AI 中文总结
本文研究承载可极化Hodge结构变分的光滑复射影簇的对数余切丛的中间双曲性，证明其各次外幂为L-丰富，给出Viehweg-丰富的判定方法及局部对称簇、五次三维簇模空间覆盖的具体结果。
AI 中文摘要
设$\bar{V}$为连通光滑复射影簇，$D \subset \bar{V}$为正规交叉除子，$\mathbb{V}$为$V := \bar{V} \setminus D$上的复可极化Hodge结构变分。假设$\mathbb{V}$的周期映射在$V$的某一点处是浸入的。我们证明，对满足$1 \leq p \leq \dim V$的任意整数$p$，向量丛$\Omega_{\bar{V}}^p(\log D)$是L-丰富的（即$\mathbb{P}\Omega_{\bar{V}}^p(\log D)$上的 tautological 线丛是丰富的）。若局部单值群是拟幂幺的，我们给出一种确定$m \in \mathbb{N}$的方法，使得当$p > m$时，$\Omega_{\bar{V}}^p(\log D)$进一步是Viehweg-丰富的。当$V$是局部对称簇时，我们明确给出$m$的最优值。我们证明，若$V$是光滑五次三维簇的精细模空间的有限平展覆盖，则结果对$m = 90$成立（此情形下$\dim V = 101$）。上述结果的证明基于对与$\Omega_{\bar{V}}^p(\log D)$相关的增广基点轨迹的研究。在局部对称簇的情形下，我们引入“高次特征子簇”，推广了Mok在$p=1$情形下定义的特征子簇，并证明它们与这些增广基点轨迹重合。
英文摘要
Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every integer $p$ with $1 \leq p \leq \dim V$, the vector bundle $Ω_{\bar{V}}^p(\log D)$ is L-big (i.e. the tautological line bundle on $\mathbb{P}Ω_{\bar{V}}^p(\log D)$ is big). If the local monodromy is quasi-unipotent, we give a method to determine an $m \in \mathbb{N}$ such that if $p > m$, then $Ω_{\bar{V}}^p(\log D)$ is moreover Viehweg-big. We give the optimal value of $m$ explicitly when $V$ is a locally symmetric variety. We prove that if $V$ is a finite étale cover of the fine moduli space of smooth quintic threefolds, the result holds for $m = 90$ (in this case $\dim V = 101$). The proof of the previous results is based on a study of an augmented base locus associated with $Ω_{\bar{V}}^p(\log D)$. In the case of locally symmetric varieties, we introduce ``higher degree characteristic subvarieties'', generalizing the characteristic subvariety defined by Mok in the case $p = 1$, and prove that they coincide with these augmented base loci.
Comments46 pages
