论文标题
复杂的准标准品种的双曲线和基本组
Hyperbolicity and fundamental groups of complex quasi-projective varieties
论文作者
论文摘要
本文研究了在线性表示$ \ varrho：π_1（x）\ to {\ rm gl} _n（\ rm gl} _n（\ mathbb {c} c} c}）$之间，研究复杂的准标准品种$ x $ $ x $ $ x $ $ x $ $ x $ $ x $ $π_1（x）$之间的关系。我们以三个部分介绍了我们的主要结果。 首先，我们表明，如果$ \ varrho $是$ \ varrho（π_1（x））$ semisimple的zariski关闭，那么对于任何$ x^σ：= x \times_σ\times_σ\ m马理bb {c} $ exists a proper Zariski closed subset $Z \subsetneqq X^σ$ such that any closed irreducible subvariety $V$ of $X^σ$ not contained in $Z$ is of log general type, and any holomorphic map from the punctured disk $\mathbb{D}^*$ to $X^σ$ with image not contained in $Z$ does not have an essential singularity at the origin.特别是，$ x^σ$中的所有曲线都在$ z $上。我们提供示例以说明这种情况的最佳性。 其次，假设$ \ varrho $是大且还原的，我们证明了$ x^σ$的广义绿色 - 格里奇斯 - 朗格。此外，如果$ \ varrho $很大，我们表明$ x^σ$的特殊子集从不同的角度捕获了$ x^σ$的非hyperbolicity locus是相等的，并且当$ x $是log General Type的情况下，此子集是正确的。 最后，我们证明，如果$ x $是campana或$ h $ - 特别的特殊准主体歧管，则$ \ varrho（π_1（x））$几乎是nilpotent。我们提供了例子，以证明该结果是尖锐的，因此修改了Campana的Abelianity猜想，以进行平滑的准标记品种。 为了证明这些定理，我们开发了非亚洲霍奇理论，几何群体理论和内凡林纳理论的新特征。获得了一些副产品。
This paper investigates the relationship between the hyperbolicity of complex quasi-projective varieties $X$ and the (topological) fundamental group $π_1(X)$ in the presence of a linear representation $\varrho: π_1(X) \to {\rm GL}_N(\mathbb{C})$. We present our main results in three parts. Firstly, we show that if $\varrho$ is bigand the Zariski closure of $\varrho(π_1(X))$ semisimple, then for any $X^σ:=X\times_σ\mathbb{C}$ where $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, there exists a proper Zariski closed subset $Z \subsetneqq X^σ$ such that any closed irreducible subvariety $V$ of $X^σ$ not contained in $Z$ is of log general type, and any holomorphic map from the punctured disk $\mathbb{D}^*$ to $X^σ$ with image not contained in $Z$ does not have an essential singularity at the origin. In particular, all entire curves in $X^σ$ lie on $Z$. We provide examples to illustrate the optimality of this condition. Secondly, assuming that $\varrho$ is big and reductive, we prove the generalized Green-Griffiths-Lang conjecture for $X^σ$. Furthermore, if $\varrho$ is large, we show that the special subsets of $X^σ$ that capture the non-hyperbolicity locus of $X^σ$ from different perspectives are equal, and this subset is proper if and only if $X$ is of log general type. Lastly, we prove that if $X$ is a special quasi-projective manifold in the sense of Campana or $h$-special, then $\varrho(π_1(X))$ is virtually nilpotent. We provides examples to demonstrate that this result is sharp and thus revise Campana's abelianity conjecture for smooth quasi-projective varieties. To prove these theorems, we develop new features in non-abelian Hodge theory, geometric group theory, and Nevanlinna theory. Some byproducts are obtained.
