[Paper Review] Thom polynomials for maps of curves with isolated singularities

Maxim Kazarian, Сергей Константинович Ландо|ArXiv.org|Jun 11, 2007

Advanced Differential Equations and Dynamical Systems7 references3 citations

TL;DR

This paper derives explicit universal formulas for Thom polynomials of maps from complex curves with isolated singularities, focusing on $A_n$ and $I_{k,l}$ types. Using equivariant cohomology and characteristic classes, it computes the cohomology classes Poincaré dual to strata of multisingularities in Hurwitz spaces, yielding closed-form expressions for double Hurwitz numbers via generating functions in Chern classes and parameters $\psi$, $\nu$, $z$, and $t_m$. The key contribution is a complete description of primitive strata in terms of finitely many basic classes, enabling explicit computation of Lyashko–Looijenga map degrees on these strata.

ABSTRACT

Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincaré dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces.

Motivation & Objective

• To compute the cohomology classes Poincaré dual to strata of functions on complex curves with isolated singularities of type $A_n$ and $I_{k,l}$.
• To derive explicit universal expressions for Thom polynomials in terms of characteristic classes, enabling computation of degrees of the Lyashko–Looijenga map on primitive strata.
• To establish a generating function framework for multisingularities in versal unfoldings of singularities, particularly for double Hurwitz numbers.
• To show that only four basic characteristic classes are sufficient to describe strata in families of meromorphic functions on curves, regardless of codimension.

Proposed method

• Uses universal residual polynomials in characteristic classes, based on Thom and Kazarian’s theory, to describe cohomology classes of singular strata in functional spaces.
• Applies the indefinite coefficients method and equivariant cohomology techniques to compute generating functions $\mathcal{N}_A$ and $\mathcal{M}_A$ for $A_n$-type and multisingularities with distinguished points.
• Computes the intersection of stratum classes with the discriminant divisor $\varepsilon = 0$ in the $I_{k,l}$-deformation space, using two different approaches to derive identities.
• Splits the singular fiber into irreducible components based on distribution of singularities across the two branches of the nodal curve, assigning multiplicities via local algebra dimensions.
• Replaces monomials in generating functions with coefficients $m+1$ to account for multiplicities of components with singularities at the node, using the dimension of $\mathbb{C}[x,y]/\langle x^a + y^b, xy \rangle$.
• Derives the generating function $\mathcal{M}_A$ for $I_{k,l}$-strata by combining contributions from splittings without and with a singularity at the node, using $\mathcal{N}_A$ and $\mathcal{M}_A$ with distinct $\nu_1, \nu_2$ parameters.

Experimental results

Research questions

• RQ1 What is the universal expression for the Thom polynomial of a map from a complex curve with a single $A_n$-type singularity and a distinguished critical value?
• RQ2 How can the cohomology class of the stratum of functions with multisingularities $A_{m_1,\dots,m_r}$ be expressed in terms of characteristic classes in the $I_{k,l}$-deformation space?
• RQ3 What is the contribution of components with a singularity at the node to the intersection of the stratum with the discriminant $\varepsilon = 0$?
• RQ4 How do the multiplicities of irreducible components in the singular fiber depend on the local singularity types at the node?
• RQ5 Can the degree of the Lyashko–Looijenga map on primitive strata be computed explicitly using generating functions in $\psi$, $\nu$, $z$, and $t_m$?

Key findings

• The generating function $\mathcal{N}_A(\psi, \nu; z; t)$ for $A_n$-type strata is derived as a product of two $\mathcal{N}_A$ functions with $\nu_1$ and $\nu_2$, corresponding to the two branches of the nodal curve.
• For strata with a singularity at the node, the contribution is given by $\mathcal{M}_A(\psi, \nu_1; z; t) \cdot \mathcal{M}_A(\psi, \nu_2; z; t)$, where $z$ marks the distinguished critical point at the node.
• The multiplicity of each component with a singularity at the node is $m+1$, equal to the dimension of the local algebra $\mathbb{C}[x,y]/\langle x^a + y^b, xy \rangle$ for $a+b = m+1$.
• The total contribution from such components is obtained by multiplying the product of $\mathcal{M}_A$ functions by $\psi z$ and replacing each $z^{m+1}$ with $(m+1)t_{m+1}$.
• The final generating function $\mathcal{M}_A$ for the $I_{k,l}$-stratum is shown to coincide with the function in Theorem 3.5, proving the formula for the cohomology class $[A_{m_1,\dots,m_r}(Y)]$.
• The paper establishes that only four basic characteristic classes are sufficient to describe all strata in the space of meromorphic functions on complex curves, regardless of codimension.