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On weak and strict relatives Kähler manifolds

Giovanni Placini

TL;DR

This work investigates when weak relatives of Kähler manifolds must be relatives. It proves that if $M_1$ is projective (admits a holomorphic immersion into $\mathbb{C}P^N$) and $M_2$ is a weak relative of $M_1$, then there exists a holomorphic isometry between ambient spaces, making $M_1$ and $M_2$ relatives. The proof relies on a de Rham decomposition of the weak relative factor, excludes Ricci-flat components via a result of Hulin, and constructs a global holomorphic isometry by adjusting factor maps. The paper also introduces strict relatives and furnishes multiple explicit examples across compact and noncompact, irreducible and reducible settings to demonstrate that related manifolds need not admit a local holomorphic immersion of one into the other, highlighting new rigidity phenomena in Kähler geometry.

Abstract

We study Kähler manifolds that are (weak) relatives, that is, Kähler manifolds which share a (locally isometric) submanifold. In particular, we prove that if two Kähler manifolds are weak relatives and one of them is projective, then they are relatives. Moreover, we introduce the notion of strict relatives Kähler manifolds and provide several nontrivial examples.

On weak and strict relatives Kähler manifolds

TL;DR

This work investigates when weak relatives of Kähler manifolds must be relatives. It proves that if is projective (admits a holomorphic immersion into ) and is a weak relative of , then there exists a holomorphic isometry between ambient spaces, making and relatives. The proof relies on a de Rham decomposition of the weak relative factor, excludes Ricci-flat components via a result of Hulin, and constructs a global holomorphic isometry by adjusting factor maps. The paper also introduces strict relatives and furnishes multiple explicit examples across compact and noncompact, irreducible and reducible settings to demonstrate that related manifolds need not admit a local holomorphic immersion of one into the other, highlighting new rigidity phenomena in Kähler geometry.

Abstract

We study Kähler manifolds that are (weak) relatives, that is, Kähler manifolds which share a (locally isometric) submanifold. In particular, we prove that if two Kähler manifolds are weak relatives and one of them is projective, then they are relatives. Moreover, we introduce the notion of strict relatives Kähler manifolds and provide several nontrivial examples.
Paper Structure (3 sections, 3 theorems, 3 equations)

This paper contains 3 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction and statements of the main results
  2. Weak relatives are relatives
  3. Stricltly related Kähler manifolds

Key Result

Theorem 3

If a projective manifold $M_1$ and a Kähler manifold $M_2$ are weak relatives, then they are relatives.

Theorems & Definitions (13)

  • Definition 1: DiscalaLoi10Relatives
  • Definition 2: DiscalaLoi10Relatives
  • Theorem 3
  • Corollary 4
  • Definition 5
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['ThmMain']}
  • Example 1
  • Example 2
  • ...and 3 more