On a multiplicative version of Mumford’s theorem

Abstract

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove a similar statement for Chow groups of arbitrary codimension, provided the variety satisfies the Lefschetz standard conjecture.

1 Introduction

Since Mumford’s famous 1969 paper [12], it is well-known that the Chow group of 0-cycles $A^nX$ on a complex variety X influences the cohomology group $H^n(X,\mathbb {Q})$:

Theorem

(Mumford [12]) Let X be a smooth complete variety of dimension n defined over $\mathbb {C}$. Suppose that $A^nX_{\mathbb {Q}}$ is supported on a divisor. Then $H^n(X,\mathbb {Q})$ is supported on a divisor, in particular $H^n(X,\mathcal O_X)=0$.

In the 1992 paper [5], Esnault, Srinivas and Viehweg study the multiplicative behaviour of the Chow ring $A^*X$ versus the multiplicative behaviour of various cohomology rings associated to X. We now state the part of their result that is relevant to us. For a given partition $n=n_1+\cdots +n_r$ (with $n_i\in \mathbb {N}_{>0}$), let us consider the following properties:

• (P1) There exists a Zariski open $V\subset X$, such that intersection product induces a surjection

  $$\begin{aligned} A^{n_1}V_{\mathbb {Q}}\otimes A^{n_2}V_{\mathbb {Q}}\otimes \cdots \otimes A^{n_r}V_{\mathbb {Q}}\ \rightarrow \ A^{n}V_{\mathbb {Q}}\ ; \end{aligned}$$

• (P2) There exists a Zariski open $V\subset X$, such that cup product induces a surjection

  $$\begin{aligned} H^{n_1}(V,\mathbb {Q})\otimes H^{n_2}(V,\mathbb {Q})\otimes \cdots \otimes H^{n_r}(V,\mathbb {Q})\ \rightarrow \ H^n(V,\mathbb {Q})/N^1\ \end{aligned}$$

  (here $N^*$ denotes the coniveau filtration);

• (P3) Cup product induces a surjection

  $$\begin{aligned} H^{n_1}(X,\mathcal O_X)\otimes H^{n_2}(X,\mathcal O_X)\otimes \cdots \otimes H^{n_r}(X,\mathcal O_X)\ \rightarrow \ H^n(X,\mathcal O_X). \end{aligned}$$

  In these terms, what Esnault, Srinivas and Viehweg prove is the following:

Theorem

(Esnault et al. [5]) Let X be a smooth complete variety of dimension n over $\mathbb {C}$. Then (P1) implies (P3), and (P2) implies (P3).

The implication from (P1) to (P3) is a kind of multiplicative variant of Mumford’s theorem, and the proof in [5] is motivated by Bloch’s proof of Mumford’s theorem using a “decomposition of the diagonal” argument ([2, 3], cf. also [4]). As noted in [5, remark 2], the generalized Hodge conjecture would imply that (P2) and (P3) are equivalent.¹

In this note, we show that the Esnault–Srinivas–Viehweg theorem can be extended from 0-cycles to arbitrary Chow groups. This is possible provided the variety X satisfies the Lefschetz standard conjecture B(X) (this is analogous to [10], where I extended Mumford’s theorem from 0-cycles to arbitrary Chow groups, provided B(X) holds):

Theorem

($=$Theorem 1) Let X be a smooth projective variety of dimension n over $\mathbb {C}$ that satisfies B(X). Suppose there exists a Zariski open $V\subset X$, and $j=j_1+\cdots + j_r$ with $j_i\in \mathbb {N}_{>0}$ such that intersection product induces a surjection

$$\begin{aligned} A^{j_1}V_{\mathbb {Q}}\otimes A^{j_2}V_{\mathbb {Q}}\otimes \cdots \otimes A^{j_r}V_{\mathbb {Q}}\ \rightarrow \ A^jV_{\mathbb {Q}}. \end{aligned}$$

Then cup product induces a surjection

$$\begin{aligned} H^{j_1}(X,\mathcal O_X)\otimes H^{j_2}(X,\mathcal O_X)\otimes \cdots \otimes H^{j_r}(X,\mathcal O_X)\ \rightarrow \ H^j(X,\mathcal O_X). \end{aligned}$$

The proof of this theorem, which is very similar to the proof given by Esnault–Srinivas–Viehweg in [5], is an exercise in using the meccano of correspondences and the Bloch–Srinivas formalism.

