arXiv: Differential Geometry | 2019
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 $\\mathrm{dim}(\\mathrm{sing}(X))=0$ or $\\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\\overline{\\partial}$ complex, denoted here $\\overline{\\eth}_{\\mathrm{rel}}$, induces a class in $K_0 (X)\\equiv KK_0(C(X),\\mathbb{C})$. A similar result, assuming $\\mathrm{dim}(\\mathrm{sing}(X))=0$, is proved also for $\\overline{\\eth}_{\\mathrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\\overline{\\partial}$ complex. We then show that when $\\mathrm{dim}(\\mathrm{sing}(X))=0$ we have $[\\overline{\\eth}_{\\mathrm{rel}}]=\\pi_*[\\overline{\\eth}_M]$ with $\\pi:M\\rightarrow X$ an arbitrary resolution and with $[\\overline{\\eth}_M]\\in K_0 (M)$ the analytic K-homology class induced by $\\overline{\\partial}+\\overline{\\partial}^t$ 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 $\\dim(V)\\leq 2$, we compare the Baum-Fulton-MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\\overline{\\partial}$ complex. We show that there is no $L^2$-$\\overline{\\partial}$ complex on $(\\mathrm{reg}(V),h)$ 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 $[\\overline{\\eth}_{\\mathrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.