Compactification of Reductive Group Schemes

TL;DR

The paper proves Česnavičius' conjecture by constructing a smooth projective $S$-scheme $\overline{G}$ with a left and right $G$-action extending translations on $G$, for any isotrivial reductive group scheme $G$ over $S$. The construction uses a Gamma-equivariant Cox–Vinberg hybrid monoid, together with a base-change compatible GIT quotient to yield $\overline{G}$, and it recovers the wonderful compactification on adjoint fibers while ensuring flatness and normal-crossing boundary behavior in the semisimple/adjoint case. Base-change compatibility and modular agreement with Shangli are established, and the approach also shows that the isotriviality hypothesis is essential by providing a non-isotrivial torus counterexample. A concrete framework for the geometry of compactifications is developed via Vinberg monoids, nondegenerate loci, and equivariant GIT, yielding a broad, uniform construction applicable to all isotrivial reductive groups.

Abstract

Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the translation actions of $\mathrm G$ on itself. This verifies a conjecture of Česnavičius (arXiv:2201.06424). When $\mathrm G$ is adjoint, we recover fiberwise the wonderful compactification. Finally, we give an example of a non-isotrivial torus admitting no equivariant compactification.

Theorems & Definitions (39)

• Conjecture 1.0.1: kestutis
• Theorem 1.0.2
• Theorem 1.0.3
• Theorem 2.1.1
• proof
• Proposition 2.2.1
• proof
• Proposition 2.2.2
• proof
• Lemma 2.2.3