Locally trivial deformations of toric varieties

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan Σ \Sigma , we construct a deformation functor Def Σ \operatorname {Def}_{\Sigma } by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, Def Σ \operatorname {Def}_{\Sigma } is isomorphic to Def X Σ ′ \operatorname {Def}’_{X_{\Sigma }} , the functor of locally trivial deformations for the toric variety X Σ X_{\Sigma } associated to Σ \Sigma . In particular, for any complete toric variety X X that is smooth in codimension 2 2 and Q \mathbb {Q} -factorial in codimension 3 3 , there exists a fan Σ \Sigma such that Def Σ \operatorname {Def}_{\Sigma } is isomorphic to Def X \operatorname {Def}_{X} , the functor of deformations of