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

TL;DR

This paper proves that on excellent surfaces with rational singularities, the constant sheaf is a dualizing complex, and explores related obstructions and applications to perverse sheaves and cohomology weights.

Contribution

It establishes the dualizing property of the constant sheaf on such surfaces and analyzes obstructions in the integral coefficients, advancing understanding of $ ext{ell}$-adic sheaves on singular surfaces.

Findings

$ ext{Q}_ ext{ell}$ is a dualizing complex on these surfaces.

Obstructions in $ ext{Z}_ ext{ell}$ relate to divisor class groups at singularities.

Applications include studying perverse sheaves and cohomology weights.

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.