Special Hermitian metrics on Oeljeklaus–Toma manifolds
Alexandra Otiman
Abstract
Oeljeklaus–Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension K $K$ of Q $\mathbb {Q}$ and a subgroup of units U $U$ . We characterize the existence of pluriclosed metrics (also known as strongly Kähler with torsion (SKT) metrics) on any OT manifold X ( K , U ) $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number b 3 $b_3$ and the Dolbeault cohomology group H ∂ ¯ 2 , 1 $H^{2,1}_{\overline{\partial }}$ . Combined with the main result in (Dubickas, Results Math. 76 (2021), 78), 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 ) $(2, 2)$ , the existence of a pluriclosed metric on X ( K , U ) $X(K, U)$ is entirely topological, namely, it is equivalent to b 3 = 2 $b_3=2$ . 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.