Toric Kato manifolds

Istrati, Nicolina; Otiman, Alexandra; Pontecorvo, Massimiliano; Ruggiero, Matteo

Journal de l’École polytechnique - Mathématiques, Volume 9 (2022), pp. 1347-1395.

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 Kähler 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 $\ge 3$ do not support pluriclosed metrics.

Nous introduisons et étudions une classe spéciale de variétés de Kato, que nous appelons variétés de Kato toriques. Leur construction est issue de la géométrie torique, étant donné que leurs revêtements universels sont des ouverts de variétés toriques de type non-fini. Cela généralise des constructions précédentes dues à Tsuchihashi et Oda. En dimension complexe $2$, on retrouve les surfaces d’Inoue proprement éclatées. Nous étudions les propriétés topologiques et analytiques des variétés de Kato toriques, et nous relions certains invariants aux données combinatoires qui viennent de la construction torique. De plus, nous produisons des familles plates de dégénérescences pour toute variété de Kato torique, qui sont essentielles pour calculer leurs nombres de Hodge. Dans la dernière partie, nous étudions la géométrie hermitienne des variétés de Kato (pas nécessairement toriques). Nous donnons une caractérisation pour l’existence de métriques localement conformes de Kähler sur toute variété de Kato. Enfin, nous montrons qu’aucune variété de Kato n’admet de métrique équilibrée, et qu’une classe très large de variétés de Kato toriques de dimension complexe $\ge 3$ n’admet pas de métrique plurifermée.

Classification: 53C55, 14F40, 14M25, 32G05, 32C35, 32L10
Keywords: Kato data, toric degeneration, Dolbeault cohomology, locally conformally Kähler metric
Mot clés : Donnée de Kato, dégénérescence torique, cohomologie de Dolbeault, métrique localement conforme de Kähler

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