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A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactifications

Gabriella Clemente

TL;DR

The paper addresses establishing a concrete K-stability criterion for Fano reductive group compactifications by computing the Donaldson–Futaki invariant for test-configurations via elementary, polyhedral methods. It translates degenerations into convex, $W$-invariant PL functions on the associated moment polytope and derives an explicit DF formula in terms of the polytope data and Duistermaat–Heckman measures. A central contribution is the reduction to a barycentric (Delcroix) criterion: KE existence is characterized by the polytope barycenter lying in the cone $2\rho+\Theta$, providing a practical, combinatorial test. The results unify reductive group compactifications with toric and spherical cases, offering a tractable route to verify KE metrics through polytope geometry.

Abstract

We present an elementary way of recovering a well-known criterion of K-stability for Fano reductive group compactifications.

A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactifications

TL;DR

The paper addresses establishing a concrete K-stability criterion for Fano reductive group compactifications by computing the Donaldson–Futaki invariant for test-configurations via elementary, polyhedral methods. It translates degenerations into convex, -invariant PL functions on the associated moment polytope and derives an explicit DF formula in terms of the polytope data and Duistermaat–Heckman measures. A central contribution is the reduction to a barycentric (Delcroix) criterion: KE existence is characterized by the polytope barycenter lying in the cone , providing a practical, combinatorial test. The results unify reductive group compactifications with toric and spherical cases, offering a tractable route to verify KE metrics through polytope geometry.

Abstract

We present an elementary way of recovering a well-known criterion of K-stability for Fano reductive group compactifications.

Paper Structure

This paper contains 8 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The Mabuchi functional
  3. K-stability
  4. The Donaldson-Futaki invariant and reductive group compactifications
  5. Reductive group compactifications
  6. The Fano condition
  7. Donaldson-Futaki invariant
  8. K-stability as a combinatorial criterion

Key Result

Theorem 1

(BMCDSTian) Let $X$ be a compact Kähler Fano manifold. Then, there exists a KE form $\omega \in 2\pi c_1(X)$ iff $(X, K^{-1}_X)$ is K-stable, where $K^{-1}_X$ is the anti-canonical line bundle of $X.$ Moreover, whenever such a form exists, it is unique up to the standard action of the identity compo

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Theorem 4