Abstract
We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those root system hypergeometric functions found by Heckman–Opdam, and in view of the work of Couwenberg–Heckman–Looijenga on the geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a \({\mathbb {C}}^{\times }\)-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we show that the space in question can be biholomorphically mapped onto a divisor complement of a ball quotient if the Schwarz conditions are invoked.
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Acknowledgements
I am grateful to my PhD supervisors: to Eduard Looijenga for his patient guidance during my PhD time, including but not only on this work; to Gert Heckman for taking me to walk around in this beautiful subject. I also thank Frits Beukers, Pierre Py, Jasper Stokman and Ian Strachan for their valuable comments on the earlier version of this paper. The referees’ helpful suggestions are appreciated as well.
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Shen, D. Geometric structures on the complement of a toric mirror arrangement. Math. Z. 300, 683–744 (2022). https://doi.org/10.1007/s00209-021-02771-1
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DOI: https://doi.org/10.1007/s00209-021-02771-1