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Standard conjecture of Künneth type with torsion coefficients. (English) Zbl 1386.14049
Let $$X$$ be a complex smooth projective variety of dimension $$n$$. Let $$cl_{X,\mathbb{Q}}^j:\text{CH}^j(X)\rightarrow H^{2j}(X,\mathbb{Q})$$ denote the cycle class map from the Chow group of codimension-$$j$$ cycles on $$X$$. For each integer $$i$$, one has the rational Künneth projector $$\pi_{X,\mathbb{Q}}^i\in H^{2n}(X\times X,\mathbb{Q})$$, which acts as 1 on $$H^i(X,\mathbb{Q})$$ and as 0 on $$H^j(X,\mathbb{Q})$$ for all $$j\neq i$$. Grothendieck’s standard conjecture of Künneth type asserts that the Künneth projectors $$\pi_{X,\mathbb{Q}}^i$$ are algebraic, namely $$\pi_{X,\mathbb{Q}}^i\in \text{Im}(\text{cl}_{X\times X,\mathbb{Q}}^n)\otimes\mathbb{Q}$$ for all $$i$$. The present paper is concerned with the question, raised by A. Venkatesh, which asks whether the mod-$$p$$-analogue of the standard conjecture holds true or not. More precisely the problem is whether or not the mod-$$p$$-Künneth projectors $$\pi_{X,\mathbb{F}_p}^i$$, which is defined similarly by replacing the coefficient $$\mathbb{Q}$$ by $$\mathbb{F}_p$$, belong to $$\text{Im}(\text{cl}_{X\times X,\mathbb{F}_p}^n)$$ for all $$i$$. One of the main result of this article is to answer the question in the negative by showing that, if $$H^*(X,\mathbb{Z})$$ has nontrivial $$p$$-torsion, then Venkatesh’s question has a negative answer for $$X$$. Furthermore he shows that for any integers $$i>0$$ and $$n\geq i+1$$, there exists a projective smooth variety $$X$$ of dimension $$n$$ such that $$\pi_{X,\mathbb{F}_p}^i$$ is not in the image of the mod-$$p$$ cycle class map. On the other hand, he looks into the question in the case of abelian varieties, and shows that Venkatesh’s question has an affirmative answer for all primes $$p$$, if $$X$$ is the Jacobian of a curve.
##### MSC:
 14C25 Algebraic cycles 14H40 Jacobians, Prym varieties 55S05 Primary cohomology operations in algebraic topology
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