A tower of coverings of quasi-projective varieties

Abstract

The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact Kähler manifold of finite volume with reasonable geometric assumptions to its universal covering. Applicable examples include moduli spaces of hyperbolic punctured Riemann surfaces and Hermitian locally symmetric spaces of finite volume.

Holomorphic Line Bundles over a Tower of Coverings

“…Part (i) follows by a straightforward normal family argument (cf. [To] [CF] [Ye3]) which we will omit here, while part (ii) is a combination of Hörmander's L 2 -estimate and Agmon estimate. For any z ∈ M, define τ j (z) = inf {dist(z, γ j z) ∶ γ j ∈ Γ j ∖ {1}}.…”

“…In particular, if E = K M is the canonical line bundle, the Bergman stability has been studied by many authors (cf. [R] [To] [O] [CF] [Ye3], etc). If one assumes the Bergman stability for E, by the standard argument in complex analysis, one can derive the higher order convergence for the Bergman metrics (cf.…”

Limits of canonical forms on towers of Riemann surfaces

“…See [Rho93] and [McM13,Appendix] for two different proofs of this result. See also [Yau86,Don96,Ohs09,Ohs10,Tre11,Yeu12,CF16] for various related results and generalizations. The purpose of this work is to prove a more general version of Date: May 9, 2019.…”