Gorenstein spherical Fano varieties

Abstract

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary $d$-dimensional $\mathbb{Q}$-factorial Gorenstein spherical Fano variety is bounded by $2d$. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical variety due to Brion.

22 pages, 3 figures

References in corpus (2)

• Localization of spherical varieties
• Homogeneous spherical data of orbits in spherical embeddings

Cited by in corpus (8)

• Kähler-Einstein metrics on group compactifications
• Kähler-Einstein metrics on smooth Fano symmetric varieties with Picard number one
• Ricci flat Kähler metrics on rank two complex symmetric spaces
• Manin's conjecture for certain spherical threefolds
• K-stability of Gorenstein Fano group compactifications with rank two
• Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII
• Momentum polytopes of projective spherical varieties and related Kähler geometry
• K-stable valuations and Calabi-Yau metrics on affine spherical varieties