# zbMATH — the first resource for mathematics

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)
Full Text: