Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 369

On the $L^2$--Stokes theorem and Hodge theory for singular algebraic varieties

D. Grieser and M. Lesch

We discuss aspects of the $L^2$--Stokes theorem on certain manifolds with singularities. We show that the $L^2$--Stokes theorem does not hold on real projective varietes, even for isolated singularities. For a complex projective variety of complex dimension n, with isolated singularities, we show that the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees $n,npm 1$. A consequence is a Hodge theorem on the operator level as well as the fact that the $L^2$--Stokes theorem holds except possibly in degrees $n-1,n$. However, in general the conjecture that the $L^2$--Stokes theorem holds on complex projective varieties remains still open

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