Cohomological constraint to deformations of compact Kähler manifolds

We prove that for every compact Kähler manifold X the cup product H ∗(X, TX)⊗ H ∗(X,Ω∗X ) → H ∗(X,Ω∗−1 X ) can be lifted to an L∞-morphism from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of linear endomorphisms of the singular cohomology of X. As a consequence we get an algebraic proof of the principle “obstructions to deformations of compact Kähler manifolds annihilate ambient cohomology”. Mathematics Subject Classification (2000): 32G05 Introduction Let X be a fixed compact Kähler manifold of dimension n and consider the graded vector space M = Hom∗C(H (X,C), H(X,C)) of linear endomorphisms of the singular cohomology of X . The Hodge decomposition gives natural isomorphisms M = ⊕