An Explicit Proof of the Generalized Gauss-Bonnet Formula

Details An Explicit Proof of the Generalized Gauss-Bonnet Formula An Explicit Proof of the Generalized Gauss-Bonnet Formula

2004-04-02

arxiv.org/abs/math/0404051v2

Mathematics

Differential Geometry

21 pages. Paper reorganized to improve exposition. To appear in Asterisque

Scientific paper

In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior algebra of E, which gives a "twisted resolution" of the structure sheaf of Z. The "generalized super-trace" of A^{2r}/r!, where r is the rank of E, is an explicit map of complexes from the twisted resolution to the Dolbeault complex of X, which represents [Z]. One may then read off the Gauss-Bonnet formula from this map of complexes.

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