Symmetric differentials on complex hyperbolic manifolds with cusps

TL;DR

This paper establishes conditions under which logarithmic and standard cotangent bundles on complex hyperbolic manifolds with cusps are big or nef, using singular metrics and curvature analysis, with applications to toroidal compactifications and ramified covers.

Contribution

It provides new curvature-based criteria for the bigness and nefness of cotangent and tangent bundles on complex hyperbolic manifolds, including explicit ramification bounds.

Findings

Logarithmic cotangent bundle is big on toroidal compactifications.

Standard tangent bundle can be nef with high enough ramification.

Effective ramification orders are independent of dimension.

Abstract

Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $X' \longrightarrow X$, \'{e}tale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that the standard tangent bundle of such a cover is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the quotient of the ball we consider.