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Non-Kählerian Compact Complex Surfaces

  • Andrei TelemanEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2246)

Abstract

This text follows the lecture series given by the author in the CIME School “Complex non-Kähler geometry” (Cetraro, July 9–13, 2018) and is dedicated to the classification of non-Kählerian surfaces. In the first three sections we present the classical theory:
  • The Enriques Kodaira classification for surfaces and the classes of non-Kählerian surfaces,

  • Class VII surfaces and their general properties,

  • Kato surfaces: construction, classification and moduli.

In Sect. 3.4 we explain the main ideas and techniques used in the proofs of our results on the existence of cycles of curves on class VII surfaces with small b2. Section 3.5 deals with criteria for the existence of smooth algebraic deformations of the singular surface obtained by contracting a cycle of rational curves in a minimal class VII surface. We included an Appendix in which we introduce several fundamental objects in non-Kählerian complex geometry (the Picard group of a compact complex manifold, the Gauduchon degree, the Kobayashi-Hitchin correspondence for line bundles, unitary flat line bundles), and we prove basic properties of these objects.

Notes

Acknowledgements

The author thanks Daniele Angella, Leandro Arosio, and Eleonora Di Nezza, the organizers of the “CIME School “Complex non-Kähler geometry”, for the invitation to give a lecture series, and to submit a written version of my lectures for publication in the proceedings of the meeting. The author is grateful to Georges Dloussky for his constant help, encouragement and collaboration, and for his useful suggestions and comments on the text.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Marseille, CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance

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