Kodaira-Spencer Formality of Products of Complex Manifolds

Abstract

AbstractWe 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.KeywordsComplex ManifoldDeformation TheoryCompact Complex ManifoldHolomorphic Tangent SheafRational Homotopy TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.