The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism

Abstract

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are: -- we introduce a generalization of the usual Mukai pairing on differential forms that applies to arbitrary manifolds; -- we give a proof of the fact that the natural Chern character map $K_0(X) \to HH_0(X)$ becomes, after the HKR isomorphism, the usual one $K_0(X) \to \bigoplus H^i(X, Ω_X^i)$; and -- we present a conjecture that relates the Hochschild and harmonic structures of a smooth space.