Special Hermitian metrics on Oeljeklaus-Toma manifolds

TLDR: In this paper, the existence of a strongly Kahler with torsion (SKT) metric on an Oeljkelaus-Toma manifold is characterized in terms of number-theoretical conditions.

Abstract: Oeljkelaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $\mathbb{Q}$ and a subgroup of units $U$. We characterize the existence of pluriclosed metrics (also known as strongly Kahler with torsion (SKT) metrics) on any OT-manifolds $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number $b_3$ and the Dolbeault cohomology group $H^{2,1}_{\overline{\partial}}$. We prove that in complex dimension 4, the existence of a pluriclosed metric on $X(K, U)$ is entirely topological, namely, it is equivalent to $b_3=2$. Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric, thus enlarging the list of examples known so far in the literature. Finally, we show that no OT-manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.