title

Relative algebro-geometric stabilities of toric manifolds

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UTF-8

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Tohoku Mathematical Journal

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citation_journal_title

Tohoku Mathematical Journal

og:type

Paper

og:url

https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-71/issue-4/Relative-algebro-geometric-stabilities-of-toric-manifolds/10.2748/tmj/1576724790.full

og:title

Relative algebro-geometric stabilities of toric manifolds

og:description

In this paper we study the relative Chow and $K$-stability of toric manifolds. 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 [34], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [13, 36]. In the end, we determine the relative $K$-stability of all toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable.

publish_date

2019-12-01T00:00:00-08:00

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