N R and Σ(1) denotes the 1-dimensional cones of Σ. We recall that F is simplicial when every σ ∈ Σ is simplicial, meaning that the minimal generators of σ are linearly independent over R [CLS11, pag.180].
Moreover, for any strongly convex cone σ ∈ N R , we denote by σ(1) its rays. Also, under our assumptions it makes sense to talk about the canonical divisor K F , which can be written as K F = − ρ∈Σ(1) D ρ (for further details, we refer the reader to [CLS11, Chapter 4]). The variety F is Fano if its anticanonical divisor −K F is ample. Note that in this case F has no torus factors. LetΩ 1 F be the sheaf of Zariski 1-differentials. Recall thatΩ 1 F is the double dual of the sheaf of Kähler differentials Ω 1 F . Moreover, as proved for instance in [PO,p. 56], the dual ofΩ 1 F and the dual of Ω 1 F are isomorphic and we denote it by The hypothesis that F is a simplicial toric variety with no torus factors is needed for the existence of a generalized Euler exact sequence [CLS11, Theorem 8.1.6], as in the case for projective spaces; namely: where CL(F ) denotes the divisor class group of F .
We follow the usual approach. Consider the generalized Euler exact sequence (2.1): Note that the divisor class group CL(F ) of F is a finitely generated abelian group that can have torsion [CLS11,p. 172]. We denote by t the rank of CL(F ). Consider the dual of the above exact sequence (2.1), i.e., apply the functor Hom OF (−, O F ) to obtain Note that the sheaf Ext 1 is a finitely generated group, the torsion subgroup is a finite abelian group, so it is a finite sum of cyclic groups of prime power order r = p h . Thus, tensoring by O F over Z, we have Moreover, the map on the LHS is an isomorphism of sheaves, we conclude that the sheaf In addition, as mentioned before, , the tangent sheaf of the Fano variety. Then, the exact sequence (2.2) reduces to Since Z is smooth, the sheaf Ω 1 Z is locally free and so Ext 1 OZ (Ω 1 Z , O Z ) = 0. Therefore, we have the usual normal exact sequence The induced exact sequence in cohomology is given by and prove that φ is smooth.
Remark 2.2. If Z is a Calabi-Yau submanifold of F , then Def Z is smooth (Bogomolov-Tian-Todorov Theorem) of dimension H 1 (Z, T Z ). Then, by the previous theorem, H F Z is also a smooth functor: the deformation space of Z inside F is smooth of dimension H 0 (Z, N Z/F ).
Remark 2.3. If dim Z = 2, then Theorem 3.5 does not hold. It is enough to consider a K3 surface in P 4 . In this case, it is not true that the morphism φ : Remark 2.4. The condition H 1 (X, T F |Z ) = 0 is not a necessary condition for the smoothness of the forgetful morphism φ : H F Z → Def Z . For example, [Se06, Example 3.4.4 (iii)], let Z ∼ = P 1 ⊂ F be a nonsingular projective curve negatively embedded in a projective nonsingular Hirzebruch surface F with Z 2 = −n < 0, n ≥ 1. Then, the exact sequence . This implies that h 0 (Z, T F |Z ) = 3. Moreover h 0 (Z, N Z|F ) = 0 and so Z is rigid in F in addition to being rigid as an abstract variety. Then, the morphism induced by φ on the tangent space is surjective, and it is injective on the obstruction spaces: they are both zero since there are no deformations. In conclusion, φ is smooth even if h 1 (Z, T F |Z ) = h 1 (Z, O Z (2) ⊕ O Z (−n)) = n − 2 can be non-zero.

A large class of examples
Let F be a simplicial toric Fano variety of dimension dim F = n + m for m ≥ 3. As in the previous section, the dual of the generalised Euler exact sequence (2.3) for F is Lemma 3.1. Let F be a toric Fano variety and Z ⊂ F a Calabi-Yau sumbanifold. Let D be a divisor such that D |Z is nef and big, then Proof. Since the divisor D |Z is nef and big and Z is a Calabi-Yau manifold (and so Remark 3.2. Let F be a toric Fano variety and Z ⊂ F a Calabi-Yau sumbanifold, such that dim Z = m. If the divisor D is such D |Z is nef and D m · Z > 0, then D |Z is nef and big [De01, Section 1.29] and so we can apply the previous Lemma 3.1. Note also that if D is nef then its restriction D |Z to Z is also nef [De01, Section 1.6].
Remark 3.3. A useful condition for nefness of a divisor D in F is the following: given a cone σ ∈ Σ, any nef divisor is linearly equivalent to a divisor of the form where a ρ = 0 if ρ ∈ σ(1) and a ρ ≥ 0 for ρ ∈ σ(1), see [CLS11,Equation 6.4.10].
