Compact convergence, deformation of the L2-(partial derivative)over-bar-complex and canonical K-homology classes

Let (X, y) be a compact, irreducible Hermitian complex space of complex dimension m and with dim(sing(X)) = 0. Let (F, tau) -> X be a Hermitian holomorphic vector bundle over X, and let us denote by (partial derivative) over bar (F,m,abs) the rolled-up operator of the maximal L-2-(partial derivative) over bar -complex of F-valued (m, center dot)-forms. Let tau : M -> 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 Khomology classes [(partial derivative) over bar (F,m,abs)] = pi*[(partial derivative) over bar (E,m)], with (partial derivative) over bar (E,m) the rolled-up operator of the L-2-(partial derivative) over bar -complex of E-valued (m, center dot)-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 dF,m,abs and (partial derivative) over bar (E,m).