Symmetric differentials on complex hyperbolic manifolds with cusps

Benoit Cadorel (1)

1 I2M - Institut de Mathématiques de Marseille

Let $(X, D)$ be a logarithmic pair, and let $h$ be a smooth metric on $T_{X \setminus D}$. We give sufficient conditions on the curvature of $h$ for $\Omega_X (\log D)$ and $\Omega_X$ to be big. As an application, we give a metric proof of the bigness of $\Omega_X (\log D)$ 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 $X′ \longrightarrow X$, étale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that $\Omega_{X′}$ 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.

Mots clés: toroidal compactifications; cotangent bundle; symmetric differentials; complex hyperbolic manifolds

Domaines: Géométrie algébrique [math.AG]; Géométrie différentielle [math.DG]

Benoit Cadorel. Symmetric differentials on complex hyperbolic manifolds with cusps. 2016. ⟨hal-01332247v1⟩