On local holomorphic maps preserving invariant (p,p)-forms between bounded symmetric domains

Abstract

Let $D,{\Omega }_1,...,{\Omega }_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into ${\Omega }_1\times ...{\Omega }_m$ preserving the invariant $(p,p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank one case for any $p$ and in the rank at least two case for certain sufficiently large $p$. The total geodesy thus follows if $D={\mathbb{B}}^n,{\Omega }_i={\mathbb{B}}^{N_i}$ for any $p$ or if $D={\Omega }_1=...={\Omega }_m$ with rank$(D)\ge 2$ and $p$ sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties $D/\Gamma$ preserving invariant $(p,p)$-forms is modular, where $\Gamma$ is a torsion free, discrete, finite co-volume subgroup of Aut$(D)$. This solves partially a problem raised by Mok.

References in corpus (1)

• Submanifolds of Hermitian symmetric spaces

Cited by in corpus (1)

• Complexity of holomorphic maps from the complex unit ball to classical domains