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

Abstract

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 Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). 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, under the assumption that the ambient variety ${{\mathbb {P}}}_\Sigma ^{2k+1}$ has Picard group ${\mathbb {Z}}$. This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

1 Introduction

A projective simplicial toric variety ${\mathbb {P}}^d_{\Sigma }$ satisfies the Hodge Conjecture, i.e., every cohomology class in $H^{p,p}({\mathbb {P}}^{d}_{\Sigma },{\mathbb {Q}})$ 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 $p<\frac{d-1}{2}$ and by the hard Lefschetz theorem also for $p>\frac{d-1}{2}$. Moreover, by Theorem 1.1 in [3], when $p=\frac{d-1}{2}$, $d=2k+1$ and ${\mathbb {P}}_{\Sigma }^{2k+1}$ is an Oda variety with an ample class $\beta$ such that $k\beta -\beta _0$ is nef, where $\beta _0$ is the anticanonical class, the Hodge conjecture with rational coefficients holds for a very general hypersurface in the linear system $|\beta |$.

The notion of Oda varieties was introduced in [2]. Let us recall that the Cox ring of a toric variety ${\mathbb {P}}_\Sigma$ is graded over the class group ${\text {Cl}}({\mathbb {P}}_\Sigma )$, and that one has an injection ${\text {Pic}}({\mathbb {P}}_\Sigma ) \rightarrow {\text {Cl}}({\mathbb {P}}_\Sigma )$.

Definition 1.1

Let ${\mathbb {P}}_\Sigma$ be a toric variety with Cox ring S. ${\mathbb {P}}_\Sigma$ is said to be an Oda variety if the multiplication morphism $S^{\alpha _1 }\otimes S^{\alpha _2} \rightarrow S^{\alpha _1+\alpha _2}$ is surjective whenever the classes $\alpha _1$ and $\alpha _2$ in ${\text {Pic}}({\mathbb {P}}_\Sigma )$ are ample and nef, respectively.

In [15] 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 $X=X_{f_1}\cap \dots \cap X_{f_s}\subset {\mathbb {P}}^{d}_{\Sigma }$ to the cohomology of a quasi-smooth hypersurface $Y\subset {\mathbb {P}}^{d+s-1}_{\Sigma }$. This allows us to prove a Noether–Lefschetz type theorem, namely:

Theorem 2.5. Let ${\mathbb {P}}^d_{\Sigma }$ be an Oda projective simplicial toric variety. For a very general quasi-smooth intersection subvariety X cut off by $f_1,\dots f_s$ such that $d+s=2(\ell +1)$ and

$$\begin{aligned} \sum _{i=1}^s \deg (f_i)- \beta _0 \end{aligned}$$

is nef, one has

$$\begin{aligned} H^{\ell +1-s,\ell +1-s}(X,{\mathbb {Q}})=i^*\left( H^{\ell +1-s,\ell +1-s}({\mathbb {P}}^d_{\Sigma },{\mathbb {Q}})\right) . \end{aligned}$$

From this one obtains the following result about the Hodge conjecture for quasi-smooth intersections.

Corollary 2.7. If ${\mathbb {P}}_{\Sigma }^d$ is an Oda projective simplicial toric variety, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety X cut off by $f_1,\dots f_s$ such that $d+s$ is even and $\sum _{i=1}^s \deg (f_i)- \beta _0$ is nef.

Let T be the open subset of $\vert \beta \vert$ corresponding to quasi-smooth hypersurfaces, and let ${\mathcal {H}}^{2k}=R^{2k}\pi _*{\mathbb {C}}\otimes _{\mathbb {C}}{\mathcal {O}}_T$ be the Hodge bundle on T; here $\pi :{\mathcal {X}} \rightarrow T$ is the tautological family on T, and $d=2k+1$. We restrict ${\mathcal {H}}^{2k}$ to a contractible open subset $U\subset T$. The bundle ${\mathcal {H}}^{2k}$ has a Hodge decomposition

$$\begin{aligned} {\mathcal {H}}^{2k} = \bigoplus _{p+q=2k}{\mathcal {H}}^{p,q} \end{aligned}$$

but this is not holomorphic. On the other hand, the bundles that make up the Hodge filtration

$$\begin{aligned} F^p{\mathcal {H}}^{2k} = \bigoplus _{p=0}^{2k}{\mathcal {H}}^{2k-p,p} \end{aligned}$$

are holomorphic; to see this one can use the period map (which in particular we write for $p=k$)

$$\begin{aligned} {\mathcal {P}}^{k,2k}:U \rightarrow {\text {Grass}}(b_k,H^{2k}(X_{u_0},{\mathbb {C}})) \end{aligned}$$

