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HKKN-stratifications in a non-compact framework

Paul-Emile Paradan, Nicolas Ressayre

TL;DR

The paper develops HKKN stratifications for a relative algebraic setting and a non-compact Kähler framework, focusing on $X\subset V\times\mathbb{P}E$ and $V\times M$ where $V$ is affine and $M$ is compact. It introduces relative semistability via the Mumford invariant $\mathbf{M}_{rel}$ and the Kempf–Ness theory adapted to non-compact settings, proving existence and uniqueness of optimal destabilizing vectors and establishing flows toward zero-moment orbits. It then constructs HKKN stratifications, analyzes strata via moment-polyhedra $\mathcal{C}(X)$ and $\Delta(X)$, and extends convexity and connectedness results for the moment map to non-compact and $B$-stable situations. The key contributions include a full relative algebraic framework, a non-compact Kähler analog of the Kirwan convexity results, and a unified description of strata, moment polytopes, and retractions in projective-over-affine contexts. These results generalize classical Brion–Guillemin–Sjamaar convexity and Kirwan-type theorems to broader non-compact and analytic settings with potential applications to quotients and stability in complex geometry.

Abstract

The aim of this paper is twofold. First, we study HKKN stratifications, both algebraically and analytically, for a Cartesian product between a vector space and a compact K{ä}hler manifold. We then use these stratifications to prove convexity properties of the moment map for non-compact analytic subsets invariant under a Borel subgroup.

HKKN-stratifications in a non-compact framework

TL;DR

The paper develops HKKN stratifications for a relative algebraic setting and a non-compact Kähler framework, focusing on and where is affine and is compact. It introduces relative semistability via the Mumford invariant and the Kempf–Ness theory adapted to non-compact settings, proving existence and uniqueness of optimal destabilizing vectors and establishing flows toward zero-moment orbits. It then constructs HKKN stratifications, analyzes strata via moment-polyhedra and , and extends convexity and connectedness results for the moment map to non-compact and -stable situations. The key contributions include a full relative algebraic framework, a non-compact Kähler analog of the Kirwan convexity results, and a unified description of strata, moment polytopes, and retractions in projective-over-affine contexts. These results generalize classical Brion–Guillemin–Sjamaar convexity and Kirwan-type theorems to broader non-compact and analytic settings with potential applications to quotients and stability in complex geometry.

Abstract

The aim of this paper is twofold. First, we study HKKN stratifications, both algebraically and analytically, for a Cartesian product between a vector space and a compact K{ä}hler manifold. We then use these stratifications to prove convexity properties of the moment map for non-compact analytic subsets invariant under a Borel subgroup.
Paper Structure (51 sections, 81 theorems, 166 equations)

This paper contains 51 sections, 81 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. HKKN-stratifications in the relative algebraic framework
  3. GIT-quotient
  4. Instability degree
  5. Optimal destabilizing $1$-parameter subgroups
  6. The case of a torus action
  7. The general case
  8. Optimal $1$-parameter subgroup and flow
  9. Algebraic Shifting Trick
  10. À la Ness
  11. With a tensor product
  12. A stratification of the nullcone
  13. A partition of the unstable locus
  14. Moment polyhedron
  15. Definition
  16. ...and 36 more sections

Key Result

Lemma 2.1

The quotient map $\pi$ is affine and for any affine open subset $U\subset X^{\rm ss}{/\space/} G$, the restriction of $\pi$ on $\pi{^{-1}}(U)$ is the affine quotient $\pi{^{-1}}(U)\longrightarrow \pi{^{-1}}(U){/\space/} G$. In particular, each fiber of $\pi$ contains a unique closed $G$-orbit in $X

Theorems & Definitions (167)

  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 157 more