论文标题
复合物与Etale Abel Jacobi图和零位基因座的代数正常函数
Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions
论文作者
论文摘要
我们证明，使用$ p $ - 亚种霍奇理论用于开放代数品种，它可以在子场$ k \ subset \ subset \ mathbb c $上平滑投射品种，该品种是有限类型的$ \ mathbb q $，如果Et eTale Abel Jacobi Map vanishes vanishes vanishes vanishes，则复杂的Abel Jacobi Map avension ables type ass by s subse。这意味着，对于平稳的投射形态，$ f：x \至s $ of $ k \ subset \ subset \ mathbb c $的光滑复杂的代数品种，该品种超过$ \ mathbb q $和$ \ mathbb q $和$ z \ in \ mathcal z^d（x d（x，x，n）^f，poartial = 0}纤维，与$ z $相关的eTale正常函数的零基因座包含在与$ z $相关的复杂正常函数的零基因座中。从斋藤或查尔斯的工作中，我们推断出与$ z $相关的复杂正常函数的零基因座定义在代数关闭$ \ bar k $ $ k $的情况下，如果与$ z $相关的eTale正常函数的零基因座不是空的。我们还证明，在$ \ m athbb q $上的有限类型的字段上，与代数周期相关的eTale正常函数的零基因座的代数结果是一个代数结果。顺便说一句，对于平滑的形态$ f：x \ to s $的平滑代数品种，在$ \ mathbb q $上面有限类型的领域，我们嵌入了$ f $的霍奇类别的hodge-tate类的座位。
We prove, using $p$-adic Hodge theory for open algebraic varieties, that for a smooth projective variety over a subfield $k\subset\mathbb C$ which is of finite type over $\mathbb Q$, the complex abel jacobi map vanishes if the etale abel jacobi map vanishes. This implies that for a smooth projective morphism $f:X\to S$ of smooth complex algebraic varieties over $k\subset\mathbb C$ which is of finite type over $\mathbb Q$ and $Z\in\mathcal Z^d(X,n)^{f,\partial=0}$ an algebraic cycle flat over $S$ whose cohomology class vanishes on fibers, the zero locus of the etale normal function associated to $Z$ is contained in the zero locus of the complex normal function associated to $Z$. From the work of Saito or Charles, we deduce that the zero locus of the complex normal function associated to $Z$ is defined over the algebraic closure $\bar k$ of $k$ if the zero locus of the etale normal function associated to $Z$ is not empty. We also prove an algebraicity result for the zero locus of an etale normal function associated to an algebraic cycle over a field of finite type over $\mathbb Q$. By the way, for a smooth morphism $f:X\to S$ of smooth algebraic varieties over a field of finite type over $\mathbb Q$, we embed the locus of Hodge-Tate classes of $f$ inside the locus of Hodge classes of $f$.
