  We give equivalent and sufficient criteria for the automorphism group of a
complete toric variety, respectively a Gorenstein toric Fano variety, to be
reductive. In particular we show that the automorphism group of a Gorenstein
toric Fano variety is reductive, if the barycenter of the associated reflexive
polytope is zero. Furthermore a sharp bound on the dimension of the reductive
automorphism group of a complete toric variety is proven by studying the set of
Demazure roots.

Algebraic Geometry;Combinatorics