The Effect on Topology of the Action of a Unipotent Group

TL;DR

This paper proves that algebraic varieties related by a unipotent group action have identical topological Euler characteristics, and under smooth conditions, share the same cohomology groups, highlighting the topological invariance of such group actions.

Contribution

It establishes the invariance of topological Euler characteristic and cohomology groups for varieties connected via unipotent group actions, extending understanding of their topological properties.

Findings

Same Euler characteristic for varieties related by unipotent group action

Equal cohomology groups when varieties are smooth and the morphism is smooth

Topological invariance under unipotent group actions

Abstract

Assume that two algebraic varieties of finite type over the complex numbers are related by a morphism whose fibers are precisely the orbits for the action of a unipotent group. We show that the two varieties have the same topological Euler characteristic. If they are smooth and the morphism is smooth, we show that the two varieties have the same cohomology groups.