Taking quotient by a unipotent group induces a homotopy equivalence

Mikhail Borovoi; Andrei Gornitskii

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arXiv · 2105.04441

Taking quotient by a unipotent group induces a homotopy equivalence

Mikhail Borovoi·Andrei Gornitskii

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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.

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BibTeX RIS

Mikhail Borovoi, Andrei Gornitskii. 2021-05-10. Taking quotient by a unipotent group induces a homotopy equivalence. https://arxiv.org/abs/2105.04441

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