Complete toric varieties with reductive automorphism group

Abstract

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.Comment: AMS-LaTeX, 20 pages with 1 figur

Lattices generated by skeletons of reflexive polytopes

“…For this we exploit a partial addition property on the set of lattice points in P , observed by the second author in [7,Proposition 4.1]. This method was applied in [8] to the set R of lattice points inside the facets of P in order to investigate the automorphism group of the Gorenstein toric Fano variety P Σ P * ,N . The elements of R are precisely the associated Demazure roots.…”

“…It has a purely combinatorial proof itself. [8,Lemma 4.8 and Corollary 4.9]. ) Let x, y ∈ ∂P ∩ M with y = −x such that x, y are not contained in a common facet.…”

Gale duality and homogeneous toric varieties

“…Example 2. The automorphism group Aut(X) of a complete toric variety X is a linear algebraic group, see [12,10,16]. It implies that X is homogeneous if and only if X is S-homogeneous.…”

“…It is well known that root subgroups are in one-to-one correspondence with Demazure roots of the fan Σ X , see [12,17,10]. Nowadays Demazure roots and their generalizations became a central tool in many research projects, see [16,15,7,8,1,5,6]. For instance, Demazure roots and Gale duality was used in [8] and [1] to describe orbits of the group Aut(X) on complete and affine toric varieties X, respectively.…”

Integral cohomology and mirror symmetry for Calabi-Yau 3-folds

“…Recently Haase and Nill proved that for a reflexive polytope ∆ * of arbitrary dimension d > 2 the sublattice in N generated by all lattice points in ∆ * ∩ N coincides with N ′ ∆ * [14]. The proof of this results uses the fact that the interior lattice points in codimension-1 faces of ∆ * are Demazure roots for the authomorphism group of the corresponding Gorenstein toric Fano variety P ∆ * [20]. In particular, this result easily implies that N = N ′ ∆ * for all reflexive polyhedra ∆ * of dimension d = 3.…”