Arnold's seminar. April 15th 2003

M. Kazarian,  LOCALIZED THOM POLYNOMIALS

Thom polynomials are the expressions for the cohomology classes dual
to singularity loci of smooth maps in terms of the Chern classes of
the source and target manifolds. Applications of the theory of Thom
polynomials to the enumerative algebraic geometry, differential
geometry and topology is well known.

In the talk we discuss the application of this theory to the singularity
theory itself. The account of relevant characteristic classes leads to
enumerative conditions for the existence of singularity classes with
prescribed symmetry.

For example, the classification of the degeneration of $1$-jets leads
to the following splittitng
   $$ H^*(BU) = \bigoplus_k H^{*-2k(k+q)}(BU(k)\times BU(k+q))$$
for every fixed  number $q$. The left hand side of the equality
is the ring of polynomials in the Chern classes. The terms entering
the right hand side are the so called Porteous-Thom classes and
their derived classes. These terms are supported on the loci of
points with singularities of fixed corank and have very explicit
geometrical meaning.

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