Thom series of contact singularities

Abstract

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes and with iterated residue identities are revealed.

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

“…Unfortunately, the explicit computation of Q A is difficult, his description does not give more information for Morin singularities, where Q k is unknown for k > 6. Rimányi and Fehér in [20] compute Thom series for further singularity classes. The structure of Thom polynomials of contact singularities was also studied in [21,22].…”

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

“…The idea to express such sums as iterated residues at infinity came from Bérczi and Szenes [2] who used it as a computational tool for Thom polynomials. This idea was implemented also by other authors, see for example Fehér and Rimányi [5,13] and Weber [14]. One can find a different approach to the residue formulas in papers of Jeffrey and Kirwan [9,10], who consider symplectic manifolds with a hamiltonian action of a compact Lie group (not necessarily abelian).…”

Thom polynomials in $\mathcal A$-classification I: counting singular projections of a surface

“…Note that the Thom polynomial can not directly be applied to ϕ (x,l,q) , because the base space F is not compact. A key idea is to introduce a certain family of ϕ over the flag manifold of (x, l) by choosing q = q (x,l) at the 'infinity on l with respect to x' so that generically q does not meet any special positions, see (5) and (6) in §3.2.…”