Juan Viu Sos (ICMC)
Wednesday, May 23rd, 16:20, Room 3-010


Title: Motivic zeta functions, orbifold motivic measures and Q-resolutions of singularities

Abstract: The \emph{motivic zeta function} $Z_{mot}(f;s)$ is a geometrical invariant associated to a complex polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$, introduced by Dener and Loeser in 1998 as a generalization of the \emph{topological zeta function} $Z_{top}(f;s)$ and the \emph{Igusa's $p$-adic zeta function} of $f$ by using Kontsevich's motivic integration theory.
The previous functions are classically computed in terms of an embedded resolution of singularities of $f^{-1}(0)\subset\mathbb{A}^n_{\mathbb{C}}$, where every exceptional divisor gives a ``pole candidate'' $s_0$ for $Z_{mot}(f;s)$ (or $Z_{top}(f;s)$), which could be not a real pole when one gets the final expression.
The \emph{Monodromy Conjecture} affirms that any pole $s_0$ gives an eigenvalue $\exp(2\pi s_0)$ of the monodromy on the cohomology of the Milnor fiber of $f^{-1}(0)$.
The latter is proved in some particular cases, but one of the main difficulties to approach this conjecture is the fact that minimal resolutions of singularities does not exist for $n>2$, the resolution models are complicated to compute and could become very complexes in terms of number of exceptional divisors and relations between them, providing a lot of ``bad pole candidates''.
In this work, we study the motivic zeta function $Z_{mot}(f;s)$ (and its specialization in $Z_{top}(f;s)$) in terms of the so-called \emph{embedded $\mathbf{Q}$-resolutions of singularities} of $f^{-1}(0)$, which are roughly embedded resolutions $\pi:X\to\mathbb{C}^n$ where the ambient space $X$ is allowed to contain abelian quotient singularities, providing a ``simpler'' model with less exceptional divisors and thus less ``bad pole candidates'' for $Z_{mot}(f;s)$.