Derived invariance of the space of holomorphic two-forms

Let Φ:Dᵇ(X)→ Dᵇ(Y) be a derived equivalence between smooth projective complex varieties, and let h=c₁(L) for an ample line bundle L on X. For z, transporting the cup-product action and evaluating at eᶻʰ gives a homomorphism of graded algebras Θ^•F,z:H^•(Y, OY)→ H^•(X, OX), where F is the cohomological transform of $Φ$. We prove that Θ^•F,i is an isomorphism in degrees one and two. In particular, X and Y have the same number of holomorphic two-forms. The proof in degree two uses the Hodge-Riemann bilinear relations on X and on X× X. We also prove derived invariance of all Hodge numbers when KX is trivial and the conjugate of a holomorphic volume form belongs to the subalgebra of H^•(X, OX) generated in degrees one and two. In even dimension, the evaluation isomorphism allows us to compare the ranks of cup product by powers of symplectic two-forms on the Hochschild diagonals. These ranks determine the Hodge numbers individually.