On the motivic measure on the space of functions (Q2471420)

The paper studies the relation between the measure on the space of function-germs and the motivic measure on the space of arcs. The author demonstrates the way to reduce the integration over the space of functions to the integration over multi-arcs. Let \(\mathbb{P}O_{(C^2,0)}\) be the projectivized space of function germs on the plane, let \(\mathcal{L}_{(C^2,0)}\) be the space of arcs and \(B\) be the space of branches at the origin (so that \(B=\mathcal{L}/\text{Aut}(C^1,0)\)). Each function-germ defines the germ of curve, thus defining the map: \(Z:\mathbb{P}O_{(C^2,0)}\to\coprod S^kB\). The measure on \(\mathbb{P}O_{(C^2,0)}\) was constructed by Campillo Delgado and Goussein-Zade (first defined for the cylindric sets then extended to the algebra of measurable sets). The motivic measure on \(\mathcal{L}\) was constructed by Kontsevich, Denef and Loeser. It descends to the measure on the space of branches \(B\). These measures extend naturally to the symmetric products: \(S^k\mathcal{L}\) and \(S^kB\). The relation between the corresponding integrals is given in Lemma 3.2. Finally, the measures on \(S^kB\) and \(\mathbb{P}O_{(C^2,0)}\) are related by the formula (Theorem 3.5): \[ \mu(N)=\int_M\mathbb{L}^{\delta(\gamma)-k-P(\gamma)}d\chi_g \] Here: \(M\subset S^kB\), \(N=Z^{-1}(M)\), \(\mathbb{L}\in K_0(\text{Var}_C)\) is the class of \(C^1\) in the Grothendieck ring of varieties, \(\delta(\gamma)\) is the delta invariant of the plane curve defined by \(\gamma\) (aka the genus discrepancy, aka the virtual number of nodes) and \(P(\gamma)\) is the kappa invariant of the plane curve (the local degree of intersection of the curve with its generic polar). Several examples are considered (the spaces of curves with ordinary multiple points, \(A_k\)'s and \(x^p+y^q\) for \(\text{gcd}(p,q)=1\) and some other integrals). Finally, the correspondence for Euler characteristic is discussed.