Segre class of smooth vector bundles over smooth manifolds?

Question

Before you read the following question, please assume I have no knowledge in algebraic geometry.

Is it possible to define Segre class of a smooth complex vector bundle over a smooth manifold by using Chern-Weil theory? That is, using connection, curvature and invariant polynoimal, etc.

I know the total Segre class of a holomorphic vector bundle over a complex manifold can be defined as the inverse of the total Chern class (of course one can translate the statement into the language of algebraic geometry). But I have never seen the appearance of Segre class in differential geometric setting. On the other hand, I don't know whether saying "we can define the total Segre form by inverting the total Chern form" makes sense or not.

Thanks~

Answer 1

It makes sense. The total Chern form in the algebra of differential forms is of the form $1+\alpha $, where $\alpha $ is nilpotent. Just define the Segre form as $1-\alpha +\alpha ^2+\ldots $

Answer 2

You might also look at Harvey and Lawson's paper "Geometric residue theorems" where they derive Chern-Weil formulas for currents representing degeneracy loci of maps between vector bundles. I can't remember if they give explicit formulas for the Segre classes but I think one could use their method to get something along those lines.