On the Hodge conjecture for quasi-smooth intersections in toric varieties

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether-Lefschetz locus, where"asymptotically"means that the degree of the hypersurface is big enough. This extendes to toric varieties Otwinowska's result in [15].


Introduction
A projective simplicial toric variety P d Σ satisfies the Hodge Conjecture, i.e., every cohomology class in H p,p (P d Σ , Q) is a linear combination of algebraic cycles. On the one hand, by the Lefschetz hyperplane theorem, the Hodge conjecture holds true for every hypersurface and p < d−1 2 and by the hyperplane Lefschetz theorem, and by Poincaré duality, also for p > d− 1 2 . Moreover, by Theorem 1.1 in [3], when p = d−1 2 , d = 2k + 1 and P 2k+1 Σ is an Oda variety with an ample class β such that kβ − β 0 is nef, where β 0 is the anticanonical class, the Hodge conjecture with rational coefficients holds for a very general hypersurface in the linear system β .
The notion of Oda varieties was introduced in [2]. Let us recall that the Cox ring of a toric variety P Σ is graded over the class group Cl(P Σ ), and that one has an injection Pic(P Σ ) → Cl(P Σ ). Definition 1.1. Let P Σ be a toric variety with Cox ring S. P Σ is said to be an Oda variety if the multiplication morphism S α 1 ⊗ S α 2 → S α 1 +α 2 is surjective whenever the classes α 1 and α 2 in Pic(P Σ ) are ample and nef, respectively.
In [14] Mavlyutov proved a Lefschetz type theorem for quasi-smooth intersection subvarieties, and moreover using the "Cayley trick" he related the cohomology of a quasi-smooth subavariety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X fs ⊂ P d Σ to the cohomology of a quasi-smooth hypersurface Y ⊂ P d+s−1 Σ . This allows us to prove a Noether-Lefschetz type theorem, namely: Theorem 2.5.Let P d Σ be an Oda projective simplicial toric variety. For a very general quasi-smooth intersection subvariety X cut off by f 1 , . . . f s such that d + s = 2(ℓ + 1) and From this one obtains the following result about the Hodge conjecture for quasi-smooth intersections.
Corollary 2.7. If P d Σ is an Oda projective simplicial toric variety, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety X cut off by f 1 , . . . f s such that d + s is even and ∑ Let T be the open subset of β corresponding to quasi-smooth hypersurfaces, and let H 2k = R 2k π * C ⊗ C O T be the Hodge bundle on T ; here π∶ X → T is the tautological family on T , and d = 2k + 1. We restrict H 2k to a contractible open subset U ⊂ T . The bundle H 2k has a Hodge decomposition H 2k = ⊕ p+q=2k H p,q but this is not holomorphic. On the other hand, the bundles that make up the Hodge filtration are holomorphic; to see this one can use the period map (which in particular we write for . This map is holomorphic (see [12] and [5,Prop. 3.4]). But, by the very definition of the period map (see also [16], Section 10.2.1 for the smooth case) where U k is the tautological bundle on the Grassmannian Grass(b k , H 2k (X u 0 , C)), so that the bundles F k H 2k are indeed holomorphic.
Pushing ahead the ideas developed in [5] and [4], let λ f be a nonzero class in the primitive cohomology H k,k (X f , Q) H k,k (P 2k+1 Σ , Q), and let U be a contractible open subset of T around f , so that H 2k U is constant. Moreover, let λ ∈ H 2k (U) be the section defined by λ f and letλ be its image in (H 2k F k H 2k )(U). One has Proposition 1.2. The local Noether-Lefschetz loci can be defined as The following result is Theorem 1.2 in [4].
Theorem. Let P 2k+1 Σ be an Oda variety with an ample class β such that kβ − β 0 = nη, where β 0 is the anticanonical class, η is a primitive ample class, and n ∈ N. Let For every positive ǫ there is a positive δ such that for every m ≥ max( 1 δ , m β ) and d ∈ [1, mδ], and every nontrivial Hodge class λ ∈ F k H 2k (U) such that Here deg V and deg X f are taken with respect to the ample divisor η, i.e., Based on this, in this paper we obtain the following result.
