Invariants via Atiyah Classes

Recently, L.Rozansky and E.Witten [RW] associated to any hyper-Kähler manifold X an invariant of topological 3-manifolds. In fact, their construction gives a system of weights c Γ (X) associated to 3-valent graphs Γ and the corresponding invariant of a 3-manifold Y is obtained as the sum c Γ (X)I Γ (Y) where I Γ (Y) is the standard integral of the product of linking forms. So the new ingredient is the system of invariants c Γ (X) of hyper-Kähler manifolds X, one for each trivalent graph Γ. They are obtained from the Riemannian curvature of the hyper-Kähler metric. In this paper we give a reformulation of the c Γ (X) in simple cohomological terms which involve only the underlying holomorphic symplectic manifold. The idea is that we can replace the curvature by the Atiyah class [At] which is the cohomological obstruction to the existence of a global holomorphic connection. The role of what in [RW] is called " Bianchi identities in hyper-Kähler geometry " is played here by an identity for the square of the Atyiah class expressing the existence of the fiber bundle of second order jets. The analogy between the curvature and the structure constants of a Lie algebra observed in [RW] in fact holds even without any symplectic structure, and we study the nonsymplectic case in considerable detail so as to make the specialization to the symplec-tic situation easier. We show, first of all, that the Atiyah class of the tangent bundle of any complex manifold X satisfies a version of the Jacobi identity when considered as an element of an appropriate operad. In particular, we find (Theorem 2.3) that for any coherent sheaf A of O X-algebras the shifted cohomology space H • −1 (X, T X ⊗ A) has a natural structure of a graded Lie algebra, given by composing the cup-product with the Atiyah class. If E is any holomorphic vector bundle over X, then H • −1 (X, E ⊗ A) is a representation of this Lie algebra. Then, we unravel the Jacobi identity to make the space of cochains with coefficients in the tangent bundle into a " Lie algebra up to higher homotopies " [S]. An algebra of this type is best described by exhibiting a complex replacing the Chevalley-Eilenberg complex for an ordinary Lie algebra. In our case this latter complex is identified with the sheaf of functions on the …