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.

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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