Kodaira-Spencer Formality of Products of Complex Manifolds

Abstract

We shall say that a complex manifold X is Kodaira-Spencer formal if its Kodaira-Spencer differential graded Lie algebra A ^0, ∗ _X (Θ _X ) is formal; if this happen, then the deformation theory of X is completely determined by the graded Lie algebra H ^∗(X, Θ _X ) and the base space of the semiuniversal deformation is a quadratic singularity. Determine when a complex manifold is Kodaira-Spencer formal is generally difficult and we actually know only a limited class of cases where this happen. Among such examples we have Riemann surfaces, projective spaces, holomorphic Poisson manifolds with surjective anchor map H ^∗(X, Ω _X ¹) → H ^∗(X, Θ _X ) [4] and every compact Kähler manifold with trivial or torsion canonical bundle, see [9] and references therein. In this short note we investigate the behavior of this property under finite products. Let X, Y be compact complex manifolds; we prove that whenever X and Y are Kähler, then X × Y is Kodaira-Spencer formal if and only if the same holds for X and Y (Corollary 7.2). A revisit of a classical example by Douady shows that the above result fails if the Kähler assumption is dropped.