On Analytic Todd Classes of Singular Varieties

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 1st part, assuming either dim(sing(X))=0 or dim(X)=2⁠, we show that the rolled-up operator of the minimal L2-Dolbeault- complex, denoted here ð_rel⁠, induces a class in K_0(X)≡KK_0(C(X),C)⁠. A similar result, assuming dim(sing(X))=0⁠, is proved also for ð_abs⁠, the rolled-up operator of the maximal L2-Dolbeault-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 the Dirac-Dolbeault operator on M⁠. In the 2nd 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 K-homology class of V with the class defined analytically through the rolled-up operator of any L2-Dolbeaul- complex. We show that there is no L2- Dolbeault-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.