where $b_k = \dim F^kH^{2k}(X_{u_0},{\mathbb {C}})$ for a fixed point $u_0\in U$; this map sends $f\in U$ to the subspace $F^kH^{2k}(X_f,{\mathbb {C}}) \subset H^{2k}(X_f,{\mathbb {C}}) = H^{2k}(X_{u_0},{\mathbb {C}})$. This map is holomorphic (see [14] and [5, Prop. 3.4]). But, by the very definition of the period map (see also [17], Section 10.2.1 for the smooth case)

$$\begin{aligned} F^k{\mathcal {H}}^{2k} \simeq ({\mathcal {P}}^{k,2k})^*{\mathcal {U}}_k, \end{aligned}$$

where ${\mathcal {U}}_k$ is the tautological bundle on the Grassmannian ${\text {Grass}}(b_k,H^{2k}(X_{u_0},{\mathbb {C}}))$, so that the bundles $F^k{\mathcal {H}}^{2k}$ are indeed holomorphic.

Pushing ahead the ideas developed in [5] and [4], let $\lambda _f$ be a nonzero class in the primitive cohomology $H^{k,k}(X_f,{\mathbb {Q}})/H^{k,k}({{\mathbb {P}}}_\Sigma ^{2k+1},{\mathbb {Q}})$, and let U be a contractible open subset of T around f, so that ${\mathcal {H}}^{2k}_{\vert U}$ is constant. Moreover, let $\lambda \in {\mathcal {H}}^{2k}(U)$ be the section defined by $\lambda _f$ and let ${\bar{\lambda }}$ be its image in $({\mathcal {H}}^{2k}/F^k{\mathcal {H}}^{2k})(U)$. One has

Proposition 1.2

The local Noether–Lefschetz loci can be defined as

$$\begin{aligned} N^{k,\beta }_{\lambda ,U}:=\{G\in U \mid {\bar{\lambda }}_{G}=0\} \end{aligned}$$

where $\beta =\deg (f)$.

The following result is Theorem 1.2 in [4].

Theorem. Let ${{\mathbb {P}}}_\Sigma ^{2k+1}$ be an Oda variety with an ample class $\beta$ such that $k\beta -\beta _0=n\eta$, where $\beta _0$ is the anticanonical class, $\eta$ is a primitive ample class, and $n\in {\mathbb {N}}$. Let

$$\begin{aligned} m_\beta = \max \{i\in {\mathbb {N}}\,\,\vert \,i\eta \le \beta \}. \end{aligned}$$

(1)

For every positive $\epsilon$ there is a positive $\delta$ such that for every $m\ge \max (\frac{1}{\delta },m_\beta )$ and ${\hat{d}}\in [1,m\delta ]$, and every nontrivial Hodge class $\lambda \in F^k{\mathcal {H}}^{2k}(U)$ such that

$$\begin{aligned} {\text {codim}}N^{k,\beta }_{\lambda ,U} \le {\hat{d}}\frac{m_{\beta }^k}{k!}, \end{aligned}$$

for every $f\in N_{\lambda ,U}^{k,\beta }$, there exists a k-dimensional variety $V\subset X_f$ with $\deg V\le (1+\epsilon ) {\hat{d}}$. Here $\deg V$ is taken with respect to the ample divisor $\eta$, i.e.,

$$\begin{aligned} \deg V = [V]\cdot \eta ^k. \end{aligned}$$

Based on this, in this paper we obtain the following result.

Theorem 4.2. Under the same hypotheses of the previous theorem, assume also that ${\text {Pic}}({{\mathbb {P}}}_\Sigma ^{2k+1})={\mathbb {Z}}$. Then, if $V\subset X_f$ is a nonempty quasi-smooth intersection subvariety of ${{\mathbb {P}}}_\Sigma ^{2k+1}$ for some $f\in N_{\lambda ,U}^{k,\beta }$, there exists $c\in {\mathbb {Q}}^*$ such that $\lambda _f=c\lambda _V$, where $\lambda _V$ is the class of V in $H_{\mathrm{prim}}^{k,k}(X_f,{\mathbb {Q}})$.

In other words, $\lambda _f$ is algebraic.

In his paper [11] A. Dan proves a form of our Theorem 4.2 for smooth hypersurfaces in odd-dimensional projective spaces ${\mathbb {P}}^{2k+1}$ which is not asymptotic. Although our result is more general in two ways, as we consider quasi-smooth intersections in toric varieties with $h^{k,k}=1$ (for instance, weighted or fake projective spaces); however, our result is asymptotic.

