Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems

TL;DR

The paper develops a general, constructive framework for building collective completely integrable systems on the duals of Lie algebras of arbitrary compact Lie groups by combining stratified gradient Hamiltonian flows with toric degenerations arising from valuations. Central to the approach is the notion of a stratified gradient flow on decomposed (possibly singular) affine varieties, whose time-1 limit lands in toric fibers, enabling the pullback of toric integrable structures to the original space. The authors introduce good valuations and a Rees-algebra construction to produce toric degenerations compatible with a given decomposition, culminating in a main existence theorem for a completely integrable system on each stratum and, when compact, on the whole space via convex polytopes and moment maps. Applications include concrete descriptions of Gromov width for coadjoint orbits and independence-of-polarization results for geometric quantization, as well as the construction of integrable systems on T^*K and the base affine space $G\!\sslash\!N$, connecting to Gelfand–Zeitlin data and crystal bases. Overall, the work extends Gelfand–Zeitlin-type phenomena to arbitrary compact groups and provides a robust, algebraic-to-symplectic pipeline for generating integrable torus actions with favorable convexity and fiber-connectedness properties.

Abstract

We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian $K$-manifold defines a Hamiltonian torus action on an open dense subset, B) if the $K$-manifold is multiplicity-free, then the resulting torus action is \textit{completely} integrable, and C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand-Zeitlin systems and Gelfand-Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map.

Figures (2)

• Figure 1: In Harada-Kaveh the time-1 limit of the gradient Hamiltonian flow defines a continuous map from the 1-fiber (left), a smooth projective variety, to the possibly singular toric fiber of the degeneration (right).
• Figure 2: In our work, the 1-fiber (left) is not necessarily smooth or projective. Instead we suppose it has a decomposition into smooth pieces (illustrated in blue and orange). We define a piecewise gradient Hamiltonian vector field which we call the stratified gradient Hamiltonian vector field. Under some assumptions the time-1 limit of its flow exists and defines a continuous map to the toric fiber (right).

Theorems & Definitions (89)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• Definition 3.4