Uniqueness of Polarization for the Autonomous 4-dimensional Painlevé-type Systems

Creators: Nakamura, Akane; Rains, Eric

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Abstract

We prove that for any autonomous 4-dimensional integral system of Painlevé type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus 2 component of the boundary of an affine patch of the Liouville torus.

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© The Author(s) 2020. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) Received: 30 October 2019. Revision received: 30 October 2019. Accepted: 06 February 2020. Published: 24 February 2020. The author would like to thank Professors Tadashi Ashikaga, Brian Conrad, Kazuki Hiroe, Yusuke Nakamura, Elena Mantovan, and Haruo Yoshida for valuable discussions and advices. This work was partially supported by a grant from the National Science Foundation [DMS-1500806 to E.R.]; The first author is grateful for Professors Toshio Oshima (with JSPS grant-in-aid for scientific research B, No. 25287017) and Yoko Umeta (with Josai University President's Fund) for supporting part of her trips. The first author is grateful for the hospitality of Caltech while she stayed there.

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