AN EXPLICIT PROOF OF THE GENERALIZED GAUSS-BONNET FORMULA

In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] is an element of Ext(r) (O-Z, Omega(tau)(X)) of a complex submanifold Z of a complex manifold X when Z is the zero locus of a real analytic section of a holomorphic vector bundle E of rank r on X. To this data we associate a super-connection A on boolean AND* E-V, which gives a "twisted resolution" T* of O-Z such that the "generalized super-trace" of 1/r! A(2r), which is a map of complexes from T* to the Dolbeault complex a(X)(r,)*, represents [Z]. One may then read off the Gauss-Bonnet formula from this map of complexes.