A G ] 2 7 A ug 2 02 1 Residual finiteness for central extensions of lattices in PU ( n , 1 ) and negatively curved projective varieties
A G ] 2 7 A ug 2 02 1 Residual finiteness for central extensions of lattices in PU ( n , 1 ) and negatively curved projective varieties
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发表时间：
2021
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通讯作者：
Matthew Stover
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作者：
Matthew Stover
研究了余紧算术格Γ<；PU(n，1)单型的循环中心扩张的剩余有限性。证明了PU(n，1)的任一连通覆盖，特别是泛覆盖中的Γ的原象是剩余有限的。这是从特征类包含在Poincarédual的H(Γ，Z)中的扩张到球商Γ\Bn上的全测地因子的扩张的剩余有限性的更一般的定理。对于n≥4，如果Γ是同余格，我们证明了与H(Γ，Z)的任何元素相关的中心扩张的剩余有限性。我们的主要应用是研究在全测地因子上分支的球商的循环覆盖的存在性。这给出了光滑射影簇允许负截面曲率的度量不同于局部对称流形的同伦的例子。这种例子的存在对于≥4的所有维度都是新的。
We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices Γ < PU(n, 1) simple type. We prove that the preimage of Γ in any connected cover of PU(n, 1), in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in H(Γ,Z) of the Poincaré duals to totally geodesic divisors on the ball quotient Γ\Bn. For n ≥ 4, if Γ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of H(Γ,Z). Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions n ≥ 4.