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On Affine Structures Which Come from Berkovich Geometry for K-trivial Finite Quotients of Abelian Varieties

Goto, Keita 京都大学 DOI:10.14989/doctor.k24384

2023.03.23

概要

1.1. At the end of the 20th century, in order to formulate what is called mirror
symmetry, several approaches have been proposed. One of them is due to
Strominger, Yau and Zaslow [SYZ96]. In op.cit., they gave a geometric
interpretation for mirror symmetry and proposed a conjecture called the SYZ
conjecture. Gross and Siebert provided an algebro-geometric interpretation
of the SYZ conjecture [GS06]. It is known as the Gross-Siebert program.
In this program, it is important to associate an integral affine manifold with
singularities (IAMS, for short) with a degeneration of polarized Calabi-Yau
manifolds, and vice versa. For a (toric) degeneration of polarized Calabi-Yau
manifolds, they extracted the polyhedral decomposition and the fan structure
for each vertex and gave an IAMS structure to the dual intersection complex
based on them, and vice versa.
Kontsevich and Soibelman associated an IAMS structure of the dual intersection complex in a non-Archimedean way [KS06]. The precise definition
will be given later (§5), but for now, we call it NA SYZ picture. ...

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Department of Mathematics, Kyoto university, Oiwake-cho, Kitashirakawa,

Sakyo-ku, Kyoto city,Kyoto 606-8285. JAPAN

Email address: k.goto@math.kyoto-u.ac.jp

...

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