It seems natural to wonder whether the converse to Theorem 1 might perhaps be true (this would be a multiplicative variant of Bloch’s conjecture). In [11], I prove this converse implication in some special cases for 0-cycles (i.e. $j=n$); the converse implication for $j\not =n$ appears to be more difficult.

Convention In this note, the word variety will refer to a quasi-projective irreducible algebraic variety over $\mathbb {C}$, endowed with the Zariski topology. A subvariety is a (possibly reducible) reduced subscheme which is equidimensional. The Chow group of j-dimensional algebraic cycles on X with $\mathbb {Q}$-coefficients modulo rational equivalence is denoted $A_jX$; for X smooth of dimension n the notations $A_jX$ and $A^{n-j}X$ will be used interchangeably. Caveat: note that what we denote $A^jX$ is elsewhere often denoted $CH^j(X)_{\mathbb {Q}}$. In an effort to lighten notation, we will often write $H^jX$ or $H_jX$ to indicate singular cohomology $H^j(X,\mathbb {Q})$ resp. Borel–Moore homology $H_j(X,\mathbb {Q})$.

For basics concerning algebraic cycles and their functorial behaviour, the curious reader is invited to consult [6]. For the formalism of correspondences, cf. [14, 15].

2 Preliminary

Let X be a smooth projective variety of dimension n, and $h\in H^2(X,\mathbb {Q})$ the class of an ample line bundle. The hard Lefschetz theorem asserts that the map

$$\begin{aligned} L^{n-i}:H^i(X,\mathbb {Q})\rightarrow H^{2n-i}(X,\mathbb {Q}) \end{aligned}$$

obtained by cupping with $h^{n-i}$ is an isomorphism, for any $i< n$. One of the standard conjectures asserts that the inverse isomorphism is algebraic.

Definition 1

(Lefschetz standard conjecture) Given a variety X, we say that B(X) holds if for all ample h, and all $i<n$ the isomorphism

$$\begin{aligned} (L^{n-i})^{-1}:H^{2n-i}(X,\mathbb {Q})\mathop {\rightarrow }\limits ^{\cong } H^i(X,\mathbb {Q}) \end{aligned}$$

is induced by a correspondence.

Remark 1

It is known that B(X) holds for the following varieties: curves, surfaces, abelian varieties [8, 9], threefolds not of general type [16], varieties motivated by a surface in the sense of Arapura [1] (this includes the Hilbert schemes of 0-dimensional subschemes of surfaces [1, Corollary 7.5]), n-dimensional varieties X which have $A_i(X)_{}$ supported on a subvariety of dimension $i+2$ for all $i\le {n-3\over 2}$ [17, Theorem 7.1], n-dimensional varieties X which have $H_i(X)=N^{\llcorner {i\over 2}\lrcorner }H_i(X)$ for all $i>n$ [18, Theorem 4.2], products and hyperplane sections of any of these [8, 9].

It is known that B(X) implies that the Künneth components

$$\begin{aligned} \pi _j\in H^{2n-j}(X)\otimes H^j(X)\subset H^{2n}(X\times X) \end{aligned}$$

of the diagonal $\Delta \subset X\times X$ are algebraic [8, 9]. Moreover, these Künneth components satisfy the following property:

Lemma 1

Let X be a smooth projective variety satisfying B(X), and let $h\in H^2(X)$ be the class of an ample line bundle. For any $j\le n$, there exists a cycle $P_j\in A^j(X\times X)$ such that

$$\begin{aligned} \pi _j=(\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(P_j)\ \ \in H^{2n}(X\times X) , \end{aligned}$$

where $\tau :Y_j\rightarrow X$ denotes the inclusion of a dimension j complete intersection of class $[Y_j]=h^{n-j}$.

Proof

As mentioned above, B(X) ensures that $\pi _j$ is algebraic [8, 9]. Consider now the isomorphism

$$\begin{aligned} L^{n-j}\times \hbox {id}:\ \ H^jX\otimes H^jX\ \xrightarrow {\cong }\ H^{2n-j}X\otimes H^jX\ \end{aligned}$$

(here we tacitly identify both sides with their images in $H^*(X\times X)$).

