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A characterization of varieties whose universal cover is the polydisk or a tube domain

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Abstract

In this article we give necessary and sufficient conditions, in terms of certain tensors called semispecial tensors, respectively slope zero tensors, in order that the universal covering of a complex projective manifold be a symmetric domain of tube type. As an application, we give precisions of a result of Kazhdan showing that a Galois conjugate of such a manifold has the same universal covering.

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Notes

  1. We are indebted to Pascal Dingoyan for providing this reference.

  2. We are indebted to Gang Tian for providing this reference.

  3. They however took for granted Yau’s wrong assertion, that if \(S^m (V_j)\) is not stable, then it should have a direct factor of rank one having the same slope.

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Acknowledgments

We would like to thank Marco Franciosi for interesting conversations which led to our present cooperation. We also thank a first referee for the reference [11] , a second one for the nice derivation of the formula for the tensor \(\tilde{\psi }\) in the case of domains of type \(IV,\) both referees for several useful comments.

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Authors and Affiliations

  1. Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, 95447 , Bayreuth, Germany

    Fabrizio Catanese

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 , Torino, Italy

    Antonio J. Di Scala

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  1. Fabrizio Catanese
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  2. Antonio J. Di Scala
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Correspondence to Fabrizio Catanese.

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This article is dedicated, with admiration (and with the friendship and gratitude of the first author), to Enrico Bombieri on the occasion of his 70th birthday.

The present work took place in the realm of the DFG Forschergruppe 790 “Classiffication of algebraic surfaces and compact complex manifolds”. The visit of the second author to Bayreuth was supported by the DFG FOR 790 The second author was also partially supported by GNSAGA (INdAM) and MIUR (PRIN07, Differential Geometry and Global Analysis), Italy.

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Catanese, F., Di Scala, A.J. A characterization of varieties whose universal cover is the polydisk or a tube domain. Math. Ann. 356, 419–438 (2013). https://doi.org/10.1007/s00208-012-0841-x

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