\(L^2\) -cohomology on the coverings of a compact complex manifold (Q1428377)

The aim of this paper is to define a natural \(L^2\)-cohomology on any unramified covering of a complex analytic space \(X\), with values in the lifting of any coherent analytic sheaf on \(X\). This \(L^2\) cohomology has been constructed independently by \textit{P. Eyssidieux} [Math. Ann. 317, 527--566 (2000; Zbl 0964.32008)]. It is seen that the usual properties of sheaf cohomology such as cohomology exact sequences or spectral sequences hold in this \(L^2\)-cohomology on \(X\). If \(X\) is projective and non-singular there are \(L^2\) vanishing theorems analogous to those of Kodaira-Serre and Kawamata-Viehweg. When \(X\) is compact it is possible to define the \(\Gamma\)-dimension for Galois coverings. This \(\Gamma\)-dimension turns out to be finite in this case. An extension of Atiyah's index theorem is given in this context.