Let be a complete -factorial toric variety. We explicitly describe the space and the cup product map 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.
"Deformations of smooth complete toric varieties: obstructions and the cup product." Algebra Number Theory 14 (4) 907 - 926, 2020. https://doi.org/10.2140/ant.2020.14.907