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On the L 2 ‐Stokes Theorem and Hodge Theory for Singular Algebraic Varieties
Author(s) -
Grieser Daniel,
Lesch Matthias
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200212)246:1<68::aid-mana68>3.0.co;2-y
Subject(s) - mathematics , pure mathematics , algebraic variety , projective variety , embedding , algebraic number , complex dimension , operator (biology) , discrete mathematics , mathematical analysis , biochemistry , chemistry , repressor , artificial intelligence , computer science , transcription factor , gene
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 ± 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.