Since we have B(X), there exists a correspondence, say $Q\in A^{j}(X\times X)$, such that

$$\begin{aligned} (L^{n-j}\times \hbox {id}) (Q\times \hbox {id})_*=\hbox {id}:\ H^{2n-j}X\otimes H^jX\ \rightarrow \ H^{2n-j}X\otimes H^jX. \end{aligned}$$

Since $\pi _j$ is algebraic,

$$\begin{aligned} P_j:= (Q\times \hbox {id})_*(\pi _j)\in A^j(X\times X) \end{aligned}$$

is still algebraic, and has the requested property. $\square$

Remark 2

Lemma 1 implies in particular that for a variety satisfying B(X), the Künneth component $\pi _j$ is represented by an algebraic cycle contained in $Y_j\times X$, for a dimension j complete intersection $Y_j$. This was also proven in [7] (and independently in [10, proof of Theorem 3.1], as I wasn’t aware of the Kahn–Murre–Pedrini reference at the time).

3 Main

We now prove the main theorem of this note:

Theorem 1

Let X be a smooth projective variety of dimension n over $\mathbb {C}$ that satisfies B(X). Suppose there exists a Zariski open $V\subset X$, and $j=j_1+\cdots + j_r$ with $j_i\in \mathbb {N}_{>0}$ such that intersection product induces a surjection

$$\begin{aligned} A^{j_1}V_{}\otimes A^{j_2}V_{}\otimes \cdots \otimes A^{j_r}V_{}\ \rightarrow \ A^jV_{}. \end{aligned}$$

Then cup product induces a surjection

$$\begin{aligned} H^{j_1}(X,\mathcal O_X)\otimes H^{j_2}(X,\mathcal O_X)\otimes \cdots \otimes H^{j_r}(X,\mathcal O_X)\ \rightarrow \ H^j(X,\mathcal O_X). \end{aligned}$$

Proof

Since B(X) holds, it follows from Lemma 1 that the Künneth component $\pi _j$ can be written

$$\begin{aligned} \pi _j=(\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(P_j)\ \ \in H^{2n}(X\times X)\ \end{aligned}$$

for some $P_j\in A^j(X\times X)$, where $\tau :Y_j\rightarrow X$ is the inclusion of a dimension j complete intersection.

Applying the Bloch–Srinivas argument, in the form of Proposition 1 below, to the cycle $P_j\in A^j(X\times X)$, we find a decomposition

$$\begin{aligned} P_j=C_1\cdot \ldots \cdot C_r+\Gamma _1+\Gamma _2\ \ \in A^j(X\times X), \end{aligned}$$

where $\Gamma _1, \Gamma _2$ are supported on $D\times X$, resp. on $X\times D$, for some divisor $D\subset X$. This induces a decomposition of the Künneth component

$$\begin{aligned} \pi _j&=(\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(C_1\cdot \ldots \cdot C_r)+ (\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(\Gamma _1+\Gamma _2)\\&=(\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(C_1\cdot \ldots \cdot C_r)+ \Gamma _1^\prime +\Gamma _2^\prime \ \ \in H^{2n}(X\times X), \end{aligned}$$

where $\Gamma _2^\prime$ is still supported on $X\times D$, and $\Gamma _1^\prime$ is supported on $Z\times X$, for some $Z\subset X$ of dimension $j-1$ (indeed, the general complete intersection $Y_j$ will be in general position with respect to D; we then define Z to be $D\cap Y_j$).

Now we consider the action of $\pi _j$ on $H^j(X,\mathcal O_X)$. Since $H^j(X,\mathcal O_X)=\hbox {Gr}^0_F H^j(X,\mathbb {C})$ (where F is the Hodge filtration), $\pi _j$ acts as the identity on $H^j(X,\mathcal O_X)$. On the other hand, it is clear that

$$\begin{aligned} (\Gamma _1^\prime )_*H^j(X,\mathcal O_X)=0 \end{aligned}$$

(by Lemma 3, the action of $\Gamma _1^\prime$ factors over $\hbox {Gr}^0_F H^j(Z,\mathbb {C})$, which is 0 for dimension reasons), and also that

$$\begin{aligned} (\Gamma _2^\prime )_*H^j(X,\mathcal O_X)=0 \end{aligned}$$

(by Lemma 3, the action of $\Gamma _2$ factors over $\hbox {Gr}^{-1}_F H^{j-2}(\widetilde{D},\mathbb {C})=0$, where $\widetilde{D}$ is a resolution of singularities of D). To finish the argument, it only remains to analyze the action

$$\begin{aligned} \bigl ((\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(C_1\cdot \ldots \cdot C_r)\bigr )_*:\ \ H^j(X,\mathcal O_X)\ \rightarrow \ H^j(X,\mathcal O_X). \end{aligned}$$

