Invariants de Von Neumann des faisceaux coherents

arXivmath/9806159MaRDI QIDQ6500944

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Abstract: Inspired by some recent work of M. Farber, W. L"uck 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).

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