In Sect. 3 we give an extension of the notion of Gorenstein ideal to Cox rings; this may have some interest on its own.

2 Very general quasi-smooth intersections

Let $f_1,\dots ,f_s$ be homogeneous polynomials in the Cox ring $S = {\mathbb {C}}[x_1,\dots ,x_n]$ of ${\mathbb {P}}_{\Sigma }^d$. Their zero locus $V(f_1,\dots ,f_s)$ defines a closed subvariety $X\subset {\mathbb {P}}^{d}_{\Sigma }$. Let $U(\Sigma )= {\mathbb {A}}^n -Z(\Sigma )$, where $Z(\Sigma )$ is the irrelevant locus, i.e., $Z(\Sigma ) = {\text {Spec}}B$, where B is the irrelevant ideal.

Definition 2.1

[15] X is a codimension s quasi-smooth intersection if $V(f_1,\dots ,f_s)\cap U(\Sigma )$ is either empty or a smooth intersection subvariety of codimension s in $U(\Sigma )$.

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 ${\mathbb {P}}_{\Sigma }^d$ as an orbifold, then an intersection of hypersurfaces $X_{f_1}\cap \dots \cap X_{f_s}$ is quasi-smooth when it is a sub-orbifold of ${\mathbb {P}}_{\Sigma }^d$ , see Prop 1.3 [15]; heuristically, “X has only singularities coming from the ambient variety.”

We also have a Lefschetz type theorem in this context.

Proposition 2.2

( [15] Proposition 1.4) Let $X\subset {\mathbb {P}}_{\Sigma }^d$ be a closed subset, defined by homogeneous polynomials $f_1,\dots f_s\in B$. Then the natural map $i^{*}: H^i({\mathbb {P}}_{\Sigma }^d)\rightarrow H^i(X)$ 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 $p\ne \frac{d-s}{2}$ every cohomology class in $H^{p,p}(X)$ is a linear combination of algebraic cycles. So let us see what happens when $p=\frac{d-s}{2}$. The idea is to relate the Hodge structure of a quasi-smooth intersection variety $X=X_{f_1}\cap \dots \cap X_{f_s}$ in ${\mathbb {P}}^d_{\Sigma }$ with the Hodge structure of a quasi-smooth hypersurface Y in a toric variety ${\mathbb {P}}^{d+s-1}_{X,\Sigma }$ whose fan depends on X and $\Sigma$.

Proposition 2.3

Let $X=X_{f_1} \cap \dots \cap X_{f_s}$ be quasi-smooth intersection subvariety in ${\mathbb {P}}_{\Sigma }^d$ cut off by homogeneous polynomials $f_1\dots f_s$. There exists a projective simplicial toric variety ${\mathbb {P}}^{d+s-1}_{X,\Sigma }$ and a quasi-smooth hypersurface $Y\subset {\mathbb {P}}^{d+s-1}_{X,\Sigma }$ such that for $p\ne \frac{d+s-1}{2}, \frac{d+s-3}{2}$

$$\begin{aligned} H^{p-1,d+s-1-p}_{\mathrm{\small prim}}(Y)\simeq H^{p-s,d-p}_{\mathrm{\small prim}}(X). \end{aligned}$$

Proof

One constructs ${\mathbb {P}}^{d+s-1}_{X,\Sigma }$ via the so-called “Cayley trick”. Let $L_1,\dots , L_s$ be the line bundles associated to the quasi-smooth hypersurfaces $X_1,\dots X_s$, and so let ${\mathbb {P}}(E)$ be the projective bundle of $E=L_1\oplus \dots \oplus L_s$. It turns out that ${\mathbb {P}}(E)$ is a $d+s-1$- dimensional projective simplicial toric variety whose Cox ring is

$$\begin{aligned} {\mathbb {C}}[x_1,\dots ,x_n,y_1,\dots y_s] \end{aligned}$$

where $S={\mathbb {C}}[x_1,\dots , x_n]$ is the Cox ring of ${\mathbb {P}}^{d}_{\Sigma }$. The hypersurface Y is cut off by the polynomial $F=y_1f_1+\dots + y_sf_s$ and is quasi-smooth by Lemma 2.2 in [15]. Moreover, combining Theorem 10.13 in [1] and Theorem 3.6 in [15], we have that

$$\begin{aligned} {H^{p-1,d+s-1-p}_{\mathrm{\small prim}}(Y)}\simeq { R(F)_{(d+s-p)\beta -\beta _0}} \simeq H^{p-s,d-p}_{\mathrm{\small prim}}(X) \end{aligned}$$

for $p\ne \frac{d+s-1}{2}, \frac{d+s-3}{2}$ as desired. $\square$

Here R(F) is the Jacobian ring of Y, i.e., the quotient of the Cox ring

$$\begin{aligned} R(F) = {\mathbb {C}}[x_1,\dots ,x_n,y_1,\dots y_s]/J(F), \end{aligned}$$

