Invariants de Von Neumann des faisceaux coherents

Abstract

Inspired by some recent work of M. Farber, W. Lück and M. Shubin on L2 homotopy invariants of infinite Galois coverings of simplicial complexes (L2 Betti numbers and Novikov-Shubin invariants), this article extends Atiyah's L2 index theory to coherent analytic sheaves on complex analytic spaces. Let $X$ be a complex analytic space with a proper cocompact biholomorphic action of a discrete group $G$. Let $F$ be a $G$-equivariant coherent analytic sheaf on $X$. We give a meaningful notion of a L2 section of $F$ on $X$. We also construct L2 cohomology groups. We prove that these L2 cohomology groups belong to an abelian category of topological $G$-modules introduced by M. Farber. On this category there are two kinds of invariants: Von Neumann dimension and Novikov-Shubin invariants. The alternating sum of the Von Neumann dimensions of the L2 cohomology groups of $F$ can be computed by an analogue of Atiyah's L2 index theorem. Novikov-Shubin invariants show up when the L2 cohomology groups are non-Hausdorff and, like in algebraic topology, are still very intriguing (and not very well understood).