Relative Algebro-Geometric stabilities of Toric Manifolds

TL;DR

This paper investigates the relative Chow and K-stability of toric manifolds, providing criteria, reduction methods, and applications to Fano threefolds, revealing instances of stability and instability in the toric setting.

Contribution

It introduces new criteria for relative K-stability and Chow stability of toric Fano manifolds, and explores their relationships through reduction techniques and applications.

Findings

Partial classification of relative K-stability for toric Fano threefolds

Identification of counter-examples with relative K-stability but Chow instability

Development of reduction methods using torus actions and degenerations

Abstract

In this paper we study the relative Chow and $K$-stability of toric manifolds in the toric sense. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [Ono13], which fits into the relative GIT stability detected by Sz\'ekelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [D02, RT07]. As applications of our main theorem, we partially determine the relative $K$-stability of toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable. In the end, we explain the erroneous parts of the published version of this article (corresponding to Sections 1-5), which provides some inconclusive results for relative $K$-stability in Table 6.