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

Let $(X,\gamma )$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}(X))=0$. Let $(F,\tau )\to X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with ${\overline{\mathrm{ð}}}_{F,m,\mathrm{a}\mathrm{b}\mathrm{s}}$ the rolled-up operator of the maximal $L^2$-$\bar{\mathrm{\partial }}$ 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 $[{\overline{\mathrm{ð}}}_{F,m,\mathrm{a}\mathrm{b}\mathrm{s}}]={\pi }_{\ast }[{\overline{\mathrm{ð}}}_{E,m}]$, with ${\overline{\mathrm{ð}}}_{E,m}$ the rolled-up operator of the $L^2$-$\bar{\mathrm{\partial }}$ 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 ${\overline{\mathrm{ð}}}_{F,m,\mathrm{a}\mathrm{b}\mathrm{s}}$ and ${\overline{\mathrm{ð}}}_{E,m}$.