Theorem 4.2. Under the same hypotheses of the previous theorem, if V ⊂ X f is a nonempty quasi-smooth intersection subvariety of P 2k+1 for some f ∈ N k,β λ,U , then there exists c ∈ Q * such that prim (X f , Q). In other words, λ f is algebraic.
In his paper [9] A. Dan proves a form of our Theorem 4.2 for smooth hypersurfaces in odd-dimensional projective spaces P 2k+1 which is not asymptotic. So our result is more general in two ways, as we consider quasi-smooth intersections in toric varieties; however, our result is asymptotic.
Acknowledgement. We thank Paolo Aluffi for useful discussions, and Antonella Grassi for developing with the first author the foundations on which this work is based. We are very thankful to the referee for her/his very careful reading, and the many suggestions and remarks which allowed us to greatly improve the presentation of this paper. The second author acknowledges support from FAPESP postdoctoral grant No. 2019/23499-7.

Very general quasi-smooth intersections
Let f 1 , . . . , f s be weighted homogeneous polynomials in the Cox ring is either empty or a smooth interesection subvariety of codimension s in U(Σ).
This notion generalizes that of smooth complete intersection in a projective space. For s = 1 it reduces to the notion of quasi-smoooth hypersurface, see Def. 3.1 in [1]. If we regard P d Σ as an orbifold, then a hypersurface X is quasi-smooth when it is a sub-orbifold of P d Σ ; heuristically, "X has only singularities coming from the ambient variety." We also have a Lefschetz type theorem in this context.
is an isomorphism for i < d − s and an injection for i = d − s. In particular, this is true if the hypersurfaces cut by the polynomials f i are ample.
Hence if p ≠ d−s 2 every cohomology class in H p,p (X) is a linear combination of algebraic cycles. So let us see what happens when p = d−s 2 . The idea is to relate the Hodge structure of a quasi-smooth intersection variety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X fs in P d Σ with the Hodge structure of a quasi-smooth hypersurface Y in a toric variety P d+s−1 X,Σ whose fan depends on X and Σ. Proposition 2.3. Let X = X 1 ∩ ⋅ ⋅ ⋅ ∩ X s be quasi-smooth intersection subvariety in P d Σ cut off by homogeneous polynomials f 1 . . . f s . There exists a projective simplicial toric variety Proof. One constructs P d+s−1 X,Σ via the so-called "Cayley trick". Let L 1 , . . . , L s be the line bundles associated to the quasi-smooth hypersurfaces X 1 , . . . X s , and so let P(E) be the projective bundle of E = L 1 ⊕ ⋅ ⋅ ⋅ ⊕ L s . It turns out that P(E) is a d + s − 1-dimensional projective simplicial toric variety whose Cox ring is The hypersurface Y is cut off by the polynomial F = y 1 f 1 + ⋅ ⋅ ⋅ + y s f s and is quasi-smooth by Lemma 2.2 in [14]. Moreover, combining Theorem 10.13 in [1] and Theorem 3.6 in [14], we have that Here R(F ) is the Jacobian ring of Y , i.e., the quotient of the Cox ring where J(F ) is the ideal generated by the derivatives of F , see [1].
Remark 2.4. With the same notation of Proposition 2.3, note that we have a well defined Moreover, by the Noether-Lefschetz theorem NL β is a countable union of closed sets ⋃ i C i and hence ⋃ φ −1 (C i ) is too. △ We have a Noether-Lefschetz type theorem, namely, Theorem 2.5. Let P d Σ be an Oda projective simplicial toric variety. Then for a very general quasi-smooth intersection subvariety X cut off by f 1 , . . . f s such that d + s = 2(k + 1) and So we get a natural generalization of the Noether-Lefschetz loci.
Definition 2.6. The Noether-Lefschetz locus NL β 1 ,...,βs of quasi-smooth intersection varieties is the locus of s−tuples (f 1 , . . . , f s ) such that Now we consider the Hodge conjecture for very general quasi-smooth intersection subvarieties in P d Σ .
