Tohoku Mathematical Journal

Relative algebro-geometric stabilities of toric manifolds

Naoto Yotsutani and Bin Zhou

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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.

Article information

Tohoku Math. J. (2), Volume 71, Number 4 (2019), 495-524.

First available in Project Euclid: 19 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 14L24: Geometric invariant theory [See also 13A50] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

extremal metrics $K$-stability Chow stability toric manifold


Yotsutani, Naoto; Zhou, Bin. Relative algebro-geometric stabilities of toric manifolds. Tohoku Math. J. (2) 71 (2019), no. 4, 495--524. doi:10.2748/tmj/1576724790.

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