Compact convergence, deformation of the L^2--complex and canonical K-homology classes

Let (X, ) be a compact, irreducible Hermitian complex space of complex dimension m and with (sing (X) ) =0. Let (F, ) X be a Hermitian holomorphic vector bundle over X, and let us denote by ₅, ₌, ₀₁ₒ the rolled-up operator of the maximal L^2 - -complex of F -valued (m, ) -forms. Let: M X be a resolution of singularities, g a metric on M, E: =^*F and: =^*. In this paper, under quite general assumptions on, we prove the following equality of analytic K -homology classes ₅, ₌, ₀₁ₒ=*₄, ₌, with ₄, ₌ the rolled-up operator of the L^2 - -complex of E -valued (m, ) -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 ₅, ₌, ₀₁ₒ and ₄, ₌.