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 map

$$\begin{aligned} \begin{array}{rcl} \phi : |\beta _1|\times \dots \times |\beta _s| &{}\rightarrow &{}|\beta |\\ (f_1,\dots ,f_s)&{}\mapsto &{} f_1y_1+\dots +f_sy_s. \end{array} \end{aligned}$$

Moreover, by the Noether-Lefschetz theorem $NL_{\beta }$ is a countable union of closed sets $\bigcup _i {C_i}$ and hence $\bigcup \phi ^{-1}(C_i)$ is too.

We have a Noether-Lefschetz type theorem, namely,

Theorem 2.5

Let ${\mathbb {P}}^d_{\Sigma }$ be an Oda projective simplicial toric variety. Then for a very general quasi-smooth intersection subvariety X cut off by $f_1,\dots f_s$ such that $d+s=2(l+1)$ and $\sum _{i=1}^s \deg (f_i)-\beta _0$ is nef, one has that

$$\begin{aligned} H^{l+1-s,l+1-s}(X,{\mathbb {Q}})=i^*\left( H^{l+1-s,l+1-s}({\mathbb {P}}^d_{\Sigma },{\mathbb {Q}})\right) \end{aligned}$$

So we get a natural generalization of the Noether-Lefschetz loci.

Definition 2.6

The Noether-Lefschetz locus $NL_{\beta _1,\dots ,\beta _s}$ of quasi-smooth intersection varieties is the locus of $s-$tuples $(f_1,\dots ,f_s)$ such that $X=X_{f_1}\cap \dots X_{f_s}$ is quasi-smooth intersection with $f_i\in |\beta _i|$ and $H^{l+1-s,l+1-s}(X,{\mathbb {Q}})\ne i^*\left( H^{l+1-s,l+1-s}({\mathbb {P}}^d_{\Sigma },{\mathbb {Q}})\right)$.

Now we consider the Hodge conjecture for very general quasi-smooth intersection subvarieties in ${\mathbb {P}}^d_{\Sigma }$.

Corollary 2.7

If ${\mathbb {P}}_{\Sigma }^d$ is a Oda projective simplicial toric variety, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety X cut off by $f_1,\dots f_s$ such that $d+s=2(l+1)$ and $\sum _{i=1}^s \deg (f_i)- \beta _0$ is nef.

Proof

First note that by Thereom 4.1 in [12] the projective simplicial toric variety ${\mathbb {P}}_{X,\Sigma }^{2l+1}$ 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 $h^{l,l}_{\mathrm{prim}}(Y)=0= h^{l+1-s,l+1-s}_{\mathrm{prim}}(X)$ or equivalently every $(l+1-s,+1-s)$ cohomology class is a linear combination of algebraic cycles. $\square$

3 Cox-Gorenstein ideals

We shall need a partial generalization of Macaulay’s theorem (see e.g. Thm. 6.19 in [18] for the classical theorem). This generalization is basically contained in the work of Cox and Cattani-Cox-Dickenstein [7, 9].

Let S be the Cox ring of a complete simplicial toric variety ${\mathbb {P}}_\Sigma$. This is graded over the effective classes in the class group ${\text {Cl}}({\mathbb {P}}_\Sigma )$ and [8]

$$\begin{aligned} S^\alpha \simeq H^0({\mathbb {P}}_\Sigma ,{\mathcal {O}}_{{\mathbb {P}}_\Sigma }(\alpha )). \end{aligned}$$

As ${\mathcal {O}}_{{\mathbb {P}}_\Sigma }(\alpha )$ is coherent and ${\mathbb {P}}_\Sigma$ is complete, each $S^\alpha$ is finite-dimensional over ${\mathbb {C}}$; in particular, $S^0\simeq {\mathbb {C}}$.

Lemma 3.1

For every effective $N\in {\text {Cl}}({\mathbb {P}}_\Sigma )$, the set of classes $\alpha \in {\text {Cl}}({\mathbb {P}}_\Sigma )$ such that $N-\alpha$ is effective is finite.

