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.Comment: 21 pages. Paper reorganized to improve exposition. To appear in
Asterisqu
