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Relative algebro-geometric stabilities of toric manifolds. (English) Zbl 07199976
Summary: 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 H. Ono [Asian J. Math. 17, No. 4, 609–616 (2013; Zbl 1297.14056)], 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 S. K. Donaldson [J. Differ. Geom. 62, No. 2, 289–349 (2002; Zbl 1074.53059)] and J. Ross and R. Thomas [J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)]. 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.
MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14L24 Geometric invariant theory
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Software:
Normaliz; polymake
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References:
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