Motivic vanishing cycles as a motivic measure (Q525690)

Let \(k\) be an algebraically closed field of characteristic \(0\) and let Var\(_k\) be the category of \(k\)-varieties, i.e. separated reduced schemes of finite type over \(k\). Fix a \(k\) variety \(S\). An \(S\)-variety is a morphism \(X\to S\) of \(k\)-varieties. The Grothendieck group \(K_0(\text{Var}_S)\) is the quotient of the free abelian group on isomorphisms classes \([X\to S]\) of \(0\) of \(S\)-varieties by the subgroup generated by \(\{[X\to S]- [(X- Y)\to S]- [Y\to S]\}\), where \(Y\subset X\) is a closed reduced subscheme. For a positive number \(n\) let \(\mu_n= \text{Spec\,}k[x]/(x^n- 1)\) the group \(k\)-variety of \(n\)th roots of unity. A good \(\mu_n\)-action on a \(k\)-variety \(X\) is such that each \(\mu_n(k)\)-orbit is contained in an affine open subset of \(X\). An \(S\) variety with a good \(\mu_n\)-action is an \(S\)-variety \(p: X\to S\) together with a good \(\mu_n\) action on \(X\). By taking the trivial action of \(\mu_n\) on \(S\) one obtains the category \(\text{Var}^{\mu_n}_S\) of \(S\)-varieties with good \(\mu_n\)-action and the Grothendieck group \(K_0(\text{Var}^{\mu_n}_S)\). Let \[ {\mathcal M}^{\widehat\mu_n}_S= K_0(\text{Var}^{\mu_n}_S)[{\mathbb{L}}^{-1}_S], \] where \({\mathbb{L}}_S= {\mathbb{L}}_{S,\mu_n}= [{\mathbb{A}}^1_S\to S]\), with respect to the trivial action of \(\mu_n\) on \({\mathbb{A}}^1_S\), and \[ K_0(\text{Var}^{\widehat\mu}_S)= \text{colim}_nK_0(\text{Var}^{\mu_n}_S);\;{\mathcal M}^{\widehat\mu}_S= \text{colim}_n\,{\mathcal M}^{\mu_n}_S. \] Let \(S={\mathbb{A}}^1_k\) and let \(V: X\to{\mathbb{A}}^1_k\in \text{Var}_{{\mathbb{A}}^1_k}\). The motivic nearby fiber \(\psi_{V,a}\) and the motivic vanishing cycle \(\phi_{V,a}\) at a point \(a\in k={\mathbb{A}}^1_k(k)\) are elements of a localization M'X, over the reduced fiber jXaj of \(V\) over \(a\). The motivic nearby fiber is additive in the Grothendieck group \(K_0(\text{Var}_{{\mathbb{A}}^1_k})\). In this paper the authors show that the motivic vanishing cycles are both additive and multiplicative. Here is their main result Theorem 1. There is a morphism \[ M: (K_0(\text{Var}_{{\mathbb{A}}^1_k}),*)\to({\mathcal M}^{\widehat\mu}_k, *) \] of \(K_0(\text{Var}_k)\)-algebras which is uniquely determined by the following property: it maps the class of each proper morphism \(V: X\to{\mathbb{A}}_k^1\) from a smooth and connected \(k\)-variety \(X\) to the sum \(\sum_{a\in k}\phi_{V,a}\) of its motivic vanishing cycles. The map \(M\) is called a motivic vanishing cycle measure in the sense that is a ring morphism from some Grothendieck ring of varieties to another ring. The multiplication \(*\) on the target is a convolution product as defined by Looijenga, while the multiplication on the source is given by \[ [X@> V>>{\mathbb{A}}^1_k] * [Y@> W>>{\mathbb{A}}^1_k]= [X\times Y@> V\otimes W>>{\mathbb{A}}^1_k] \] .