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On the two types of affine structures for degenerating Kummer surfaces – non-Archimedean vs Gromov-Hausdorff limits –.(English)Zbl 1523.14015

The paper compares two integral affine manifolds attached to degenerations of Kummer surfaces, which are special types of \(K3\) surfaces. Such surfaces are constructed from an abelian surface by taking the quotient by \(\mathbb{Z}/2\mathbb{Z}\) acting by multiplication by \(-1\) and resolving the \(16\) singularities, coming from the \(2\)-torsion points.
Motivated by the Strominger-Yau-Zaslow conjucture in mirror symmetry, to any degeneration of a Calabi-Yau, in particular, to a degeneration of a Kummer surface, one can attach two integreal affine manifolds with singularities. The first one, constructed by Kontsevich-Soibelman, using non-Archimedean techniques, can also be described as a dual intersection complex of a dlt model of the degeneration [J. Nicaise andC. Xu, Am. J. Math. 138, No. 6, 1645–1667 (2016;Zbl 1375.14092)]. The second integral affine manifold is defined as the Gromov-Hausdorff limit of the Ricci flat metric on the Camabi-Yau, which exists by Yau’s proof of the Calabi conjecture.
Relating these two constructions, one coming from algebraic geometry and the other from Riemannian geometry is a challenging task. In general, it is an open conjecture of Kontsevich-Soibelman that these two integral affine manifolds are the same. The author proves this conjecture for degenerations of Kummer surfaces.
The proof uses the construction of Kummer surfaces as quetients of abelian surfaces. In particular, it uses the work of Künnemann on degenerations of abelian varieties, to construct nice degenerations of Kummer surfaces, with good enough properties so that one can understand the structure of the dual intersection complex [K. Künnemann, Duke Math. J. 95, No. 1, 161–212 (1998;Zbl 0955.14017)].

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14G22 Rigid analytic geometry
14T20 Geometric aspects of tropical varieties

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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