Families of Riemann surfaces, uniformization and arithmeticity. (English) Zbl 1386.32015
Let \(C\) be a Riemann surface. The uniformization theorem says that the universal cover of \(C\) is conformally equivalent to only one of these Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. On the other hand, there seems to be a plenty number of possible universal covers for a complex surface. In this work, the authors consider holomorphic families of Riemann surfaces \(V\longrightarrow C\), such that both base and fibers are Riemann surfaces of finite hyperbolic type \((g,n)\), i.e., \(2-2g-n<0\), where \(g\) denotes the genus and \(n\) the number of punctures. A such family is also said to be algebraic. In the first part, they characterize completely those simply connected and bounded domains \(\mathcal{B}\) of \(\mathbb C^2\) arising as the universal covers of holomorphic families of Riemann surfaces. It turns out that a bounded and simply connected domain \(\mathcal{B}\) of \(\mathbb C^2\) is the universal cover of a (non-trivial) algebraic family of Riemann surfaces if and only if \(\mathcal{B}\) arises as the graph of a non-trivial holomorphic motion \(W\) of the unit circle \(\Delta\times \mathbb S^1\longrightarrow \mathbb C\), which was first introduced by Mañe, Sad and Sullivan. In the second part, the authors turn their attention to those families which are also arithmetic. A holomorphic familiy of Riemann surfaces is said to be arithmetic if the base and the total space are defined over an algebraic number field. Also in this case, the authors give a complete characterization of those domains arising as the universal covers of such families. They finally apply their results to characterize arithmeticity of Kodaira fibrations \(S\longrightarrow C\), that are holomorphic families of compact Riemann surfaces over a compact Riemann surface \(C\). It was already known that the base \(C\) and the fibers of such families are of genus at least two and three, respectively, in particular \(S\) must be a minimal projective surface of general type. For such families, the authors show that arithmeticity depends only on the biholomorphic class of their universal covers.
Reviewer: Gianluca Faraco (Bologna)
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |
| 14J29 | Surfaces of general type |
Keywords:
holomorphic families of Riemann surfaces; holomorphic motions; uniformization of complex surfacesReferences:
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