Cohomological constraint on deformations of compact Kaehler manifolds

We prove that for every compact K\"ahler manifold $X$ the cup product \[H^*(X,T_X)\otimes H^*(X,\Omega_X^*)\to H^*(X,\Omega_X^{*-1})\] can be lifted to an $L_{\oo}$-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 K\"{a}hler manifolds annihilate ambient cohomology''.