Thom series of contact singularities. (English) Zbl 1264.32023

Fehér, L. M.; Rimányi, R.

Ann. Math. (2) 176, No. 3, 1381-1426 (2012).

In this well-written paper the authors develop a general method to compute Thom polynomials. A Thom polynomial \(\text{Tp}(\eta)\) of a set of points \(\eta(F)\) with a certain kind of singularity \(\eta\) in the source manifold of a holomorphic map \(F: N\to P\) is a multivariate polynomial depending only on \(\eta\) such that the cohomology class of the closure of \(\eta(F)\) is the specialisation of \(\text{Tp}(\eta)\) at the characteristic classes \(c_i(N)\), \(F^*(c_i(P))\). It can be considered as \(\mathrm{GL}(n)\times \mathrm{GL}(p)\)-equivariant cohomology class. A contact class of singularities is specified by an Artinian algebra \(\mathcal Q\) and two integers \(n\) and \(p\). For fixed \(\mathcal Q\) the Thom polynomials \(\text{Tp}_{\mathcal Q}(n,p)\) can be organised into a formal power series in infinitely many variables.
The method of computation is based on the ideas in the paper of G. Bérczi and A. Szenes [Ann. Math. (2) 175, No. 2, 567–629 (2012; Zbl 1247.58021)]. It uses a version of the Berline-Vergne-Atiyah-Bott equivariant localisation formula, for an equivariant cohomology class on the total space of a vector bundle over a compact singular base space. In this case the vector bundle is obtained by a partial resolution of contact invariant subvarieties of \(J^k(n,p)\), originally due to J. Damon. The form of the localisation formula implies different stabilisation properties of Thom polynomials, including the \(d\)-stability property of Thom series. The input of the formula for a fixed \(\mathcal Q\) is a finite set of Euler classes inside a Grassmannian. But the Euler classes in question are difficult to calculate directly. One of the strategies of this paper is to use the localisation formula backwards. In this way new Thom series are computed, but there are still many open problems, some of them listed at the end of the paper, like what is the Thom series of \(A_n\) for \(n>6\).

Reviewer: Jan Stevens (Göteborg)

MSC:

32S20	Global theory of complex singularities; cohomological properties
58K30	Global theory of singularities
58K65	Topological invariants on manifolds

Keywords:

contact singularities; equivariant localization; partial resolution; Thom polynomial

Citations:

Zbl 1247.58021

References:

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