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 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ 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 2nd 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 $(\operatorname{reg}(V),h)$ 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 $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

A note on higher Todd genera of complex manifolds

“…is precisely [ð M ], the analytic K-homology class associated to the operator ∂ + ∂ t over M . See [2],…”

A Dolbeault–Hilbert complex for a variety with isolated singular points

“…In [2], Andersson and Samuelsson gave a resolution of the structure sheaf by certain currents on X, that are smooth on X reg . After this paper was written, Bei and Piazza posted [10], which also has a proof of Proposition 5.1.…”