Compact convergence, deformation of the L²-∂̄-complex and canonical K-homology classes

Let (X,γ) be a compact, irreducible Hermitian complex space of complex dimension m and with (sing(X))=0 . Let (F,τ)→ X be a Hermitian holomorphic vector bundle over X , and let us denote by ̄F,m,abs the rolled-up operator of the maximal L² - ∂̄ -complex of F -valued (m,•) -forms. Let π:M→ 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 [̄F,m,abs]=π*[̄E,m] , with ̄E,m the rolled-up operator of the L² - ∂̄ -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 ̄F,m,abs and ̄E,m .