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Lagrangian loci in moduli of abelian surfaces

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Title
Lagrangian loci in moduli of abelian surfaces
Author(s)
Jun-Muk Hwang
Publication Date
2022-09
Journal
European Journal of Mathematics, v.8, no.3, pp.972 - 984
Publisher
Springer Science and Business Media Deutschland GmbH
Abstract
We show that any smooth surface germ in the moduli of abelian surfaces arises from a Lagrangian fibration of abelian surfaces. By Donagi–Markman’s cubic condition, the key issue of the proof is to find a suitable affine structure with a compatible cubic form on the base space of the family. We achieve this by analyzing the properties of cubic forms in two variables and proving the existence of the solution of the resulting partial differential equations by the Cauchy–Kowalewski Theorem. Modifying the argument, we show also that a smooth curve germ in the moduli of abelian surfaces arises from a Lagrangian fibration if and only if the curve is a null curve with respect to the natural holomorphic conformal structure on the moduli of abelian surfaces.
URI
https://pr.ibs.re.kr/handle/8788114/12858
DOI
10.1007/s40879-021-00476-7
ISSN
2199-675X
Appears in Collections:
Center for Complex Geometry (복소기하학 연구단) > 1. Journal Papers (저널논문)
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