Corollary 2.7. If P d Σ is a Oda projective simplicial toric variety, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety X cut off by Proof. First note that by Thereom 4.1 in [11] the projective simplicial toric variety P 2k+1 X,Σ is Oda and since X is very general the quasi-smooth hypersurface Y is very general as well. So applying the Noether-Lefschetz theorem one has that h k,k prim (Y ) = 0 = h k+1−s,k+1−s prim (X) or equivalently every (k + 1 − s, k + 1 − s) cohomology class is a linear combination of algebraic cycles.

Cox-Gorenstein ideals
We shall need a partial generalization of Macaulay's theorem (see e.g. Thm. 6.19 in [17] for the classical theorem). This generalization is basically contained in the work of Cox and Cattani-Cox-Dickenstein [8,6].
Let S be the Cox ring of a complete simplicial toric variety P Σ . This is graded over the effective classes in the class group Cl(P Σ ) and [7] S α ≃ H 0 (P Σ , O P Σ (α)).
As O P Σ (α) is coherent and P Σ is complete, each S α is finite-dimensional over C; in particular, S 0 ≃ C. Lemma 3.1. For every effective N ∈ Cl(P Σ ), the set of classes α ∈ Cl(P Σ ) such that N − α is effective is finite.
Proof. Since the torsion submodule of Cl(P Σ ) is finite, we may assume that Cl(P Σ ) is free. Then the exact sequence 0 → M → Div T (P Σ ) → Cl(P Σ ) → 0 splits, and we may identify Cl(P Σ ) with a free subgroup of Div T (P Σ ), generated by a subset {D 1 , . . . , D r } of T-invariant divisors. A class in Cl(P Σ ) is effective if and only its coefficients on this basis are nonnegative, whence the claim follows.
We shall give a definition of Cox-Gorenstein ideal of the Cox rings which generalizes to toric varieties the definition given by Otwinowska in [15] for projective spaces. Let B ⊂ S be the irrelevant ideal, and for a graded ideal I ⊂ B, denote by V T (I) the corresponding closed subscheme of P Σ .
there exists a C-linear form Λ ∈ (S N ) ∨ such that for all α ∈ Cl(P Σ )

the natural bilinear morphism
is nondegenerate whenever α and N − α are effective.
Proof. 1. From eq. (2) we see that the sequence , wherex,ȳ are pre-images of x, y in S. One easily checks that this is well defined and that via the isomorphism R N ≃ k it coincides with the pairing (3). Now if x ∈ R α and Φ(x, y) = 0 for all y ∈ R N −α then Λ(xȳ) = 0 for all y ∈ S N −α so thatx ∈ I α , i.e., x = 0.
Let f 0 , . . . , f d be homogeneous polynomials, f i ∈ S α i , where d = dim P Σ and each α i is ample, and let N = ∑ i α i − β 0 , where β 0 is the anticanonical class of P Σ . Assume that the f i have no common zeroes in P Σ , i.e., V T (I) = ∅ if I = (f 0 , . . . , f d ).
In [1,8,6] it is shown that for each G ∈ S N one can define a meromorphic d-form ξ G on P Σ by letting where Ω is a Euler form on P Σ . The form ξ G determines a class in H d (P Σ , ω), where ω is the canonical sheaf of P Σ (the sheaf of Zariski d-forms on P Σ ), and in turn the trace morphism Tr P Σ ∶ H d (P Σ , ω) → C associates a complex number to G, so we can define Λ ∈ (S N ) ∨ as Finally, we can prove a toric version of Macaulay's theorem. Proof. By Prop. 3.13 in [6] the map Λ establishes an isomorphism R N ≃ C. Hence, if f ∈ S α is such that Λ(f g) = 0 for all g ∈ S N −α , then f g ∈ I N , which implies f ∈ I α . On the other hand, it is clear that Λ(f g) = 0 if f ∈ I α and g ∈ S N −α .
