贝林森-霍奇猜想的一个相对版本

Rob de Jeu, James D. Lewis, D. Patel
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引用次数: 2

摘要

设k是复数的代数闭子域,X是定义在k上的一个变种。贝林森-霍奇猜想的一个版本似乎可以通过仔细的检验,即Betti循环类映射cl_{r,m}: H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r)))是满射,在m=0的情况下等价于霍奇猜想。现在考虑k上光滑拟射影变体的光滑和适当映射\rho: X -> S。我们为一般纤维构造了这个猜想的一个版本,期望相应的循环类映射是满射的。在X是乘积的情况下,我们提供了一些证据来支持这一点,映射是对一个因子的投影,m=1。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A relative version of the Beilinson-Hodge conjecture
Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map \rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.
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