论文标题
增加了支持Hodge结构的变化的品种的双波动性与水平结构的变化
Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures
论文作者
论文摘要
查看有限的典型一致性覆盖了一个复杂的代数$ x $的$ x（p）$，配备了各种整体两极化的霍奇结构，其周期映射为准芬矿，我们表明，所有$ x（p）$ in $ x（p）$ ins $ x的最低曲线都包含在$ x（p）$ x（p）中的最小值$ x $ x（$ x）$ x（p）$ x（p）$ x（p）$ x（p）。例如，这适用于Shimura品种，曲线的模量空间，Abelian品种的模量空间，Calabi-Yau品种的模量空间，并且在许多情况下可以有效。证明大致如下。我们首先证明了Arakelov不平等的概括，这对于高维代数品种上的Hodge结构的任何变化有效，这暗示$ X $的子变量的双波利度受单线束的积极性控制。然后，我们通常表明，通过使用在Zariski密集开放的子集中定义的本地系统的级别结构定义的$ \ bar x $有限盖，可以使越来越积极的正常线条捆绑$ \ bar x $变得越来越积极。
Looking at the finite étale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all curves contained in $X(p)$ and the minimal volume among all subvarieties of $X(p)$ tend to infinity with $p$. This applies for example to Shimura varieties, moduli spaces of curves, moduli spaces of abelian varieties, moduli spaces of Calabi-Yau varieties, and can be made effective in many cases. The proof goes roughly as follows. We first prove a generalization of the Arakelov inequalities valid for any variation of Hodge structures on higher-dimensional algebraic varieties, which implies that the hyperbolicity of the subvarieties of $X$ is controlled by the positivity of a single line bundle. We then show in general that a big line bundle on a normal proper algebraic variety $\bar X$ can be made more and more positive by going to finite covers of $\bar X$ defined using level structures of a local system defined on a Zariski-dense open subset.
