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July 2021 Symmetric differentials on complex hyperbolic manifolds with cusps
Benoît Cadorel
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J. Differential Geom. 118(3): 373-398 (July 2021). DOI: 10.4310/jdg/1625860621

Abstract

Let $(X,D)$ be a logarithmic pair, and let $h$ be a smooth metric on $T_{X \setminus D}$. We give a simple criterion on the curvature of $h$ for the bigness of $\Omega_X(\operatorname{log} D)$ or $\Omega_X$. As an application, we obtain a metric proof of the bigness of $\Omega_X(\operatorname{log} D)$ on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of $\Omega_X$ in the more specific case of the ball. We obtain effective ramification orders for a cover $X^\prime \to X$, étale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that $\Omega_{X^\prime}$ is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the ball quotient we consider.

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Benoît Cadorel. "Symmetric differentials on complex hyperbolic manifolds with cusps." J. Differential Geom. 118 (3) 373 - 398, July 2021. https://doi.org/10.4310/jdg/1625860621

Information

Received: 27 March 2018; Accepted: 14 December 2018; Published: July 2021
First available in Project Euclid: 10 July 2021

Digital Object Identifier: 10.4310/jdg/1625860621

Rights: Copyright © 2021 Lehigh University

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Vol.118 • No. 3 • July 2021
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