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A structure theorem along fibers of extreme points of the momentum polytope

Peter Heinzner, Christian Zöller

TL;DR

The paper develops a structure theorem for invariant compact submanifolds of Kähler manifolds under the action of a complex reductive group, by analyzing the fibers over extreme points of the momentum polytope. Central ingredients include restricted momentum maps, a Luna-type Slice Theorem, and a parabolic-subgroup framework which yields a local product decomposition along strata indexed by extreme points. The main result provides a fiberwise decomposition near points mapping to an extreme momentum value, with an ineffectivity group controlling the fibers and a negatively weighted representation specifying the transverse directions; in the complex case, this specializes to a global, equivariant local normal form, recovering and extending the Brion–Luna–Vust local structure in the projective setting. The framework has concrete consequences for quotients, properness of actions, and the structure of compact orbits, and it recovers BLV in projective space as a special case with explicit holomorphic bundle models.

Abstract

Let G be a complex reductive Lie group acting on a compact Kähler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an associated open dense subset of X, which is invariant under a parabolic subgroup Q of G. We prove a Q-equivariant product decomposition for the Q-action on this subset and discuss some applications of the result. We show a similar statement for real reductive subgroups of G for the restricted momentum map.

A structure theorem along fibers of extreme points of the momentum polytope

TL;DR

The paper develops a structure theorem for invariant compact submanifolds of Kähler manifolds under the action of a complex reductive group, by analyzing the fibers over extreme points of the momentum polytope. Central ingredients include restricted momentum maps, a Luna-type Slice Theorem, and a parabolic-subgroup framework which yields a local product decomposition along strata indexed by extreme points. The main result provides a fiberwise decomposition near points mapping to an extreme momentum value, with an ineffectivity group controlling the fibers and a negatively weighted representation specifying the transverse directions; in the complex case, this specializes to a global, equivariant local normal form, recovering and extending the Brion–Luna–Vust local structure in the projective setting. The framework has concrete consequences for quotients, properness of actions, and the structure of compact orbits, and it recovers BLV in projective space as a special case with explicit holomorphic bundle models.

Abstract

Let G be a complex reductive Lie group acting on a compact Kähler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an associated open dense subset of X, which is invariant under a parabolic subgroup Q of G. We prove a Q-equivariant product decomposition for the Q-action on this subset and discuss some applications of the result. We show a similar statement for real reductive subgroups of G for the restricted momentum map.
Paper Structure (8 sections, 24 theorems, 24 equations)

This paper contains 8 sections, 24 theorems, 24 equations.

Table of Contents

  1. Basic properties of the momentum map
  2. The Slice Theorem
  3. Parabolic subgroups
  4. Extreme points
  5. Proof of the structure theorem
  6. Quotients
  7. Compact orbits
  8. The projective case

Key Result

Theorem 1

Let $X$ be a $G$-invariant compact submanifold of $Z$ and let $\sigma$ be an extreme point of the convex envelope $E$ of $\mu_{\mathfrak{ p}}(X)$. Then for any $\beta\in C_{\sigma}^{K^\sigma}$ there is a normal compatible subgroup $I^\beta$ of $G^\beta$ containing $\exp{\mathbb R}\beta$ such that $G such that $p^{\beta_-}\circ \Phi=p_U$, where $p_U$ is the projection onto $U$.

Theorems & Definitions (42)

  • Theorem
  • Theorem
  • Theorem
  • Remark 1.1
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Corollary 1.4
  • Remark 1.5
  • Proposition 1.6
  • ...and 32 more