GÖKOVA GEOMETRY / TOPOLOGY PROCEEDINGS

Published in Proceedings of Gökova Geometry-Topology Conference 2018-2019
Title Topological torus fibrations on Calabi-Yau manifolds via Kato-Nakayama spaces
Author Hulya Arguz
Abstract
A large class of complex algebraic varieties admit degenerations into toric log Calabi-Yau spaces, formed by unions of toric varieties glued along toric strata. Such degenerations were introduced by Gross and Siebert as toric degenerations. This paper is an expository article on toric degenerations of Calabi-Yau manifolds and Kato-Nakayama spaces. We first review the combinatorial data used to reconstruct a toric degeneration from a toric log Calabi-Yau space \(X_0\), via the Gross-Siebert reconstruction algorithm. We then explain how one can understand the total space on a topological level from the Kato-Nakayama space of \(X_0\), which is defined in terms of this combinatorial data. We illustrate through a concrete example, focusing on a degeneration of \(K3\)-surfaces, that the Kato-Nakayama space of \(X_0\) defines a homeomorphism to the total space restricted to the inverse image of a circle on the base. We also investigate torus fibrations on the general fiber by further analysis of the topology of the Kato-Nakayama space. The proofs of the results presented here appear in joint work with Bernd Siebert [1].
Pages176-200
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Last updated: October 2021
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