Kwok Kin Wong

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 Carath\'eodory 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 Shimura varieties, non-arithmetic
ball quotients, moduli space of hyperbolic Riemann surfaces, etc.

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