Corollary 3.4. Let F be a toric Fano variety of dimension dim F = n + m and denote by Z ⊂ F a Calabi-Yau smooth variety of dimension dim Z = m. Suppose that Z is a complete intersection of very ample divisors. Then, for any divisor D such that D |Z is nef, we have Proof. Suppose that Z is a complete intersection of very ample divisors. Then, there exist n very ample divisors N j , for j = 1, . . . , n, such that Z = Y 1 · · · Y n , where Y j is an element in the linear system |N j |.
In particular, we have where the last equality follows from the Nakai-Moishezon Theorem [De01, Theorem 1.21], indeed N 1 is ample and (D m · N 2 · · · N n ) has dimension 1. Then, the conclusion follows by Remark 3.2 and Lemma 3.1.
Theorem 3.5. Let F be a simplicial toric Fano variety and Z ⊂ F a Calabi-Yau submanifold of dim Z = m, with m ≥ 3. Suppose that for all ρ ∈ Σ(1) the divisor D ρ satisfies the following vanishing Then, the forgetful morphism φ : H F Z → Def Z is smooth.
Proof. By tensoring with O Z the dual of the generalized Euler exact sequence for F (2.3), we obtain Example 3.6. Let F be a toric Fano variety of dimension dim F = n + m and denote by Z ⊂ F a Calabi-Yau submanifold of dimension dim Z = m ≥ 3. Suppose that Z is a complete intersection of very ample divisors, such that D ρ |Z is nef for all ρ ∈ Σ(1). Then, by Corollary 3.4 we have that Corollary 3.7. Let F be a simplicial toric Fano variety and Z ⊂ F a Calabi-Yau sumbanifold of dim Z = m. Let D ρ be the divisor associated with ρ ∈ Σ(1) and assume further that D ρ |Z is nef and D m ρ · Z > 0, for all ρ ∈ Σ(1). Then, Proof. It is enough to apply Lemma 3.1 and Theorem 3.5.
Corollary 3.8. Let F be a simplicial toric Fano variety of dimension dim F = n + m and denote by Z ⊂ F a Calabi-Yau submanifold of dimension dim Z = m ≥ 3. Suppose that Z is a complete intersection of very ample divisors, such that D ρ |Z is nef for all ρ ∈ Σ(1).
It is enough to apply Theorem 3.5 and Example 3.6.

Hodge numbers of Calabi-Yau varieties
In this section, we are interested in computing Hodge numbers of Calabi-Yau submanifolds Z of a toric Fano variety F , in particular for the case investigated in the previous section, i.e., whenever H 1 (Z, T F |Z ) = 0. Recall that the Hodge numbers of Z are defined as h i,j (Z) = dim C H j (Z, Ω i Z ) and they satisfy the Hodge duality Under the assumption that Z is a Calabi-Yau submanifolds of a simplicial toric Fano variety F such that H 1 (Z, T F |Z ) = 0, we can estimate the Hodge numbers of Z.
where t is the rank of CL(F ).
Proof. The exact sequence (2.5) induces the following exact sequence in cohomology: Then, by the long exact sequence associated with the Euler exact sequence 3.1 restricted to Z, we obtain where t is the rank of CL(F ). Hence, Remark 4.2. In the setup above of a smooth Calabi-Yau submanifold Z in a simplicial toric Fano variety F , the previous proposition provides the dimension of the moduli space at the point corresponding to Z, that is smooth of dimension H 1 (Z, T Z ).
Proposition 4.3. Let F be a simplicial toric Fano variety of dimension dim F = n + m and denote by Z ⊂ F a Calabi-Yau submanifold of dimension dim Z = m ≥ 3, that is the complete intersection of n very ample divisors, such that D ρ |Z is nef for all ρ ∈ Σ(1). Then, h 1,1 = dim H 1 (Z, Ω 1 Z ) = t, where t is the rank of CL(F ).
