论文标题
K3类型的杂货结构有理表面
Hodge Structures of K3 type of bidouble covers of rational surfaces
论文作者
论文摘要
Bidouble Cover是一个平面$ G：= \ left（\ Mathbb {z}/2 \ Mathbb {z} \ right）^2 $ -galois cover $ x \ rightarrow y $。在这种情况下，存在三个中间人$ y_1，y_2 $和$ y_3 $，它们对应于三个子组$ \ mathbb {z}/2 \ mathbb {z} \ leq g $。在本文中，我们考虑以下情况：$ y $将是理性的表面，$ y_i $将是$ p_g = 0 $的表面，或者是K3表面。这些假设将使我们能够对覆盖面$ x $的重量2 hodge结构进行强有力的控制。特别是，如果$ y $很少，我们将所有覆盖物分类为这些属性，以获取$ x $的表面$ x $，$ p_g（x）= 1,2,3 $。此外，我们将讨论无限的Torelli物业，Chow Groups和Chow Motive，以及Tate和Mumford-Tate的猜想，价格为$ x $。我们还引入了另一种称为迭代的budouble盖的结构，这使我们能够获得较高值$ p_g $的表面，我们仍然可以对重量2 Hodge结构进行强有力的控制。
A bidouble cover is a flat $G:=\left(\mathbb{Z}/2\mathbb{Z}\right)^2$-Galois cover $X \rightarrow Y$. In this situation there exist three intermediate quotients $Y_1,Y_2$ and $Y_3$ which correspond to the three subgroups $\mathbb{Z}/2\mathbb{Z} \leq G$. In this paper we consider the following situation: $Y$ will be a rational surface and $Y_i$ will be either a surface with $p_g=0$ or a K3 surface. These assumptions will enable us to have a strong control on the weight 2 Hodge structure of the covering surface $X$. In particular, we classify all covers with these properties if $Y$ is minimal, obtaining surfaces $X$ with $p_g(X)=1,2,3$. Moreover, we will discuss the Infinitesimal Torelli Property, the Chow groups and Chow motive, and the Tate and Mumford-Tate conjectures for $X$. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of $p_g$ for which we still have a strong control on the weight 2 Hodge structure.
