Locally Trivial Deformations of Toric Varieties

TL;DR

This paper investigates locally trivial deformations of toric varieties using combinatorial methods, establishing isomorphisms between deformation functors and providing criteria for unobstructed deformations.

Contribution

It introduces a combinatorial construction of deformation functors for toric varieties and applies this to classify unobstructed cases and compute deformation spaces.

Findings

Constructed a deformation functor ech cochains for fans

Established isomorphisms between combinatorial and geometric deformation functors

Classified toric threefolds with unobstructed deformation spaces

Abstract

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $\Sigma$, we construct a deformation functor $\mathrm{Def}_\Sigma$ by considering \v{C}ech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, $\mathrm{Def}_\Sigma$ is isomorphic to $\mathrm{Def}'_{X_\Sigma}$, the functor of locally trivial deformations for the toric variety $X_\Sigma$ associated to $\Sigma$. 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 $\Sigma$ such that $\mathrm{Def}_\Sigma$ 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 $\mathrm{Def}_\Sigma$ 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.