Locally Trivial Deformations of Toric Varieties

Abstract

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $Σ$, we construct a deformation functor $\mathrm{Def}_Σ$ by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}'_{X_Σ}$, the functor of locally trivial deformations for the toric variety $X_Σ$ associated to $Σ$. In particular, for any complete toric variety $X$ that is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$, there exists a fan $Σ$ such that $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}_X$, the functor of deformations of $X$. We apply these results to give a new criterion for a smooth complete toric variety to have unobstructed deformations, and to compute formulas for higher order obstructions, generalizing a formula of Ilten and Turo for the cup product. We use the functor $\mathrm{Def}_Σ$ to explicitly compute the deformation spaces for a number of toric varieties, and provide examples exhibiting previously unobserved phenomena. In particular, we classify exactly which toric threefolds arising as iterated $\mathbb{P}^1$-bundles have unobstructed deformation space.