K\"{a}hler Soliton Surfaces Are Generically Toric

TL;DR

This paper proves that generic four-dimensional Kähler gradient Ricci solitons admit a toric symmetry, showing they are effectively toric under broad conditions, which links geometric properties to integrable Hamiltonian systems.

Contribution

It demonstrates that, under generic assumptions, Kähler gradient Ricci solitons in four dimensions possess a toric symmetry, extending the understanding of their geometric structure.

Findings

All known complete examples are toric.

The soliton admits a toric action under generic non-degeneracy and properness conditions.

The system is an integrable Hamiltonian system with a toric symmetry.

Abstract

Let $(M, g, \omega, f, \lambda)$ be a K\"{a}hler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\SS$. While another article addresses the case that these functions are functionally dependent, this one considers the independent case. The main result states that the soliton admits a toric action under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function $f$ is proper. Then there is an effective, completely integrable Hamiltonian toric $\mathbb{T}^2$- action on $(M, \omega)$.