Journal of the Mathematical Society of Japan

A necessary condition for Chow semistability of polarized toric manifolds

Hajime ONO

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Let $\Delta \subset R^n$ be an $n$-dimensional Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(C^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we show that if $(X_\Delta,L_\Delta^i)$ is Chow semistable then the sum of integer points in $i\Delta$ is the constant multiple of the barycenter of $\Delta$. Using this result we get a necessary condition for the polarized toric manifold $(X_\Delta,L_\Delta)$ being asymptotically Chow semistable. Moreover we can generalize the result in [4] to the case when $X_\Delta$ is not necessarily Fano.

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J. Math. Soc. Japan Volume 63, Number 4 (2011), 1377-1389.

First available in Project Euclid: 27 October 2011

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Mathematical Reviews number (MathSciNet)

Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Chow semistability polarized toric manifold


ONO, Hajime. A necessary condition for Chow semistability of polarized toric manifolds. Journal of the Mathematical Society of Japan 63 (2011), no. 4, 1377--1389. doi:10.2969/jmsj/06341377.

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