Using Lemmas 2 and 3, we find an inclusion

$$\begin{aligned} \bigl ((\tau \times \hbox {id})_*(\tau \times \hbox {id})^*(C_1\cdot \ldots \cdot C_r)\bigr )_*H^j(X,\mathcal O_X) \subset \bigl (C_1\cdot \ldots \cdot C_r\bigr )_*\hbox {Gr}_F^{n-j} H^{2n-j}(X,\mathbb {C}). \end{aligned}$$

Using Lemma 4, we find that

$$\begin{aligned} \bigl (C_1\cdot \ldots \cdot C_r\bigr )_*\hbox {Gr}_F^{n-j} H^{2n-j}(X,\mathbb {C})\!\subset \! \hbox {Im}\bigl (H^{j_1}(X,\mathcal O_X)\!\otimes \cdots \otimes \!H^{j_r}(X,\mathcal O_X) \!\rightarrow \!H^j(X,\mathcal O_X)\!\bigr ) , \end{aligned}$$

and so we are done. $\square$

Proposition 1

(Bloch–Srinivas style) Let X be a smooth projective variety of dimension n. Suppose there exists a Zariski open $V\subset X$, and $j=j_1+\cdots + j_r$ with $j_i\in \mathbb {N}_{>0}$ such that intersection product induces a surjection

$$\begin{aligned} A^{j_1}V_{}\otimes A^{j_2}V_{}\otimes \cdots \otimes A^{j_r}V_{}\ \rightarrow \ A^jV_{}. \end{aligned}$$

Then for any $a\in A^j(X\times X)$, there exists a decomposition

$$\begin{aligned} a= C_1\cdot \ldots \cdot C_r+\Gamma _1+\Gamma _2\ \ \in A^j(X\times X), \end{aligned}$$

where $C_i\in A^{j_i}(X\times X)$, and $\Gamma _1, \Gamma _2$ are supported on $D\times X$ (resp. on $X\times D$), for some divisor $D\subset X$.

Proof

To be sure, this is a variant of the argument of [4], exploiting the fact that $\mathbb {C}$ is a universal domain. Let $D_1\subset X$ denote the complement of V. Taking the smallest possible field of definition, we can suppose everything (X, V and the cycle a) is defined over a field $k\subset \mathbb {C}$ which is finitely generated over its prime subfield. Then the inclusion $k(X)\subset \mathbb {C}$ (which comes from $\mathbb {C}$ being a universal domain) induces an injection

$$\begin{aligned} A^j(X_{k(X)})\ \rightarrow \ A^j(X_{\mathbb {C}}) \end{aligned}$$

[3, Appendix to Lecture 1]. On the other hand,

$$\begin{aligned} A^j(X_{k(X)})_{}= \varinjlim A^j(X\times U)_{}, \end{aligned}$$

where the limit is taken over opens $U\subset X$ [3, Appendix to Lecture 1].

Given the cycle $a\in A^j(X\times X)$, consider the restriction

$$\begin{aligned} a_{restr}\ \ \in A^j(X_{k(X)}). \end{aligned}$$

The assumption implies there exist cycles $c_i\in A^{j_i}(X_{\mathbb {C}})$ such that

$$\begin{aligned} a_{restr}=c_1\cdot \ldots \cdot c_r+a_0\ \ \in A^j(X_{\mathbb {C}}), \end{aligned}$$

where $a_0$ is supported on $D_1$. Now, we extend k so that the cycles $c_i$ are also defined over k (and k is still finitely generated over its prime subfield, so that $k(X)\subset \mathbb {C}$). Then using the injection $A^j(X_{k(X)})\ \rightarrow \ A^j(X_{\mathbb {C}})$ cited above, we obtain the decomposition

$$\begin{aligned} a_{restr}=c_1\cdot \ldots \cdot c_r+a_0\ \ \in A^j(X_{k(X)}). \end{aligned}$$

