Special Hermitian metrics on Oeljeklaus–Toma manifolds

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 Kähler 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 𝜕 . 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), 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.

On the pluriclosed flow on Oeljeklaus–Toma manifolds

“…The existence of pluriclosed metrics on Oeljeklaus-Toma manifolds was studied in [1,8,18]. In particular, from [1] it follows the following result.…”

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature, and Oeljeklaus–Toma manifolds

“…It was shown in [6] that they do not admit any analytic hypersurfaces and have algebraic dimension zero. It was shown in [36] that OT manifolds do not admit any balanced metrics but always have locally conformally balanced metrics. See [2,3,20,25,34,35,37,[62][63][64] for recent progress and open problems on OT manifolds.…”