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Affine structures and non-archimedean analytic spaces

Maxim Kontsevich·Yan Soibelman
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Abstract

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from "flat pieces" an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group $SO(1,18)$ by piecewise-linear transformations on the 2-dimensional sphere $S^2$ equipped with naturally defined singular affine structure.

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Maxim Kontsevich, Yan Soibelman. 2004-06-28. Affine structures and non-archimedean analytic spaces. https://arxiv.org/abs/math/0406564

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