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The universal covers of hypertoric varieties and Bogomolov’s decomposition

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Abstract

In this paper, we study the (singular) universal cover of an affine hypertoric variety. We show that it is given by another affine hypertoric variety, and taking the universal cover corresponds to taking the simplification of the associated hyperplane arrangement. Also, we describe the fundamental group of the regular locus of an affine hypertoric variety in general. In the latter part, we show that the hamiltonian torus action is block indecomposable if and only if \({\mathbb {C}}^*\)-equivariant symplectic structures on the associated hypertoric variety are unique up to scalar. In particular, we establish the analogue of Bogomolov’s decomposition for hypertoric varieties, which is proposed by Namikawa for general conical symplectic varieties. As a byproduct, we show that if two affine (or smooth) hypertoric varieties are \({\mathbb {C}}^*\)-equivariant isomorphic as varieties, then they are also the hamiltonian torus action equivariant isomorphic as symplectic varieties. This implies that the combinatorial classification actually gives the classification of these varieties up to \({\mathbb {C}}^*\)-equivariant isomorphisms.

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Acknowledgements

The author wishes to express his gratitude to his supervisor Yoshinori Namikawa for stimulating discussions. He is also greatly indebted to Hiraku Nakajima for giving some comments on his presentation, which leads to start this project. He is grateful to Ryo Yamagishi for spending much time to discuss with him after sharing ideas during Winter School on Poisson Structures in Algebraic Geometry at KIAS. The author is also grateful to Nicholas Proudfoot for sharing his nice idea on a computation of the fundamental group (in Remark 6.6). He also wishes to express his thanks to Masahiko Yoshinaga for letting him know the reference [5]. He is also grateful to Makoto Enokizono for pointing out an error in the previous version. The author is partially supported by Grant-in-Aid for JSPS Fellows 19J11207.

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  1. Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8522, Japan

    Takahiro Nagaoka

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Nagaoka, T. The universal covers of hypertoric varieties and Bogomolov’s decomposition. Math. Z. 300, 2533–2569 (2022). https://doi.org/10.1007/s00209-021-02860-1

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