Proof

Since the torsion submodule of ${\text {Cl}}({\mathbb {P}}_\Sigma )$ is finite, we may assume that ${\text {Cl}}({\mathbb {P}}_\Sigma )$ is free. Then the exact sequence

$$\begin{aligned} 0 \rightarrow M \rightarrow {\text {Div}}_{{\mathbb {T}}}({\mathbb {P}}_\Sigma ) \rightarrow {\text {Cl}}({\mathbb {P}}_\Sigma ) \rightarrow 0 \end{aligned}$$

splits, and we may identify ${\text {Cl}}({\mathbb {P}}_\Sigma )$ with a free subgroup of ${\text {Div}}_{{\mathbb {T}}}({\mathbb {P}}_\Sigma )$, generated by a subset $\{D_1,\dots ,D_r\}$ of ${\mathbb {T}}$-invariant divisors. A class in ${\text {Cl}}({\mathbb {P}}_\Sigma )$ is effective if and only its coefficients on this basis are nonnegative, whence the claim follows. $\square$

We shall give a definition of Cox-Gorenstein ideal of the Cox rings which generalizes to toric varieties the definition given by Otwinowska in [16] for projective spaces. Let $B\subset S$ be the irrelevant ideal, and for a graded ideal $I\subset B$, denote by $V_{{\mathbb {T}}}(I)$ the corresponding closed subscheme of ${\mathbb {P}}_\Sigma$.

Definition 3.2

A graded ideal I of S contained in B is said to be a Cox-Gorentstein ideal of socle degree $N\in {\text {Cl}}({\mathbb {P}}_\Sigma )$ if

• 1. there exists a ${\mathbb {C}}$-linear form $\Lambda \in (S^N)^\vee$ such that for all $\alpha \in {\text {Cl}}({\mathbb {P}}_\Sigma )$

  $$\begin{aligned} I^\alpha =\{f\in S^\alpha \,\vert \, \Lambda (fg) = 0 \ \hbox {for all} \ g\in S^{N-\alpha }\}; \end{aligned}$$

  (2)

• 2. $V_{{\mathbb {T}}}(I)=\emptyset$.

Remark 3.3

Cox-Gorenstein ideals need not be Artinian. Property 2 in this definition replaces that condition.

Proposition 3.4

Let $R=S/I$. If I is Cox-Gorenstein then

• 1. $\dim _{\mathbb {C}}R^N = 1$;

• 2. the natural bilinear morphism

  $$\begin{aligned} R^\alpha \times R^{N-\alpha } \rightarrow R^N\simeq {\mathbb {C}}\end{aligned}$$

  (3)

  is nondegenerate whenever $\alpha$ and $N-\alpha$ are effective.

Proof

• 1. From eq. (2) we see that the sequence

  $$\begin{aligned} 0 \rightarrow I^N \rightarrow S^N \xrightarrow {\Lambda } {\mathbb {C}}\rightarrow 0 \end{aligned}$$

  is exact.

• 2. Define $\Phi :R^\alpha \times R^{N-\alpha } \rightarrow {\mathbb {C}}$ as $\Phi (x,y) = \Lambda ({{\bar{x}}}{{\bar{y}}})$, where ${{\bar{x}}}$, ${{\bar{y}}}$ are pre-images of x, y in S. One easily checks that this is well defined and that via the isomorphism $R^N\simeq k$ it coincides with the pairing (3). Now if $x\in R^\alpha$ and $\Phi (x,y) = 0$ for all $y\in R^{N-\alpha }$ then $\Lambda ({{\bar{x}}}{{\bar{y}}}) =0$ for all ${{\bar{y}}}\in S^{N-\alpha }$ so that ${{\bar{x}}} \in I^\alpha$, i.e., $x=0$. $\square$

Let $f_0,\dots ,f_d$ be homogeneous polynomials, $f_i\in S^{\alpha _i}$, where $d=\dim {\mathbb {P}}_\Sigma$ and each $\alpha _i$ is ample, and let $N = \sum _i\alpha _i-\beta _0$, where $\beta _0$ is the anticanonical class of ${\mathbb {P}}_\Sigma$. Assume that the $f_i$ have no common zeroes in ${\mathbb {P}}_\Sigma$, i.e., $V_{{\mathbb {T}}}(I)=\emptyset$ if $I=(f_0,\dots ,f_d)$.

In [1, 7, 9] it is shown that for each $G \in S^N$ one can define a meromorphic d-form $\xi _G$ on ${\mathbb {P}}_\Sigma$ by letting

$$\begin{aligned} \xi _G = \frac{G\,\Omega }{f_0\cdots f_d} \end{aligned}$$

where $\Omega$ is a Euler form on ${\mathbb {P}}_\Sigma$. The form $\xi _G$ determines a class in $H^d({\mathbb {P}}_\Sigma ,\omega )$, where $\omega$ is the canonical sheaf of ${\mathbb {P}}_\Sigma$ (the sheaf of Zariski d-forms on ${\mathbb {P}}_\Sigma$), and in turn the trace morphism ${\text {Tr}}_{{\mathbb {P}}_\Sigma }:H^d({\mathbb {P}}_\Sigma ,\omega )\rightarrow {\mathbb {C}}$ associates a complex number to G, so we can define $\Lambda \in (S^N)^\vee$ as

$$\begin{aligned} \Lambda (G) = {\text {Tr}}_{{\mathbb {P}}_\Sigma }([\xi _G])\in {\mathbb {C}}. \end{aligned}$$

(4)

Finally, we can prove a toric version of Macaulay’s theorem.