Let $C_i\in A^{j_i}(X\times X)$ be any cycle restricting to $c_i$, and let $\Gamma _1$ be any cycle restricting to $a_0$. Then using the limit property cited above, we find that the difference

$$\begin{aligned} a-C_1\cdot \ldots \cdot C_r-\Gamma _1 \ \ \in A^j(X\times X) \end{aligned}$$

restricts to 0 in $A^j(X\times U)$, for some open $U\subset X$. This means there exists a divisor $D_2\subset X$ and a cycle $\Gamma _2$ supported on $D_2$ such that

$$\begin{aligned} a= C_1\cdot \ldots \cdot C_r+\Gamma _1+\Gamma _2\ \ \in A^j(X\times X). \end{aligned}$$

Taking D a divisor containing both $D_1$ and $D_2$, this proves the proposition. $\square$

Lemma 2

Let $f:Y\rightarrow X$ be a proper morphism of smooth projective varieties, where $\dim X=n$ and $\dim Y=m$. Let $C\in A^j(X\times X)$. Then

$$\begin{aligned} \bigl ( (f\times \hbox {id})^*C\bigr )_*= C_*f_*:\ \ H^iY\ \rightarrow \ H^{i+2(j-m)}X. \end{aligned}$$

Proof

This is purely formal, and surely well-known. Let $p_1, p_2:X\times X\rightarrow X$ denote projection on the first (resp. second) factor. Let $q_1, q_2$ denote projections from $Y\times X$ to Y (resp. to X). For $a\in H^iY$, we have

$$\begin{aligned} \bigl ( (f\times \hbox {id})^*C\bigr )_*(a)&= (q_2)_*\bigl ( (q_1)^*(a)\cdot (f\times \hbox {id})^*(C)\bigr )\\&= (p_2)_*\bigl ( (f\times \hbox {id})_*((q_1)^*(a)\cdot (f\times \hbox {id})^*(C)) \bigr ) \\&=(p_2)_*\bigl ( (f\times \hbox {id})_*(q_1)^*(a)\cdot C\bigr )\\&=(p_2)_*\bigl ( (p_1)^*f_*(a)\cdot C\bigr )=: C_*f_*(a)\ \ \in H^{i+2(j-m)}X.\\ \end{aligned}$$

Lemma 3

Let $f:Y\rightarrow X$ be as in Lemma 2. Let $D\in A^j(Y\times X)$. Then

$$\begin{aligned} \bigl ( (f\times \hbox {id})_*D\bigr )_*= D_*f^*:\ \ H^iX\ \rightarrow \ H^{i+2(j-m)}X. \end{aligned}$$

Proof

Just as Lemma 2, this is surely well-known. Keeping the notation of Lemma 2, for $b\in H^iX$ we have

$$\begin{aligned} \bigl ( (f\times \hbox {id})_*D\bigr )_*(b)&= (p_2)_*\bigl ( (p_1)^*(b)\cdot (f\times \hbox {id})_*(D)\bigr )\\&= (p_2)_*\bigl ( (f\times \hbox {id})_*\bigl ( (f\times \hbox {id})^*(p_1)^*(b)\cdot D\bigr )\bigr )\\&=(p_2)_*(f\times \hbox {id})_*\bigl ( (q_1)^*f^*(b)\cdot D\bigr )\\&= (q_2)_*\bigl ( (q_1)^*f^*(b)\cdot D\bigr )=: D_*f^*(b)\ \ \in H^{i+2(j-m)}X. \\ \end{aligned}$$

$\square$

Lemma 4

(Esnault–Srinivas–Viehweg [5]) Let X be a smooth projective variety of dimension n. Let $C_i\in A^{j_i}(X\times X), i=1,\ldots ,r$, with $j=j_1+\cdots +j_r$. Then

$$\begin{aligned}\ (C_1\cdot \ldots \cdot C_r)_*\hbox {Gr}_F^{n-j}H^{2n-j}(X,\mathbb {C}) \!\subset \!\hbox {Im}\Bigl ( H^{j_1}(X,\mathcal O_X)\!\otimes \!\cdots \!\otimes \!H^{j_r}(X,\mathcal O_X) \!\rightarrow \!H^j(X,\mathcal O_X)\Bigr )\!. \end{aligned}$$

