Carathéodory Geometry, Hyperbolicity and Rigidity

Date/heure
24 juin 2024
14:00 - 15:00

Lieu
Salle de conférences Nancy

Oratrice ou orateur
Kwok-Kin Wong

Catégorie d'évènement
Séminaire de géométrie complexe

Résumé

We discuss some recent results concerning complex manifolds whose
universal coverings admit many bounded holomorphic functions.

Let $X$ be a quasi-projective manifold whose universal covering $M$ is
a strongly Carathéodory hyperbolic manifold. We will see that any
(quasi-)projective subvariety of $X$ is of (log-)general type. The
result is consistent with the prediction of a conjecture of Lang. We
will also see that $M$ has many interesting geometric and analytic
properties. Examples of $X$ include finite volume quotients of bounded
symmetric domains, moduli space of hyperbolic Riemann surfaces, etc.

Next we consider holomorphic maps $f:S=\mathrm{\Omega }/\mathrm{\Gamma }\to N$
from a finite volume quotient of bounded symmetric domain $Omega$ of
rank $\ge 2$ to a complex manifold $N$, where the universal covering
$\tilde{N}$ of $N$ has sufficiently many bounded holomorphic
functions. We will see that the inverse $F^{-1}$ of the lifting
$F:\mathrm{\Omega }\to \tilde{N}$ of $f$ extends to a bounded
holomorphic map $R:\tilde{N}\to {\mathbb{C}}^n$. This gives
another proof that $F$ must be a holomorphic embedding and lead to
certain rigidity result when $N$ satisfies some natural additional
geometric properties.