  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).