Theorem 3.5

The linear map defined in Eq. (4) satisfies the condition in Definition 3.2. Therefore, the ideal $I=(f_0,\dots ,f_d)$ is a Cox-Gorenstein ideal of socle degree N.

Proof

By Theorem 6 in [7] the map $\Lambda$ establishes an isomorphism $R^N \simeq {\mathbb {C}}$. Hence, if $f\in S^\alpha$ is such that $\Lambda (fg)=0$ for all $g\in S^{N-\alpha }$, then $fg\in I^N$, which implies $f\in I^\alpha$. On the other hand, it is clear that $\Lambda (fg)=0$ if $f\in I^\alpha$ and $g\in S^{N-\alpha }$. $\square$

Another example is given in terms of toric Jacobian ideals. For every ray $\rho \in \Sigma (1)$ we shall denote by $v_\rho$ its rational generator, and by $x_\rho$ the corresponding variable in the Cox ring. Recall that d is the dimension of the toric variety ${\mathbb {P}}_\Sigma$, while we denote by $r=\#\Sigma (1)$ the number of rays. Given $f\in S^\beta$ one defines its toric Jacobian ideal as

$$\begin{aligned} J_0(f) = \left( x_{\rho _1} \frac{\partial f }{\partial x_{\rho _1}}, \dots , x_{\rho _r} \frac{\partial f }{\partial x_{\rho _r}} \right) . \end{aligned}$$

We recall from [1] the definition of nondegenerate hypersurface and some properties (Def. 4.13 and Prop. 4.15).

Definition 3.6

Let $f\in S(\Sigma )^\beta$, with $\beta$ an ample Cartier class. The associated hypersurface $X_f$ is nondegenerate if for all $\sigma \in \Sigma$ the affine hypersurface $X_f\cap O(\sigma )$ is a smooth codimension one subvariety of the orbit $O(\sigma )$ of the action of the torus ${\mathbb {T}}^d$.

Proposition 3.7

• 1. Every nondegenerate hypersurface is quasi-smooth.

• 2. If f is generic then $X_f$ is nondegenerate.

The following is part of Prop. 5.3 in [9], with some changes in the terminology.

Proposition 3.8

Let $f\in S(\Sigma )^\beta$, and let $\{\rho _1,\dots ,\rho _d\}\subset \Sigma (1)$ be such that $v_{\rho _1},\dots ,v_{\rho _d}$ are linearly independent.

• 1. The toric Jacobian ideal of f coincides with the ideal

  $$\begin{aligned} \left( f, x_{\rho _1} \frac{\partial f }{\partial x_{\rho _1}}, \dots , x_{\rho _d} \frac{\partial f }{\partial x_{\rho _d}} \right) . \end{aligned}$$

• 2. The following conditions are equivalent:

  • (a) f is nondegenerate;

  • (b) the polynomials $x_{\rho _i} \frac{\partial f }{\partial x_{\rho _i}}$, $i=1,\dots ,r$, do not vanish simultaneously on $X_f$;

  • (c) the polynomials f and $x_{\rho _i} \frac{\partial f }{\partial x_{\rho _i}}$, $i=1,\dots ,d$,do not vanish simultaneously on $X_f$.

• 3. If moreover $\beta$is ample and f is nondegenerate, then $J_0(f)$ is a Cox-Gorenstein ideal of socle degree $N = (d+1){\beta} -{\beta _0}$, where $\beta _0$is the anticanonical class of ${\mathbb {P}}_\Sigma ^d$.

4 Asymptotic Hodge conjecture

In this section we prove Theorem 4.2. Let us recall part of the notation and assumptions of [4]. Let ${{\mathbb {P}}}_\Sigma ^{2k+1}$ be an Oda variety with an ample Cartier class $\beta$ such that $k\beta -\beta _0=n\eta$, where $\beta _0$ is the anticanonical class, $\eta$ is a primitive ample class and $n\in {\mathbb {N}}$.

