Taking quotient by a unipotent group induces a homotopy equivalence

TL;DR

This paper proves that under certain conditions, taking the quotient of a complex algebraic variety by a unipotent group results in a homotopy equivalence between the original and quotient spaces, with real case extensions.

Contribution

It establishes conditions under which the quotient map by a unipotent group induces a homotopy equivalence on complex points and surjective homotopy equivalences on real points.

Findings

Quotient by a unipotent group induces a homotopy equivalence on complex points.

The induced map on real points is surjective and a homotopy equivalence on connected components.

Results apply when the variety and group actions satisfy smoothness and orbit dimension conditions.

Abstract

Let U be a unipotent group over the field of complex numbers C, acting on a complex algebraic variety X. Assume that there exists a surjective morphism of complex algebraic varieties f: X --> Y whose fibres are orbits of U. We show that if X and Y are smooth and all orbits of U in X have the same dimension, then the induced map on C-points X(C) --> Y(C) is a homotopy equivalence. Moreover, if U, X, Y, and f are defined over the field of real numbers R, then the induced map on R-points X(R) --> Y(R) is surjective and induces homotopy equivalences on connected components.