Vol. 313, No. 2, 2021

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The universal Poisson deformation of hypertoric varieties and some classification results

Takahiro Nagaoka

Vol. 313 (2021), No. 2, 459–508
DOI: 10.2140/pjm.2021.313.459
Abstract

In this paper, we study and describe the universal Poisson deformation space of hypertoric varieties concretely. In the first application, we show that affine hypertoric varieties as conical symplectic varieties are classified by the associated regular matroids (this is a partial generalization of the result by Arbo and Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose dimension vectors have all coordinates equal to one are isomorphic to each other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as quiver varieties and give some examples of 8-dimensional hypertoric varieties which cannot be raised as such quiver varieties. In the second application, we compute explicitly the number of all projective crepant resolutions of some 4-dimensional hypertoric varieties by using the combinatorics of hyperplane arrangements.

Keywords
hypertoric variety, universal Poisson deformation, classification, counting crepant resolutions, hyperplane arrangement
Mathematical Subject Classification 2010
Primary: 14B07, 14E15, 14M25, 52B40, 52C35
Milestones
Received: 14 April 2019
Revised: 21 February 2021
Accepted: 22 May 2021
Published: 12 October 2021
Authors
Takahiro Nagaoka
Research Institute for Mathematical Sciences
Kyoto university
Japan