On the L–Stokes theorem and Hodge theory for singular algebraic varieties

We discuss aspects of the L–Stokes theorem on certain manifolds with singularities. We show that the L–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, n ± 1. A consequence is a Hodge theorem on the operator level as well as the fact that the L–Stokes theorem holds except possibly in degrees n − 1, n. However, in general the conjecture that the L–Stokes theorem holds on complex projective varieties remains still open. 1991 Mathematics Subject Classification. 58A (32S)