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# Mathematical Research Letters

## Volume 24 (2017)

### Number 6

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

Pages: 1875 – 1895

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n6.a15

#### Author

#### Abstract

Let $D, \Omega_1, \dotsc , \Omega_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $\Omega_1 \times \dotsm \times \Omega_m$ 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 $D = \Omega_1 = \dotsm = \Omega_m$ with $\mathrm{rank}(D) \geq 2$ 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 $\mathrm{Aut}(D)$.

Supported in part by National Science Foundation grant DMS-1412384.

Received 23 February 2016

Accepted 18 July 2016

Published 29 January 2018