On analytic Todd classes of singular varieties

Abstract

Let (X, h) be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of (X, h). In the fist part, assuming either dim(sing(X)) = 0 or dim(X) = 2, we show that the rolled-up operator of the minimal L 2 -∂ complex, denoted here ð rel , induces a class in K0(X) ≡ KK0(C(X), C). A similar result, assuming dim(sing(X)) = 0, is proved also for ð abs , the rolled-up operator of the maximal L 2 -∂ complex.We then show that when dim(sing(X)) = 0 we have [ð rel ] = π * [ðM ] with π : M → X an arbitrary resolution and with [ðM ] ∈ K0(M ) the analytic K-homology class induced by ∂ + ∂ t on M . In the second part of the paper we focus on complex projective varieties (V, h) endowed with the Fubini-Study metric. First, assuming dim(V ) ≤ 2, we compare the Baum-Fulton-MacPherson Khomology class of V with the class defined analytically through the rolled-up operator of any L 2 -∂ complex. We show that there is no L 2 -∂ complex on (reg(V ), h) whose rolled-up operator induces a K-homology class that equals the Baum-Fulton-MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on V the push-forward of [ð rel ] in the K-homology of the classifying space of the fundamental group of V is a birational invariant.