Flat Bundles Over Some Compact Complex Manifolds
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Flat Bundles Over Some Compact Complex Manifolds Fusheng Deng1
· John Erik Fornæss2
Received: 8 January 2019 © Mathematica Josephina, Inc. 2019
Abstract We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kähler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat P1 bundle over it and a Stein domain with real analytic boundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions. Keywords Flat fiber bundle · Hopf surface · Stein neighborhood basis · Hyperconvex domain Mathematics Subject Classification 32L05
1 Introduction The aim of the present note is to construct flat bundles over some compact complex manifolds such that the total spaces can be used as examples for some problems in several complex variables. It is known that there exists a bounded pseudoconvex domain D in Cn (n > 1) with smooth boundary such that its closure has no pseudoconvex (or Stein) neighborhood basis [3]. But if a bounded pseudoconvex domain in Cn has real analytic boundary, its closure has a pseudoconvex neighborhood basis [4]. So it is natural to ask whether
B
Fusheng Deng [email protected] John Erik Fornæss [email protected]
1
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
2
Department for Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway
123
F. Deng, J. E. Fornæss
the closure of a bounded pseudoconvex or Stein domain with real analytic boundary in a complex manifold has a pseudoconvex neighborhood basis. In [5], Diederich and Fornæss constructed a domain with real analytic Levi-flat boundary in the total space B of a flat P1 bundle over a Hopf surface such that its closure does not have a pseudoconvex neighborhood basis. The domain mentioned in the previous paragraph is a flat disc bundle over the Hopf surface. Note that the Hopf surface is not Kähler. It is natural to ask whether admits a Kähler metric. It is observed in [5] that is biholomorphic to the product A × C2 \{0}, where A is an annulus. By a basic result from complex geometry, admits a complete Kähler metric. In this note, inspired by the work in [5], we present a general framework to construct flat fiber bundles over complex manifolds, based on discrete group actions. We then show some basic properties of these bundles and develop an idea of duality, with the product structure of mentioned above as a special case. As one of the applications, we construct a flat C∗ -bundle over the Hopf surface such that the total space has a complete Kähler metric. In the above example, B is not projective and is not Stein. For any compact Riemannian surface of positive genus, we
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