The $\ell$-adic Dualizing Complex on an Excellent Surface with Rational Singularities

Abstract

In this article, we show that if $X$ is an excellent surface with rational singularities, the constant sheaf ${\mathbb{Q}}_{\ell }$ is a dualizing complex. In coefficient ${\mathbb{Z}}_{\ell }$, we also prove that the obstruction for ${\mathbb{Z}}_{\ell }$ to become a dualizing complex lying on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of $\ell$-adic cohomology groups on such surfaces.

arXiv commentTotally rewritten. The title is changed.