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

Abstract

Let $(X,γ)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,τ)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with $\overlineð_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex of $F$-valued $(m,\bullet)$-forms. Let $π:M\rightarrow X$ be a resolution of singularities, $g$ a metric on $M$, $E:=π^*F$ and $ρ:=π^*τ$. In this paper, under quite general assumptions on $τ$, we prove the following equality of analytic $K$-homology classes $[\overlineð_{F,m,\mathrm{abs}}]=π_*[\overlineð_{E,m}]$, with $\overlineð_{E,m}$ 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 \cite{KuSh} and provides an explicit homotopy between the even unbounded Fredholm modules induced by $\overlineð_{F,m,\mathrm{abs}}$ and $\overlineð_{E,m}$.