Units in Number Fields Satisfying a Multiplicative Relation with Application to Oeljeklaus–Toma Manifolds

Special Hermitian metrics on Oeljeklaus–Toma manifolds

“…Thus, $$X(K, U)$$ provides the first example of OT manifold endowed with a pluriclosed metric. Remark In the recent work [7] it was shown that for any $$s \geqslant 2$$, there exists a number field $$K$$ of type $$(s, s)$$ and an admissible positive units group $$U$$ satisfying the pluriclosed condition. Therefore, OT manifolds provide examples of pluriclosed metrics in arbitrary even complex dimension.…”

“…Remark 3.2.2. In the recent work [7] it was shown that for any 𝑠 ⩾ 2, there exists a number field 𝐾 of type (𝑠, 𝑠) and an admissible positive units group 𝑈 satisfying the pluriclosed condition. Therefore, OT manifolds provide examples of pluriclosed metrics in arbitrary even complex dimension.…”

“…Equivalently, it can be described as a Hermitian metric for which there exists a covering of the manifold with open sets $$(U_i)_i$$ and smooth functions $$f_i$$ on $$U_i$$ such that $$e^{-f_i}\Omega$$ is a Kähler metric on $$U_i$$. It was proven in [21] and [6, Appendix] that the existence of an lcK metric on a generic $$X(K, U)$$ is equivalent to the following arithmetic condition: $$\begin{equation} |\sigma _{s+1}(u)|= \cdots = |\sigma _{s+t}(u)|, \qquad \forall u \in U, \end{equation}$$which can independently be studied as a number theory problem (see, for instance, [6, 28]) and automatically reveals that all $$X(K, U)$$ of type $$(s, 1)$$ carry an lcK metric. As a matter of fact, the translation of geometrical properties of $$X(K, U)$$ in a number‐theoretical language has been a recurrent theme in various instances of their study (see also [4] and [1]).…”

“…Moreover, in Subsection (3.1), we give an explicit example of OT manifold of complex dimension 4 carrying a pluriclosed metric, which was communicated to me by Matei Toma. The recent work [7] shows that in arbitrary complex dimension, there exists an admissible pair $$(K, U)$$ satisfying condition (2), thus providing examples of pluriclosed OT manifolds in any even complex dimension. They represent rather exotic examples, expanding the list of examples known so far in the literature, which are specific cases of nilmanifolds (see [11]), solvmanifolds in complex dimension 3 (see [10]), connected sum of certain product of spheres (see [14]), compact Lie groups (see [24] and [17] for a detailed proof) and simply connected examples in arbitrary complex dimension arising from A. Swann's twist construction (see [25]).…”