Invariants de Von Neumann des faisceaux analytiques cohérents

Abstract

In order to study the group of \(L^2\) holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's \(L^2\) index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf \({\cal F}\) on a n-dimensionnal smooth projective variety \(X^{(n)}\) and each Galois infinite unramified covering \(\pi:\tilde X \to X\), whose Galois group is denoted by \(\Gamma\), \(L^2\) cohomology groups denoted by \(H^q_2(\tilde X,{\cal F})\) are attached, such that: