On local holomorphic maps between Kähler manifolds preserving $(p,p)$-forms

Abstract

We study local holomorphic maps between Kähler manifolds preserving $(p,p)$-forms. In this direction, we prove that any such local holomorphic map $F$ is a holomorphic isometry up to a scalar constant provided that $p$ is strictly less than the complex dimension of the domain of $F$. We then study local holomorphic maps between finite dimensional complex space forms preserving invariant $(p,p)$-forms. It was proved by Calabi that there does not exist a local holomorphic isometry between complex space forms $M$ and $N$ provided that $M$ and $N$ are of different types. In this article, we generalize this result to local holomorphic maps between complex space forms $M$ and $N$ preserving invariant $(p,p)$-forms whenever $M$ and $N$ are of different types except for the case where the universal covers of $M,N$ are biholomorphic to ${\mathbb{C}}^m,{\mathbb{P}}^n$, respectively and $2\le p=m<n$. We also obtain some results in more general settings, including the study on indefinite Kähler manifolds and relatives for Kähler manifolds.

Accepted for publication in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

References in corpus (2)

• Holomorphic isometries into homogeneous bounded domains
• Submanifolds of some Hartogs domain and the complex Euclidean space