Seminar in Real and Complex Geometry

Thursday, July 4, 2024, 16:15-17:45, online via zoom

László Fehér
(Eötvös Loránd University, Budapest)

Thom polynomials of real singularities


Thom polynomials are designed to solve enumerative problems. The theory of Thom polynomials of complex singularities is well established. A theorem of Borel and Haeiger allows us to translate the complex results to mod 2 results for real singularities, which leads to solutions for mod 2 enumerative problems. The theory of integer valued Thom polynomials of real singularities is not very well understood. I will talk about some important examples calculated jointly with András Szenes. Al sample result is tp(A4(2l-1)) = (p_l)^2 + 3\sum_{i=1}^l 4^{i-1}p_{l-i}p_{l+i} where p_i denote the Pontryagin classes and A4(2l .. 1) is the Thom-Boardman class \Sigma^{1;1;1;1} in relative codimension 2l-1. Notice the similarity with Ronga's formula for the Thom polynomial of the cusp (or A2 or \Sigma^{1;1} singularities in the complex case. These results lead to non-trivial lower bounds for enumerative problems. Another direction to find new results is to stay in the mod 2 world but enhance the Thom polynomials. In the complex case the Segre-Schwartz-MacPherson Thom polynomials were introduced by Ohmoto Toru. In a joint work with Ákos Matszangosz we introduced the real version, the Segre-Stiefel-Whitney Thom polynomials. These allow us for example to find obstructionsand geometric meaning for them for the existence of Morin or fold maps of real projective spaces into a Euclidean space. As an example of these geometric interpretations of obstructions we can calculate the modulo 2 Euler class of certain degeneracy loci of generic smooth maps.
Joint work with András Szenes and Ákos Matszangosz.