Compactification of Reductive Group Schemes

1. Problem Statement and Context

The central problem addressed in this paper is the existence of equivariant compactifications for isotrivial reductive group schemes over an arbitrary base scheme $S$.

• Definitions:
  • A reductive group scheme $G$ over $S$ is isotrivial if it becomes split (isomorphic to a Chevalley group) after a finite étale cover of $S$.
  • An equivariant compactification $\bar{G}$ is a projective $S$-scheme containing $G$ as a dense open subscheme, equipped with left and right $G$-actions extending the natural translation actions on $G$.
• The Conjecture: Česnavičius [Čes22] conjectured that every isotrivial reductive group scheme over a Noetherian base admits such a compactification.
• Previous Work:
  • The case for isotrivial tori was established by Česnavičius.
  • For adjoint groups, Li [Li25] confirmed the conjecture using a variant of the Artin–Weil method, relying on the existence of a "big cell" (an open dense subscheme isomorphic to affine space).
• The Gap: Li's method relies heavily on the "big cell" structure, which is specific to adjoint groups and does not generalize easily to non-adjoint cases. Furthermore, no general construction existed for non-adjoint isotrivial groups over arbitrary bases.
• Counter-example Context: It was unknown whether the isotriviality assumption was necessary. The paper aims to verify the conjecture for all isotrivial groups and demonstrate that the assumption is essential by constructing a non-isotrivial torus with no such compactification.

2. Methodology

The author employs a combination of Vinberg monoids, Geometric Invariant Theory (GIT), and descent theory.

A. Vinberg Monoids (The Core Tool)

The construction is inspired by Martens–Thaddeus [MT16], who used Vinberg monoids over algebraically closed fields of characteristic 0. Nath generalizes this to arbitrary bases and non-split groups.

• Construction: For a split reductive group $G$, the Vinberg monoid $V_G$ is constructed as the spectrum of the Rees algebra associated with a canonical multi-filtration on the coordinate ring $\mathcal{O}(G)$.
• Properties: $V_G$ is an affine monoid scheme containing $G$ as its unit group. It admits an "abelianization" map $a: V_G \to T^+_{ad}$ (where $T^+_{ad}$ is a toric embedding of the adjoint torus).
• Non-degenerate Locus: A dense open subscheme $V^\circ_G \subset V_G$ is identified, which is smooth and stable under the $G \times G$ action.
• Descent: For non-split (isotrivial) groups, the construction is performed on the split form and then descended to the base $S$ using étale descent, leveraging the fact that the group is isotrivial.

B. Cox-Vinberg Hybrid Construction

To handle the compactification, the author combines the Vinberg monoid with a Cox construction (toric geometry):

1. Fan Selection: A $\Gamma$-stable fan $\Sigma$ is chosen in the cocharacter lattice (where $\Gamma$ is the Galois group of the splitting cover). This fan is a subdivision of the dominant chamber ensuring smoothness and projectivity.
2. Hybrid Scheme: A new scheme $V_{G, \beta}$ is defined as a fiber product involving the Vinberg monoid and an affine space $A^I_S$ determined by the fan.
3. GIT Quotient: The compactification $\bar{G}$ is obtained as a Geometric Invariant Theory quotient:
   $$\bar{G} = V^\circ_{G, \beta} //_{\rho} \mathbb{G}_m^I$$
   where $\rho$ is a suitable linearization. The quotient is taken with respect to a torus action, ensuring the result is a projective scheme.

C. Verification of Properties

• Smoothness: Proven by showing the existence of an open cell (isomorphic to $U^- \times T \times U^+$) whose $G \times G$-saturation covers the scheme, combined with the smoothness of the underlying toric scheme.
• Projectivity: Established via the valuative criterion for properness, utilizing the Cartan decomposition of points over discrete valuation rings (DVRs).

3. Key Contributions and Results

Theorem 1.2: General Isotrivial Case

Statement: For any isotrivial reductive group scheme $G$ over a scheme $S$, there exists a smooth, projective, finitely presented $S$-scheme $\bar{G}$ containing $G$ as a fiberwise-dense open subscheme, equipped with compatible left and right $G$-actions.

• Significance: This fully verifies Česnavičius's conjecture for all isotrivial reductive groups, extending previous results limited to tori or adjoint groups. The construction works over arbitrary bases and commutes with base change.

Theorem 1.3: The Semisimple Case

Statement: If $G$ is semisimple, the author constructs two compactifications: $\bar{G}$ (for $G$) and $\bar{G}_{ad}$ (for the adjoint quotient $G_{ad}$).

• Relationship: There is an equivariant morphism $\bar{G} \to \bar{G}_{ad}$ extending the central isogeny.
• Normalization: For every geometric point $s$, the fiber $(\bar{G})_s$ is the normalization of $(\bar{G}_{ad})_s$ in $G_s$.
• Boundary Structure: The boundary $\bar{G}_{ad} \setminus G_{ad}$ is a union of smooth relative effective Cartier divisors with relative normal crossings.
• Recovery: When $S$ is a field and $G$ is adjoint, $\bar{G}_{ad}$ recovers the classical wonderful compactification of de Concini–Procesi.

Theorem 1.4: Necessity of Isotriviality

Statement: There exists a non-isotrivial torus $T$ over a nodal cubic curve $S$ (constructed via Galois descent with infinite monodromy) that admits no projective equivariant compactification.

• Methodology: The proof uses Brion's results on the linearization of line bundles to reduce the problem to toric schemes over discrete valuation rings. It shows that the monodromy prevents the existence of a compatible compactification with normal fibers.
• Significance: This proves that the isotriviality assumption in the conjecture is not merely a technical convenience but a necessary condition.

4. Technical Significance

1. Generalization of Wonderful Compactification: The paper successfully extends the theory of wonderful compactifications (previously known for adjoint groups over fields) to non-adjoint groups and arbitrary base schemes.
2. New Construction Technique: By moving away from the "big cell" approach (which fails for non-adjoint groups) to the Vinberg monoid + GIT approach, the author provides a robust framework for compactifying reductive groups that are not necessarily split or adjoint.
3. Base Change Compatibility: The constructions are shown to commute with arbitrary base change, making them applicable in families and arithmetic geometry contexts.
4. Boundary Geometry: The paper provides a precise description of the boundary divisors, showing they form a normal crossing divisor, which is crucial for applications in intersection theory and moduli problems.
5. Counter-Example: The construction of the non-isotrivial torus without a compactification clarifies the limits of the theory, highlighting the role of monodromy in the geometry of group schemes.

Conclusion

Ayan Nath's paper resolves a significant open problem in the theory of algebraic groups by constructing smooth projective equivariant compactifications for all isotrivial reductive group schemes. The work unifies the theory of Vinberg monoids with GIT over general bases and establishes the sharpness of the isotriviality condition through a rigorous counter-example. This provides a foundational tool for further research in arithmetic geometry, moduli of bundles, and the study of group schemes over non-trivial bases.