#### Vol. 14, No. 4, 2020

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Deformations of smooth complete toric varieties: obstructions and the cup product

### Nathan Ilten and Charles Turo

Vol. 14 (2020), No. 4, 907–926
##### Abstract

Let $X$ be a complete $ℚ$-factorial toric variety. We explicitly describe the space ${H}^{2}\left(X,{\mathsc{𝒯}}_{X}\right)$ and the cup product map ${H}^{1}\left(X,{\mathsc{𝒯}}_{X}\right)×{H}^{1}\left(X,{\mathsc{𝒯}}_{X}\right)\to {H}^{2}\left(X,{\mathsc{𝒯}}_{X}\right)$ 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.

##### Keywords
deformation theory, toric varieties, cup product
##### Mathematical Subject Classification 2010
Primary: 14M25
Secondary: 14B12, 14D15