Affine structures and non-Archimedean analytic spaces. (English) Zbl 1114.14027

Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-4076-2/hbk). Progress in Mathematics 244, 321-385 (2006).
The authors explore the role of integral affine structures in mirror symmetry. One way to see an integral affine structure on a real \(n\)-manifold is as an atlas of charts whose transition functions take values in \(\text{GL}(n,\mathbb{Z})\times\mathbb{R}^n\). The paper starts with the following observation: Suppose that \(X\) is a symplectic manifold, and \(X_0\to B_0\) is a smooth proper map, with the property that the Poisson bracket vanishes on pullbacks of continuous functions on \(B_0\). Then by the Liouville integrability theorem, there are distinguished coordinates on \(B\), which are unique up to affine transformations, whence define an affine structure on \(B_0\). This applies, for example, for fibrations \(X\to\mathbb{P}^1\) on complex \(K3\) surfaces after discarding singular fibers. It leads to an affine structure on some Zariski open subset \(B_0\) of the Riemann sphere \(B=\mathbb{P}^1\).
The goal of the authors is to reconstruct, in situations as above, the original fibration \(X\to B\) from the integral affine structure on \(B_0\). To do so, they develop for integral affine structures a theory of singular extensions over boundary points. Moreover, they replace schemes by nonarchimedean analytic spaces over ground fields \(K\) that are complete, nonarchimedean local fields of residue characteristic zero.
One main result of the paper is the following: Let \(B\) be a compact oriented real 2-manifold endowed with a \(K\)-affine structure defined on the complement of a finite subset \(B_0\). Then, under suitable assumptions on the affine structure near the boundary points, there exists a compact \(K\)-analytic surfaces \(X\), endowed with an analytic 2-form, and a continuous proper Stein map \(f:X\to B\) that induces over \(B_0\) the given \(K\)-affine structure.
For the entire collection see [Zbl 1083.00015].


14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14G22 Rigid analytic geometry
32Q25 Calabi-Yau theory (complex-analytic aspects)
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