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

Shan Tai Chan

TL;DR

The paper generalizes local holomorphic map rigidity from Kähler to Hermitian settings, showing that any germ $F$ with $F^*\omega_N^p=\lambda\omega_M^p$ must satisfy $m\le n$ and, when $p<m$, be a local holomorphic isometry up to a positive scalar. It then establishes broad nonexistence theorems under curvature hypotheses, using CR geometry on unit sphere bundles and the Umehara algebra to derive obstructions. The results cover maps into semi-negative or period-domain targets and extend to bounded symmetric domains and compact-type Hermitian symmetric manifolds via canonical embeddings. Taken together, the work deepens our understanding of rigidity and nonexistence for $(p,p)$-form preserving holomorphic maps between Hermitian and symmetric spaces, with implications for mappings among symmetric spaces and projective varieties.

Abstract

In this article, we generalize some results in Chan-Yuan [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 26 (2025), 619--644] to local holomorphic maps between Hermitian manifolds preserving $(p,p)$-forms. In particular, we obtain further rigidity theorems and non-existence theorems for such maps.

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

TL;DR

The paper generalizes local holomorphic map rigidity from Kähler to Hermitian settings, showing that any germ with must satisfy and, when , be a local holomorphic isometry up to a positive scalar. It then establishes broad nonexistence theorems under curvature hypotheses, using CR geometry on unit sphere bundles and the Umehara algebra to derive obstructions. The results cover maps into semi-negative or period-domain targets and extend to bounded symmetric domains and compact-type Hermitian symmetric manifolds via canonical embeddings. Taken together, the work deepens our understanding of rigidity and nonexistence for -form preserving holomorphic maps between Hermitian and symmetric spaces, with implications for mappings among symmetric spaces and projective varieties.

Abstract

In this article, we generalize some results in Chan-Yuan [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 26 (2025), 619--644] to local holomorphic maps between Hermitian manifolds preserving -forms. In particular, we obtain further rigidity theorems and non-existence theorems for such maps.

Paper Structure

This paper contains 7 sections, 6 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rigidity theorem for local holomorphic maps between Hermitian manifolds preserving $(p,p)$-forms
  4. Non-existence Theorems
  5. The unit sphere bundles on Hermitian manifolds
  6. The Umehara algebra on bounded symmetric domains and its applications
  7. Local holomorphic maps between Hermitian symmetric manifolds of the compact type and bounded symmetric domains

Key Result

Proposition 3.1

Let $(M,\omega_M)$ and $($$N$, $\omega_N$$)$ be Hermitian manifolds of finite complex dimensions $m$ and $n$ respectively, where $\omega_M$ and $\omega_N$ denote the corresponding Hermitian forms. Let $F:(M;x_0)\to (N;F(x_0))$ be a germ of holomorphic map such that $F^*\omega_N^p = \lambda \omega_M^

Theorems & Definitions (12)

  • Proposition 3.1: $\text{cf.\,Chan-Yuan }CY25$
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 4.4
  • proof
  • Remark 4.5
  • ...and 2 more