Affine vector fields on compact pseudo-Kähler manifolds

Abstract

It is known that a Killing field on a compact pseudo-Kähler manifold is necessarily (real) holomorphic, as long as the manifold satisfies some relatively mild additional conditions. We provide two further proofs of this fact and discuss the natural open question whether the same conclusion holds for affine -- rather than Killing -- vector fields. The question cannot be settled by invoking the Killing case: Boubel and Mounoud [Trans.Amer. Math. Soc. 368, 2016, 2223--2262] constructed examples of non-Killing affine vector fields on compact pseudo-Riemannian manifolds. We show that an affine vector field v is necessarily symplectic, and establish some algebraic and differential properties of the Lie derivative of the metric along v, such as its being parallel, antilinear and nilpotent as an endomorphism of the tangent bundle. As a consequence, the answer to the above question turns out to be `yes' whenever the underlying manifold admits no nontrivial holomorphic quadratic differentials, which includes the case of compact almost homogeneous complex manifolds with nonzero Euler characteristic.