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

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

1999-02-09

arxiv.org/abs/math/9902062v2

Math. Nachr. 246/247 (2002), 68--82

Mathematics

Differential Geometry

17 pages; revised version; to appear in Math. Nachrichten

Scientific paper

For a projective algebraic variety $V$ with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of $V$. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees $n,n\pm 1$, where $n$ is the complex dimension of $V$. We also prove a Hodge theorem on the operator level and the $L^2$--Stokes theorem outside the degrees $n-1,n$. We show that the $L^2$-Stokes theorem may fail to hold in the case of real algebraic varieties, and also discuss the $L^2$-Stokes theorem on more general non-compact spaces.

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