Invariants de Von Neumann des faisceaux analytiques cohérents

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:

1. The \(H^q_2(\tilde X,{\cal F})\) underly a cohomological functor on the abelian category of coherent analytic sheaves on X.

2. If \({\cal F}\) is locally free, \(H_2^0(\tilde X,{\cal F})\) is the group of \(L^2\) holomorphic sections of the pull-back to \({\tilde X}\) of the holomorphic vector bundle underlying \({\cal F}\).

3. \(H^q_2(\tilde X,{\cal F})\) belongs to a category of \(\Gamma\)-modules on which a dimension function \(\dim_{\Gamma}\) with real values is defined.

4. Atiyah's \(L^2\) index theorem holds [Ati76]:

\[ \sum_{q=0}^{n}(-1)^q \dim_{\Gamma}H^q_2(\tilde X,{\cal F})=\sum_{q=0}^{n}(-1)^q \dim H^q( X,{\cal F}). \]

The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96] of an abelian category \(E_f(\Gamma)\) containing every closed \(\Gamma\)-submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods to study \(L^2\) Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar rôle here. We first construct a \(L^p\)-cohomology theory (\(p\in[1,\infty]\)) for coherent analytic sheaves on a complex space endowed with a proper action of a group \(\Gamma\) such that conditions 1-2 are fulfilled. The \(L^p\)-cohomology on the Galois covering \({\tilde X} \to X\) of a coherent analytic sheaf \({\cal F}\) onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of \({\cal F}\) by the locally constant sheaf on X associated to the left regular representation of the discrete group \(Gal({\tilde X}/ X)\) in the space of \(L^p\) functions on \(Gal({\tilde X}/ X)\). Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category of \(E_f(\Gamma)\) and are a good model of the Čech complex calculating \(L^2\)-cohomology. Using this we prove that \(H^q_2(\tilde X,{\cal F}) \in E_f(\Gamma)\), if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title of the article, \(L^2\) Betti numbers and Novikov-Shubin invariants of \(H^q_2(\tilde X,{\cal F})\) are the Von Neumann invariants of the coherent analytic sheaf \({\cal F}\). We also make the connection with Atiyah's \(L^2\)-index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.

Authors and Affiliations

1. Laboratoire Emile Picard, CNRS UMR 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France (e-mail: eyssi@picard.ups-tlse.fr) , , , , , , FR

   Philippe Eyssidieux

Received: 30 October 1998 / Published online: 8 May 2000

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