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Algebro-geometric semistability of polarized toric manifolds. (English) Zbl 1297.14056
Let \(X\subset\mathbb P^N\) be a smooth projective toric variety of dimension \(n\). The author compares different notions of stability for \(X\), and gives a criterion for Chow semistability of \(X\) in terms of the moment (or Newton) polytope \(\Delta_X\subset \mathbb R^n\) of \(X\).
The main ingredient is a description of the secondary polytope of \(X\) in terms of certain piecewise-linear concave functions \(g\) on \(\Delta_X\) given in [I. M. Gelfand et al., Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser. (1994; Zbl 0827.14036)]. These functions are used as test functions by the author to prove that \(X\) is Chow-semistable with respect to the standard action of a complex torus \((\mathbb C^*)^N\) on \(\mathbb P^N\) if and only if the average value of every test function \(g\) over the whole \(\Delta_X\) is greater than or equal to the average value of \(g\) over the set \(\Delta_X\cap\mathbb Z^n\). In particular, restricting this criterion to linear test functions \(g\), the author gets a necessary condition previously obtained in [H. Ono, J. Math. Soc. Japan 63, No. 4, 1377–1389 (2011; Zbl 1230.14069)].
As an application of the criterion the author shows that asymptotical semistability of \(X\) implies its \(K\)-semistability, which is a partial case of a result of J. Ross and R. Thomas [J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)].

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14L24 Geometric invariant theory
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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