Characteristic cycle and the Euler number of a constructible sheaf on a surface

TL;DR

This paper introduces a new way to understand constructible sheaves on surfaces by defining their characteristic cycle and relating it to the Euler number and vanishing cycles through intersection theory.

Contribution

It defines the characteristic cycle for constructible sheaves on surfaces and establishes its relation to the Euler number and vanishing cycles via intersection numbers.

Findings

Characteristic cycle defined for constructible sheaves on surfaces.

Intersection number with the zero section equals the Euler number.

Total dimension of vanishing cycles computed as an intersection number.

Abstract

We define the characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. We prove that the intersection number with the 0-section equals the Euler number and that the total dimension of vanishing cycles at an isolated characteristic point is also computed as an intersection number.