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

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,aftertheHKRisomorphism,theusualone$K_0(X) \to \bigoplus H^i(X, \Omega_X^i)$; and – we present a conjecture that relates the Hochschild and harmonic structures of a smooth space.