Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps

TL;DR

This paper investigates the positivity properties of the cotangent bundle of complex hyperbolic manifolds with cusps, establishing conditions under which these bundles are ample or semi-ample, with implications for subvarieties and their volumes.

Contribution

It provides new positivity results for the cotangent bundle of complex hyperbolic manifolds with cusps, depending on cusp depth, and explores their geometric consequences.

Findings

m ig( ext{logarithmic cotangent bundle}ig) is ample for small rational r>0.

m ig( ext{logarithmic cotangent bundle}ig) is ample modulo divisor D.

Subvarieties intersecting the boundary have increasing minimal volume in tower coverings.

Abstract

Let $\overline{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X=\mathbb{B}^n/\Gamma$ with the boundary divisor $D=\overline{X}\setminus X$. The main goal of this paper is to find the positivity properties of $\Omega^{1}_{\overline{X}}$ and $\Omega^{1}_{\overline{X}}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $\Omega^{1}_{\overline{X}}\big(\log(D)\big) \langle -r D \rangle$ is ample for all sufficiently small rational numbers $r >0$, and $\Omega^{1}_{\overline{X}}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4\pi$, then $\Omega^{1}_{\overline{X}}$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \overline{X}$ intersecting $\overline{X}$ has ample $K_{V}$. Finally, we show that the minimum volume of subvarieties of $\overline{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X.$