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 $H^2\left(X,{\mathcal{𝒯}}_X\right)$ and the cup product map $H^1\left(X,{\mathcal{𝒯}}_X\right)\times H^1\left(X,{\mathcal{𝒯}}_X\right)\to H^2\left(X,{\mathcal{𝒯}}_X\right)$ 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.