We need to define a pre-order in the group

$$\begin{aligned} N^1({{\mathbb {P}}}_\Sigma ^{2k+1}) = {\text {Pic}}({{\mathbb {P}}}_\Sigma ^{2k+1})\otimes {\mathbb {Q}}/ \text {numerical equivalence}, \end{aligned}$$

by letting $\alpha < \alpha '$ if $\alpha '-\alpha$ is an effective class.

Let $X_f\in \vert \beta \vert$ be a quasi-smooth hypersurface in the Noether-Lefschetz locus associated to a nontrival Hodge class $\lambda \in F^k{\mathcal {H}}^{2k}(U)$. Again, its degree is computed by intersecting with the ample class $\eta$, i.e., $\deg X_f = [X_f]\cdot \eta$. Let r be number of rays of $\Sigma$, so that $r\ge 2(k+1)$. 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:

• $\deg V\le 2\delta m_\beta$ with $0< \delta < \frac{1}{4(r-(k+1))}$ (the number $m_\beta$ was defined in Eq. (1));

• the graded ideals $I_V$ and

  $$\begin{aligned} E=\{g\in S^{\bullet } \mid \sum _{i=1}^b \lambda _i \int _{{\text {Tub}}\gamma _i} \frac{gh\Omega _0}{f^{k+1}}=0 \ \text{ for } \text{ all }\ h\in S^{N-\bullet } \} , \end{aligned}$$

  (5)

  coincide in degree less than or equal to $\left( m_{\beta }-2-(r-j)\deg V\right) \eta$ for some j, with $0<j<r$. Here ${\text {Tub}}(-)$ is the adjoint of the residue map, and $N=(k+1)\beta -\beta _0$ is the socle degre of the Cox-Gorenstein ideal E, while

  $$\begin{aligned} \lambda _f =\left( \sum _{i=1}^b\,\lambda _i\gamma _i\right) ^{pd} \end{aligned}$$

  is the Poincaré dual of some rational combination of the homology cycles $\gamma _i$ generating $H_{2k}(X_f,{\mathbb {Q}})$. Moreover, via the isomorphism $T_fU\simeq S^{\beta }$, the degree $\beta$ summand $E^\beta$ of E is identified with the tangent space $T_fN^{k,\beta }_{\lambda ,U}$ to the Noether-Lefschetz locus, so that $E^\beta$ contains the degree $\beta$ part $J(f)^\beta$ the Jacobian ideal of f.

Lemma 4.1

The toric Jacobian ideal $J_0(f)$ is contained in E.

Proof

$J_0(f)\subset J(f)$, so that $J_0(f)^\beta \subset J(f)^\beta \subset E^\beta$, and since $J_0(f)$ is generated in degree $\beta$, one has $J_0(f)\subset E$. $\square$

We denote by $\lambda _V$ the class of V in $H^{k,k}_{\mathrm{prim}}(X_f,{\mathbb {Q}})$. In the following theorem we assume that ${\text {Pic}}({{\mathbb {P}}}_\Sigma ^{2k+1})=1$, i.e., that ${{\mathbb {P}}}_\Sigma ^{2k+1}$ is a (possibly fake) weighted projective space [6, 13] (cf. [10] Exer. 5.1.13). This implies that $h^{p,p}({{\mathbb {P}}}_\Sigma ^{2k+1})=1$ for all p.

Theorem 4.2

If V is a quasi-smooth intersection subvariety, there exists $c\in {\mathbb {Q}}^*$ such that $\lambda _f=c\lambda _V$.

Proof

We divide the proof in three steps.

Step I: $\lambda _V \ne 0$. For clarity, for every cohomology class of a subvariety we denote in the cohomology of which ambient variety we consider it (so we write $[V]_{X_f}$ and $[V]_{{{\mathbb {P}}}_\Sigma ^{2k+1}}$). Since $V\subset X_f$ is a regular embedding we have

$$\begin{aligned} \begin{array}{ccl} [V]^2_{X_f}&{}=&{}\int _{V} c_k(N_{V/X_f})=\int _V \left[ c(N_{V/{{\mathbb {P}}}_\Sigma ^{2k+1}}) / c(N_{X_f/{{\mathbb {P}}}_\Sigma ^{2k+1}|V})\right] _k\\[8pt] &{}=&{} \int _{{{\mathbb {P}}}_\Sigma ^{2k+1}} [V]_{{{\mathbb {P}}}_\Sigma ^{2k+1}} \cup \Xi _k \end{array} \end{aligned}$$

