Hypertoric varieties (or toric hyperkähler varieties) are hyperkähler analogues of toric varietes. They come with a symplectic form \(\omega\) on their regular locus. Such affine varieties \((Y,\omega)\) are called “conical symplectic varieties”. Their fundamental geometric properties, deformation theory, birational geometry have been extensively studied. However, as the author suggests, notions of “universal coverings” have not been well-studied.
In [Y. Namikawa, Kyoto J. Math. 53, No. 2, 483–514 (2013; Zbl 1277.32029)] the author asks whether for a concial symplectic variety \((Y,\omega)\) the fundamental group \(\pi_1\left(Y_{\operatorname{reg}}\right)\) is finite. Some partial results in this direction are later given by Namikawa.
Now, assuming this fundamental group is finite, Nagaoka defines in the present article the (singular) universal covering of a conical symplectic variety \((Y,\omega)\). In this way one obtains a new example of a conical symplectic variety.
One of the main motivations of the present article is the problem of describing these universal coverings and the fundamental groups \(\pi_1\left(Y_{\operatorname{reg}}\right)\).
Another reason to study universal coverings comes from an of analogue of the Bogomolov’s decomposition, which asks if one can decompose the universal covering \(\left(\overline{Y}, \overline{\omega}\right)\) of \((Y, \omega)\) into a product \(\prod_i\left(Y_i, \omega_i\right)\) of irreducible conical symplectic varieties. Here, \(\omega_i\) is the unique conical symplectic structure on \(Y_i\) up to scalar. Such a decomposition result is conjectured by Namikawa for general conical symplectic varieties.
The present article gives a complete answer to the first problem for affine hypertoric varieties. It describes the universal covering of an affine hypertoric variety in terms of the combinatorics of the associated hyperplane arrangement. This is also an affine hypertoric variety. Also a description of \(\pi_1\left(Y_{\operatorname{reg}}\right)\) is given.
In the last section, the space of homogeneous \(2\)-forms on decomposable hypertoric varieties is determined. As an application, the author obtains the analogue of Bogomolov’s decomposition for hypertoric varieties and refined classification results.