论文标题
Betti Moduli空间的完整性
Integrality of the Betti moduli space
论文作者
论文摘要
If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely irreducible $\ell$-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification.有限呈现的小组在扭转特征和等级$ r $方面是微弱的组成部分，如果一旦有一个不可还原等级$ r $ r $复杂的线性表示，那么对于任何$ \ ell $，就有一个绝对不可约束的等级$ r $，而确定性的一个给定的字符，该字符定义了$ \ bar \ bar \ bar {\ mathbbbbbb {z Z}}}}}}我们证明，该特性是一个有限呈现的组成为平滑QUSI-projementive复合物品种的基本群体的新障碍。证明是通过De Deligne的同伴（L. L. L. L. L. L. L. Langlands计划）和通过de Jong的猜想（$ \ ell \ ell \ ge 3 $）的几何Langlands计划而存在的。我们还定义了弱算术复杂的局部系统，并表明它们在Betti模量中是Zariski致密。最后，我们证明我们的方法给出了Corlette-T的算术证明。 Mochizuki定理使用了驯服的纯虚构谐波指标，此后通过半简单复杂局部系统的两个光滑复杂的代数品种之间的态度下拉是半简单的。 V2：纠正了定理7.3证明中的裁判中指出的一个错误。 最终版本：出现在Transactions AMS中
If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely irreducible $\ell$-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank $r$ if once there is an irreducible rank $r$ complex linear representation, then for any $\ell$, there is an absolutely irreducible one of rank $r$ and determinant this given character which is defined over $\bar{ \mathbb{Z}}_\ell$. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth qusi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for $\ell \ge 3$). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, after which the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. v2: a mistake pointed out by the kind referee in the proof of Theorem 7.3 is corrected. Final version: appears in Transactions AMS
