1998/06/30 by Philippe Eyssidieux, Eyssidieux, Philippe
Mathematics · #14C30 (Primary) 32J25 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C30 #msc:32J25
paper · pdf · doi:10.48550/arxiv.math/9806159
Latex2e, 46 pages, French
arxiv created 1998/06/30 · arxiv updated 2009/11/30
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).