Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product

Abstract

Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.

arXiv comment19 pages, 4 figures; v2 minor revisions

References in corpus (7)

• Mirror Symmetry and Fano Manifolds
• K-Stability for Fano Manifolds with Torus Action of Complexity One
• Deformations of Rational T-Varieties
• Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes
• Deformations of Calabi-Yau hypersurfaces arising from deformations of toric varieties
• Deformations of Smooth Toric Surfaces
• Deformations of extremal toric manifolds

Cited by in corpus (2)

• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• Foldable fans, cscK surfaces and local K-moduli