Another example is given in terms of toric Jacobian ideals. For every ray ρ ∈ Σ(1) we shall denote by v ρ its rational generator, and by x ρ the corresponding variable in the Cox ring. Recall that d is the dimension of the toric variety P Σ , while we denote by r = #Σ(1) the number of rays. Given f ∈ S β one defines its toric Jacobian ideal as We recall from [1] the definition of nondegenerate hypersurface and some properties (Def. 4.13 and Prop. 4.15). Definition 3.6. Let f ∈ S(Σ) β , with β an ample Cartier class. The associated hypersurface X f is nondegenerate if for all σ ∈ Σ the affine hypersurface X f ∩ O(σ) is a smooth codimension one subvariety of the orbit O(σ) of the action of the torus T d .
2. If f is generic then X f is nondegenerate.
The following is part of Prop. 5.3 in [8], with some changes in the terminology. 1. The toric Jacobian ideal of f coincides with the ideal 2. The following conditions are equivalent: (a) f is nondegenerate;

Asymptotic Hodge conjecture
Let us recall part of the notation and assumptions of [4]. Let P 2k+1 Σ be an Oda variety with an ample Cartier class β such that kβ − β 0 = nη, where β 0 is the anticanonical class, η is a primitive ample class and n ∈ N. Let X f ∈ β be a quasi-smooth hypersurface in the Noether-Lefschetz locus associated to a nontrival Hodge class λ ∈ F k H 2k (U). Let r be number of rays of Σ, so that r ≥ 2(k + 1). Assuming that n is big enough, it follows from Proposition 4.7 or Theorem 6.1 in [4] that there exists a k-dimensional subvariety V of X f satisfying the following conditions: • deg V ≤ min{2δm β , d} with 0 < δ < 1 4(r−(k+1)) (the number m β was defined in Eq. (1)); • the graded ideals I V and coincide in degree less than or equal to (m β − 2 − (r − j) deg V ) η for some j, with 0 < j < r. Here Tub(−) is the adjoint of the residue map, and N = (k + 1)β − β 0 is the socle degre of the Cox-Gorenstein ideal E, while is the Poincaré dual of some rational combination of the homology cycles γ i generating H 2k (X f , Q). Moreover, via the isomorphism T f U ≃ S β , the degree β summand E β of E is identified with the tangent space T f N k,β λ,U to the Noether-Lefschetz locus, so that E β contains the degree β part J(f ) β the Jacobian ideal of f .
We denote by λ V the class of V in H k,k prim (X f , Q).
Theorem 4.2. If V is a smooth intersection subvariety, there exists c ∈ Q * such that Proof. We divide the proof in three steps.
Step I: λ V ≠ 0. Since V ⊂ X f is a regular embedding we have which implies that β = deg(X f ) is a proper divisor of deg V , which is a contradiction, so that Step I is proved.
Step II. Let E alg and E be the Cox-Gorenstein ideal associated to λ V and λ f , respectively, as in equation (5).
To prove the theorem it is enough to show that E = E alg . Note that I V + J 0 (f ) is contained in E and E alg . Moreover, since V ⊂ X f , and f is quasi-smooth, there exist K 1 , . . . K k+1 ∈ B such that f = A 1 K 1 + . . . A k+1 K k+1 and (A 1 , . . . , A k+1 , K 1 , . . . K k+1 ) is a Cox-Gorenstein ideal with socle degree N ; this will follow from the next step, which concludes the proof.
Let us see that K j ∈ I for every j ∈ {1, . . . , k + 1}. Let M ∈ Mat(r × (k + 1)) be the matrix Since V is a smooth complete intersection subvariety, it follows that J is base point free, and therefore it contains a complete intersection Cox-Gorenstein ideal J ′ by the toric Macaulay theorem, Theorem 3.5. Since J is generated in degree less than or equal to deg V ⋅ η k , we can take J ′ with the same property. It follows that soc(J ′ ) ≤ 2(k + 1)(deg V )η − β 0 ≤ 2rm β δη − β 0 .
On the other hand if K j ∉ I then (I ∶ K j ) contains a Cox-Gorenstein ideal with socle degree N − deg K j ≥ N − β = kβ − β 0 ; then comparing the above two inequalities and keeping in mind that r ≥ 2(k + 1), we get δ > 1 2r > 1 4(r − (k + 1)) , which is absurd.