An Explicit Proof of the Generalized Gauss-Bonnet Formula
Differential Geometry
2009-05-28 v2 Algebraic Geometry
Abstract
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.
Cite
@article{arxiv.math/0404051,
title = {An Explicit Proof of the Generalized Gauss-Bonnet Formula},
author = {Henri Gillet and Fatih Unlu},
journal= {arXiv preprint arXiv:math/0404051},
year = {2009}
}
Comments
21 pages. Paper reorganized to improve exposition. To appear in Asterisque