Palestras e Seminários

23/05/2018

16:00

Sala 3-010

Palestrante: Juan Viu Sos

Responsável: Pedro Benedini Riul (Este endereço de email está sendo protegido de spambots. Você precisa do JavaScript ativado para vê-lo.)

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Resumo: 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 $Z_{mot}(f;s)$.

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