# Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product

## TL;DR

This paper provides a combinatorial description of the deformation space and cup product for complete $Q$-factorial toric varieties, demonstrating that some smooth projective toric threefolds have obstructed deformations.

## Contribution

It explicitly describes the second cohomology and cup product map for these varieties and shows that obstructions to deformations can occur in smooth projective cases.

## Key findings

- Explicit combinatorial description of $H^2(X,T_X)$
- Identification of non-vanishing cup product in examples
- Existence of obstructed deformations in smooth projective toric threefolds

## Abstract

Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ 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.