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Complete toric varieties with reductive automorphism group. (English) Zbl 1091.14011
The author studies the automorphism group of a complex complete toric variety, in particular, of a Gorenstein toric Fano variety. He proves the following sufficient conditions for the automorphism group $$\text{ Aut}(X)$$ to be reductive:
Let $$X$$ be a complete toric variety. If the group of automorphisms of the associated fan has only the origin as a fixed point, then $$\text{ Aut}(X)$$ is reductive.
Let $$X$$ be a Gorenstein toric Fano variety. If the barycentre of the associated reflexive polytope is the origin, then $$\text{ Aut}(X)$$ is reductive.
Using the previous work of Y. Matsushima [Nagoya Math. J. 11, 53–60 (1957; Zbl 0099.37501)] and X.-J. Wang, X. Zhu [Adv. Math. 188, No.1, 87–103 (2004; Zbl 1086.53067)] on the existence of an Einstein-Kähler metric on a smooth Fano manifold, the following result is obtained: Let $$X$$ be a nonsingular toric Fano variety with associated reflexive polytope $$P$$. Then $$X$$ admits an Einstein-Kähler metric if and only if the barycentre of $$P$$ is the origin.
The following upper bound on the dimension of $$\text{ Aut}(X)$$ is obtained: If $$\text{ Aut}(X)$$ is reductive, $$\dim X=d\geq 3$$, and $$X$$ is not a product of projective spaces, then $$\dim \text{ Aut}(X)\leq d^2-2d+4$$. The set of Demazure roots of a complete toric variety plays a crucial role in the author’s arguments.

##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J45 Fano varieties 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
##### Keywords:
toric variety; lattice polytope; Fano variety; algebraic group
##### Citations:
Zbl 0099.37501; Zbl 1086.53067
PALP
Full Text:
##### References:
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