Compact convergence, deformation of the $L^2$-$\overline{\partial }$-complex and canonical $K$-homology classes

Abstract

Let $(X,\gamma )$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\dim (\mathrm{sing}(X))=0$. Let $(F,\tau )\to X$ be a Hermitian holomorphic vector bundle over $X$, and let us denote by ${\overline{ð}}_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial }$-complex of $F$-valued $(m,\bullet )$-forms. Let $\pi :M\to X$ be a resolution of singularities, $g$ a metric on $M$, $E:={\pi }^{\ast }F$ and $\rho :={\pi }^{\ast }\tau$. In this paper, under quite general assumptions on $\tau$, we prove the following equality of analytic $K$-homology classes $[{\overline{ð}}_{F,m,\mathrm{abs}}]={\pi }_{\ast }[{\overline{ð}}_{E,m}]$, with ${\overline{ð}}_{E,m}$ the rolled-up operator of the $L^2$-$\overline{\partial }$-complex of $E$-valued $(m,\bullet )$-forms on $M$. Our proof is based on functional analytic techniques developed in Kuwae and Shioya (2003) and provides an explicit homotopy between the even unbounded Fredholm modules induced by ${\overline{ð}}_{F,m,\mathrm{abs}}$ and ${\overline{ð}}_{E,m}$.