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

Ilten, Nathan; Turo, Charles

doi:10.2140/ant.2020.14.907

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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

DOI: 10.2140/ant.2020.14.907

Abstract

Let $X$ be a complete $ℚ$ -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.

Keywords

deformation theory, toric varieties, cup product

Mathematical Subject Classification 2010

Primary: 14M25

Secondary: 14B12, 14D15

Milestones

Received: 2 January 2019

Revised: 25 November 2019

Accepted: 6 February 2020

Published: 21 June 2020

Authors

Nathan Ilten
Simon Fraser University Burnaby BC Canada

Charles Turo
Simon Fraser University Burnaby BC Canada

Vol. 14, No. 4, 2020