Real loci in (log-) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations

TL;DR

This paper explores the topology of real loci in toric degenerations of complex varieties using Kato-Nakayama spaces, providing explicit descriptions and examples including K3-surfaces and local projective planes.

Contribution

It introduces a method to describe the topology of real loci in toric degenerations via Kato-Nakayama spaces and extends log geometry to real structures with explicit combinatorial descriptions.

Findings

Topology of real loci described via Kato-Nakayama spaces.

Explicit combinatorial descriptions of real loci as bundles.

Examples include real degenerations of K3-surfaces and local P^2.

Abstract

We study the real loci of toric degenerations of complex varieties with reducible central fibre. We show that the topology of such degenerations can be explicitly described via the Kato-Nakayama space of the central fibre as a log space. We furthermore provide generalities of real structures in log geometry and their lift to Kato-Nakayama spaces. A key point of this paper is a description of the Kato-Nakayama space of a toric degeneration and its real locus, both as bundles determined by combinatorial data. We provide several examples including real toric degenerations of K3-surfaces and a toric degeneration of local $\textbb{P}^2$.