The Multiplicity Polar Theorem, Collections of 1-forms, and Chern Numbers

Picture : 

Extending invariants of geometry/topology from smooth spaces to singular spaces is a long term project involving many authors. The calculation of the contributions of the singularities to the invariants is important in this program. If the contributions can be connected to an ideal, then the multiplicity of the ideal can be used for computations. If the invariant is connected with the degeneracy of sections of vector bundles, then the multiplicity of a module is used. 

If the module is not of finite colength, then the multiplicity is not defined. If the invariant is connected with sections of vector bundles, then the analogous vector bundles may not be defined at the singularities. In this talk, joint work with N. Grulha Jr., we show how these problems can be addressed in calculating the Chern obstruction of collections of differential forms on the germ of a singular complex analytic space. We use the multiplicity of pairs of modules to deal with the first obstacle, and the Nash modification to deal with the second.