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

TL;DR

This paper investigates local holomorphic maps between bounded symmetric domains that preserve invariant (p,p)-forms, showing they extend to algebraic maps under certain conditions, leading to implications for geometric and algebraic structures.

Contribution

It establishes conditions under which local holomorphic maps preserving invariant forms extend to algebraic maps, partially solving a problem posed by Mok.

Findings

Maps extend to algebraic maps in rank one cases for any p

Maps extend to algebraic maps in higher rank for sufficiently large p

Results imply modularity of algebraic correspondences preserving invariant forms

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$(D)\geq 2$ 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.