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

Let $(X,\gamma)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,\tau)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with $\overline{\eth}_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex of $F$-valued $(m,\bullet)$-forms. Let $\pi:M\rightarrow X$ be a resolution of singularities, $g$ a metric on $M$, $E:=\pi^*F$ and $\rho:=\pi^*\tau$. In this paper, under quite general assumptions on $\tau$, we prove the following equality of analytic $K$-homology classes $[\overline{\eth}_{F,m,\mathrm{abs}}]=\pi_*[\overline{\eth}_{E,m}]$, with $\overline{\eth}_{E,m}$ 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 $\overline{\eth}_{F,m,\mathrm{abs}}$ and $\overline{\eth}_{E,m}$.