On metric and cohomological properties of Oeljeklaus-Toma manifolds

TL;DR

This paper investigates the metric and cohomological structures of Oeljeklaus-Toma manifolds, providing explicit cohomology formulas and characterizations of pluriclosed metrics through number-theoretic and cohomological methods.

Contribution

It offers a detailed analysis of the differential form complex, Bott-Chern cohomology, and conditions for pluriclosed metrics on Oeljeklaus-Toma manifolds, including explicit cohomology computations.

Findings

Characterization of pluriclosed metrics via cohomology

Explicit formulas for Dolbeault cohomology

Proof that certain Hermitian metrics cannot exist

Abstract

We study metric and cohomological properties of Oeljeklaus-Toma manifolds. In particular, we describe the structure of the double complex of differential forms and its Bott-Chern cohomology and we characterize the existence of pluriclosed (aka SKT) metrics in number-theoretic and cohomological terms. Moreover, we prove they do not admit any Hermitian metric $\omega$ such that $\partial \overline{\partial} \omega^k=0$, for $2 \leq k \leq n-2$ and we give explicit formulas for the Dolbeault cohomology of Oeljeklaus-Toma manifolds admitting pluriclosed metrics.