Proof. As above, since Z is a Calabi-Yau manifold of dimension m, we have that Ω m Z ∼ = O Z and so T Z ∼ = Ω m−1 Z . Therefore, The submanifold Z is the complete intersection of n very ample divisors N 1 , · · · , N n , i.e., Z = N 1 · · · · · N n . In particular, where in the last equality we use the Kodaira vanishing (the restriction of an ample line bundle to a closed subscheme is still ample). The long exact sequence in cohomology associated with (2.5) implies that Finally, by the long exact sequence associated with the Euler exact sequence 3.1 restricted to Z, we obtain · · · → ρ∈Σ(1) Corollary 3.4 implies H j (Z, O F (D ρ ) ⊗ O Z ) = 0, for all j > 0 and all ρ ∈ Σ(1); therefore Remark 4.4. If Z is a Calabi-Yau submanifold of dimension dim Z = 3, that is a complete intersection of n very ample divisors in a simplicial toric Fano variety F of dimension dim F = n + 3, such that D ρ |Z is nef for all ρ ∈ Σ(1), then we can describe the Hodge diamond of X. Indeed, by the previous proposition we computed h 1,1 and by Proposition 4.1 we can compute h 1,2 (Z) = h 2,1 (Z) = dim H 1 (Z, Ω 2 Z ) = dim H 1 (Z, T Z ).  1, 1, 1, a 3 , . . . , a n ) be the weighted projective spaces for n ≥ 3, and a i ≥ 0, for all i ≥ 3. According to [Ko06,Claim 37 ] or [Se06, Example 3.1.25], we have T 1 P = 0 and H 1 (P, T P ) = 0 and so the local deformations of P are trivial.
These varieties are all examples of smooth subvarieties with trivial canonical bundle in a toric Fano variety with Picard rank one (that is not smooth except the cases of projective spaces P(1, 1, 1, 1, 1) and P (1, 1, 1, 1, 1, 1)). Moreover, if Z is any of these Calabi-Yau threefolds we have H 1 (Z, T P |Z ) = 0. Indeed, the generalised Euler exact sequence (2.3) for P = P (1, 1, 1, a 3 , . . . , a n ) is and it restrict to Considering the long exact sequence associated with (4.1), it is enough to prove For the weighted hypersurface case Z = X d , we tensorize the exact sequence with O P (a) and we conclude the vanishing (4.2), since H i (P, O P (n)) = 0 for all n ∈ Z and i = 0, i a i + 3 [Do82, Section 1.4]. For Z any of the above codimension 2 weighted threefolds complete intersection, similar computations prove the vanishing H 1 (Z, T P |Z ) = 0. Therefore, as proved in Theorem 2.1 the forgetful functor φ : H P Z → Def Z is smooth and so the functor H P Z is smooth at the corresponding point. Remark 4.6. A we also noted in Remark 4.4, we can compute the Hodge numbers of these Calabi-Yau weighted complete intersections. By Proposition 4.3, we have h 1,1 = dim H 1 (Z, Ω 1 Z ) = 1, and by Proposition 4.1, we have

4.2.
Examples of complete intersection Calabi-Yau fourfolds. Let Z be a smooth Calabi-Yau fourfold in F ; let j : Z → F be a closed embedding of Z in F . Suppose further that Z is a complete intersection of n very ample divisors N 1 , . . . , N n so that dim(F ) = n + 4. Let us analyse h i,j (Z) = dim C H j (Z, Ω i Z ). By Proposition 4.3, we have h 1,1 (Z) = t = rankCL(F ). Next, let us compute h 1,2 (Z) = h 2,1 (Z).
Proposition 4.7. Let F be a simplicial toric Fano variety of dimension dim F = n + m and denote by Z ⊂ F a Calabi-Yau submanifold of dimension dim Z = m ≥ 4, that is the complete intersection of n very ample divisors, such that D ρ |Z is nef for all ρ ∈ Σ(1). Then, h 1,2 = h 2,1 = 0.
Proof. The proof goes as in Proposition 4.3. Since Z is a Calabi-Yau manifold of dimension m, we have that Ω m Z ∼ = O Z and so T Z ∼ = Ω m−1 Z . Therefore, As in the proof of Propostion 4.3, H j (X, N Z/F ) = 0 for all j > 0. The long exact sequence in cohomology associated with (2.5) implies that and so, since m ≥ 4, H m−2 (X, where t is the rank of CL(F ).  p 1 (Z) = −2c 2 (Z), p 2 (Z) = 2c 4 (Z) + c 2 2 (Z). Hence we obtain the statement.
Remark 4.9. Assume Z is a smooth Calabi-Yau fourfold in F ; let j : Z → F be closed embedding of Z in F . Suppose further that Z is the complete intersection of n very ample divisors N 1 , . . . , N n so that dim(F ) = n + 4, then the Chern classes of Z can be computed in terms of the Chern classes of the bundles N j . More precisely, the following recursive relations can be deduced from the exact sequence defining the normal bundle, namely: In [GH87a,GH87b,GHL14], the authors carry out computations on Chern classes and Hodge numbers for Calabi-Yau fourfold that are complete intersection in product of projective spaces.