Cursos de Inverno em Matemática do IST 

23/02/2015, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Ngaiming Mok, University of Hong Kong

Complex geometry on bounded symmetric domains and their quotient spaces (I)

Bounded symmetric domains $\Omega\Subset\mathbb{C}^N$ are important for many areas of Mathematics, including Complex Analysis, Differential Geometry, Algebraic Geometry and Number Theory. They carry Bergman metrics which are of nonpositive holomorphic bisectional curvature. Their quotient manifolds are thus equipped with canonical Kähler metrics and many of these quotient manifolds are of importance as they are moduli spaces of different algebro-geometric problems. In this series of lectures we will examine complex-geometric problems on bounded symmetric domains in their Harish-Chandra realizations and on their finite-volume quotient manifolds. Especially, we will discuss

1. the phenomenon of metric rigidity for Hermitian metrics of nonpositive curvature on irreducible finite-volume quotients $X_\Gamma := \Omega/\Gamma$ of rank $\geq 2$,
2. geometric structures and rigidity on germs of measure-preserving holomorphic maps from an irreducible bounded symmetric domain into its Cartesian products,
3. holomorphic isometries from complex unit balls into bounded symmetric domains $\Omega$ of rank $\geq 2$,
4. total geodesy of Zariski closures of images of algebraic sets on bounded symmetric domains $\Omega$ under the universal covering map into their finite-volume quotients $X_\Gamma$.