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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)
Software:
PALP
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References:
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