(6)

where $\Xi _k$ is the component in $H^{k,k}({{\mathbb {P}}}_\Sigma ^{2k+1})$ of

$$\begin{aligned} \Xi = \frac{\prod _i (1+A_i)}{1 + [X_f]_{{{\mathbb {P}}}_\Sigma ^{2k+1}}}; \end{aligned}$$

here $A_1,\dots ,A_{k+1}$ are the classes in ${\text {Cl}}({{\mathbb {P}}}_\Sigma ^{2k+1})$ of the hypersurfaces that cut the quasi-smooth intersection subvariety V. The claim is proved by contradiction: if $[V]_{X_f}$ is the restriction of a class in $H^{k,k}({{\mathbb {P}}}_\Sigma ^{2k+1})$, i.e.,

$$\begin{aligned} {[}V]_{X_f} =b \, \eta ^k_{\vert X_f} \end{aligned}$$

for some b, then comparing this with (6) we obtain

$$\begin{aligned} \deg V = m_k \,\deg X_f, \end{aligned}$$

(7)

where $m_k$ is defined by $\Xi _k =m_k\,\eta ^k$. But (7) cannot be true when $\deg X_f$ is big enough.

Step II. Let $E_{\mathrm{alg}}$ and E be the Cox-Gorenstein ideals associated to $\lambda _V$ and $\lambda _f$, respectively, as in Eq. (5). To prove the theorem it is enough to show that $E=E_{\mathrm{alg}}$. Note that $I_V+J_0(f)$ is contained in E and $E_{\mathrm{alg}}$. Moreover, since $V\subset X_f$, and f is quasi-smooth, there exist $K_1,\dots K_{k+1}\in B$ such that $f=A_1K_1+\dots A_{k+1}K_{k+1}$ and $(A_1,\dots ,A_{k+1}, K_1,\dots K_{k+1})$ is a Cox-Gorenstein ideal with socle degree N; this will follow from the next step, which concludes the proof.

Step III. It is enough to show that every Cox-Gorenstein ideal ${\mathcal {I}}$ of socle degree N containing $I_V+J_0(f)$ also contains $(A_1,\dots ,A_{k+1}, K_1,\dots K_{k+1})$. By assumption

$$\begin{aligned} \left( A_j,j\in \{1,\dots , k+1\}, \sum _{j=1}^{k+1}x_i\frac{\partial A_j}{\partial x_i}K_j, i\in {1,\dots , r} \right) \subset {\mathcal {I}}. \end{aligned}$$

Let us see that $K_j\in {\mathcal {I}}$ for every $j\in \{1,\dots ,k+1\}$. Let $M\in {\text {Mat}}(r\times (k+1))$ be the matrix $[x_i\frac{\partial A_j}{\partial x_i}]$ and K the column $(K_j)_{j\in \{1,\dots ,k+1\}}$. Let $I\subset \{1,\dots r\}$ with cardinality $k+1$ and let $M_I$ be the matrix obtained extracting the $i\in I$-arrows of M. We have that $\sum _{j=1}^{k+1}x_i\frac{\partial A_j}{\partial x_i}K_j=(MK)_i=(M_IK)_i$; multiplying by the adjoint of $M_I$ we get that $\det (M_I)K_j\in {\mathcal {I}}$ for all $j\in \{1,\dots k+1\}$. On one hand the ideal $({\mathcal {I}}:K_j)$ contains the ideal

$$\begin{aligned} {\mathcal {J}}=I_V+\left\langle \det M_I\,\vert \, I\subset \{1,\dots ,r\},\ \# I = k+1\right\rangle . \end{aligned}$$

Since V is a smooth complete intersection subvariety, it follows that ${\mathcal {J}}$ is base point free, and therefore it contains a complete intersection Cox-Gorenstein ideal ${\mathcal {J}}'$ by the toric Macaulay theorem, Theorem 3.5. Since ${\mathcal {J}}$ is generated in degree less than or equal to $(\deg V) \eta$, we can take ${\mathcal {J}}'$ with the same property. It follows that

$$\begin{aligned} soc({\mathcal {J}}') \le 2(k+1)(\deg V)\eta - \beta _0 \le 2rm_\beta \delta \eta -\beta _0. \end{aligned}$$

On the other hand if $K_j\notin {\mathcal {I}}$ then $({\mathcal {I}}: K_j)$ contains a Cox-Gorenstein ideal with socle degree

$$\begin{aligned} N-\deg K_j\ge N-\beta = k\beta -\beta _0; \end{aligned}$$

then comparing the above two inequalities and keeping in mind that $r\ge 2(k+1)$, we get

$$\begin{aligned} \delta \ge \frac{1}{2r} \ge \frac{1}{4(r-(k+1))}, \end{aligned}$$

which is absurd. $\square$

On behalf of all authors, the corresponding author states that there is no conflict of interest.