AI 中文总结
针对亏格7的素法诺三维簇，构造其乘积上的非挠2-闭链，证明该簇不具备Shen-Vial意义下的乘法周Künneth分解，同时指出所有法诺三维簇在代数等价下均有该分解。
AI 中文摘要
设Y为亏格7的非常一般的素法诺三维簇，我们构造Y×Y上一个显式2-闭链，它在周群A⁴(Y×Y)中是Abel-Jacobi平凡但非挠的。由此可知，Y不具有Shen-Vial意义下的乘法周Künneth分解。我们还证明，任何法诺三维簇在代数等价意义下都具有乘法周Künneth分解。
英文摘要
Let $Y$ be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on $Y\times Y$ that is Abel-Jacobi trivial but non-torsion in the Chow group $A^4(Y\times Y)$. As a consequence, $Y$ does not admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-Künneth decomposition modulo algebraic equivalence.
CommentsTo appear in Journal of Pure and Applied Algebra, 16 pages, comments very welcome
