Killing fields on compact pseudo-K\"ahler manifolds

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.