Proof

This is shamelessly plagiarized from [5], who prove the $j=n$ case. Let $C\in A^j(X\times X)$ be any correspondence. The crucial observation is that the action

$$\begin{aligned} C_*:\ \ \hbox {Gr}_F^{n-j}H^{2n-j}(X,\mathbb {C})\ \rightarrow \ \hbox {Gr}_F^0 H^j(X,\mathbb {C}) \end{aligned}$$

only depends on the image of C under the composite map

$$\begin{aligned} \iota :\ \ A^j(X\times X)\ \rightarrow \ H^{2j}(X\times X,\mathbb {C})\ \rightarrow \ H^jX\otimes H^jX\ \rightarrow \ \hbox {Gr}_F^j H^jX\otimes \hbox {Gr}_F^0 H^jX\ \end{aligned}$$

(Here the second map is given by the Künneth decomposition, and the last map is induced by projection on the appropriate summands of the Hodge decomposition). Indeed, suppose $C\in A^j(X\times X)$ is such that $\iota (C)=0$, i.e. the Künneth part of type $H^jX\otimes H^jX$ of C is contained in

$$\begin{aligned} \bigoplus _{i<j} \hbox {Gr}_F^i H^jX\otimes \hbox {Gr}_F^{j-i} H^jX\ \subset \ \hbox {Gr}_F^j (H^jX\otimes H^jX). \end{aligned}$$

Then, for $a\in \hbox {Gr}_F^{n-j}H^{2n-j}(X,\mathbb {C})$ we find that

$$\begin{aligned} (p_1)^*(a)\cdot C\ \ \in \bigoplus _{i<j} \hbox {Gr}_F^{n-j+i} H^{2n}X\otimes \hbox {Gr}_F^{j-i}H^jX=0, \end{aligned}$$

and hence

$$\begin{aligned} C_*(a)=0\ \ \in H^j(X,\mathbb {C}). \end{aligned}$$

Next, we apply this observation to

$$\begin{aligned} C=C_1\cdot \ldots \cdot C_r\ \ \in A^j(X\times X), \end{aligned}$$

with $C_i\in A^{j_i}X$. The Hodge decomposition then gives that

$$\begin{aligned} \iota (C)= & {} \iota (C_1)\cdot \ldots \cdot \iota (C_r)\in \ \hbox {Im}\bigl (\hbox {Gr}_F^{j_1} H^{j_1}X\otimes \cdots \otimes \hbox {Gr}_F^{j_r} H^{j_r}X\bigr )\\&\otimes \, \hbox {Im}\bigl (\hbox {Gr}_F^{0} H^{j_1}X\otimes \cdots \otimes \hbox {Gr}_F^{0} H^{j_r}X\bigr )\subset \hbox {Gr}_F^j H^jX\otimes \hbox {Gr}_F^0 H^jX . \end{aligned}$$

This proves the Lemma: suppose

$$\begin{aligned} \iota (C)=\sum _k C^k_{left}\otimes C^k_{right}\ \ \in \hbox {Gr}_F^j H^jX\otimes \hbox {Gr}_F^0 H^jX. \end{aligned}$$

Then reasoning as above, we find that

$$\begin{aligned} (\iota (C))_*(a)= \sum _k (p_2)_*\left( \left( a\cup C^k_{left}\right) \otimes C^k_{right} \right) =\sum _k \alpha _k C^k_{right}\ \ \in \hbox {Gr}_F^0 H^jX, \end{aligned}$$

where the $\alpha _k$ are complex numbers (this is because $H^{2n}X$ is one-dimensional and generated by the class of a point). $\square$

Remark 3

It is mainly the contrapositive of Theorem 1 that is useful (this is another remark made in [5] for their theorem). Indeed, suppose X and $j=j_1+\cdots +j_r$ are such that

$$\begin{aligned} H^{j_1}(X,\mathcal O_X)\otimes H^{j_2}(X,\mathcal O_X)\otimes \cdots \otimes H^{j_r}(X,\mathcal O_X)\ \rightarrow \ H^j(X,\mathcal O_X)\ \end{aligned}$$

is not surjective (for example, because

$$\begin{aligned} \prod _{i=1}^r \dim H^{j_i}(X,\mathcal O_X)< \dim H^j(X,\mathcal O_X)\ ). \end{aligned}$$

Then by Theorem 1, likewise

$$\begin{aligned} A^{j_1}X_{}\otimes \cdots \otimes A^{j_r}X_{}\ \rightarrow \ A^jX_{}\ \end{aligned}$$

fails to be surjective (and the same holds for any open $V\subset X$).