Special Hermitian metrics on Oeljeklaus-Toma manifolds

TL;DR

This paper characterizes when Oeljeklaus-Toma manifolds admit pluriclosed metrics using number-theoretical conditions, provides explicit examples, and explores their metric properties, including the absence of balanced metrics.

Contribution

It offers a number-theoretical characterization of pluriclosed metrics on OT manifolds and constructs explicit examples in arbitrary dimensions.

Findings

Pluriclosed metrics exist on OT manifolds under specific number-theoretical conditions.

In complex dimension 4, existence of pluriclosed metrics is topologically determined by $b_3=2$.

No OT manifold admits balanced metrics, but they all have locally conformally balanced metrics.

Abstract

Oeljeklaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $Q$ and a subgroup of units $U$. We characterize the existence of pluriclosed metrics (also known as strongly K\" ahler with torsion (SKT) metrics) on any OT manifold $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number $b_3$ and the Dolbeault cohomology group $H^{2,1}_{\overline{\partial}}$. Combined with the main result in [D20], these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type $(2, 2)$, 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. Finally, we show that no OT manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.