Killing fields on compact pseudo-K\"ahler manifolds

TL;DR

This paper proves that Killing fields on compact pseudo-Kähler ddbar manifolds are necessarily holomorphic, extending previous results and clarifying issues in earlier proofs, with implications for the geometry of such manifolds.

Contribution

It establishes the holomorphicity of Killing fields on compact pseudo-Kähler ddbar manifolds and discusses the validity of prior claims, including a correction to Yamada's argument.

Findings

Killing fields are necessarily holomorphic on these manifolds

The proof extends to real dimension four without the ddbar assumption

Clarification and critique of Yamada's previous argument

Abstract

We show that a Killing field on a compact pseudo-K\"ahler ddbar manifold is necessarily (real) holomorphic. Our argument works without the ddbar assumption in real dimension four. The claim about holomorphicity of Killing fields on compact pseudo-K\"ahler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada's argument is incomplete.