Toric Kato manifolds

TL;DR

This paper introduces toric Kato manifolds, a new class constructed via toric geometry, and explores their topological, analytical, and Hermitian geometric properties, including invariants, degenerations, and metric structures.

Contribution

It defines toric Kato manifolds, generalizes previous constructions, and analyzes their geometric and topological properties, including metric restrictions and degenerations.

Findings

Toric Kato manifolds generalize known constructions and include Inoue surfaces in dimension 2.

They have computable invariants linked to toric combinatorics.

Certain metric structures, like balanced and pluriclosed metrics, do not exist on these manifolds.

Abstract

We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any toric Kato manifold, which serve as an essential tool in computing their Hodge numbers. In the last part, we study the Hermitian geometry of Kato manifolds. We give a characterization result for the existence of locally conformally K\"ahler metrics on any Kato manifold. Finally, we prove that no Kato manifold carries balanced metrics and that a large class of toric Kato manifolds of complex dimension $\geq 3$ do not support pluriclosed metrics.