Locally trivial deformations of toric varieties

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