Osaka Journal of Mathematics

Facets of secondary polytopes and chow stability of toric varieties

Naoto Yotsutani

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Chow stability is one notion of Mumford's geometric invariant theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope [7]. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This essentially means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.

Article information

Osaka J. Math., Volume 53, Number 3 (2016), 751-765.

First available in Project Euclid: 5 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]


Yotsutani, Naoto. Facets of secondary polytopes and chow stability of toric varieties. Osaka J. Math. 53 (2016), no. 3, 751--765.

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