Towards the Classification of Class VII Surfaces

Chapter
First Online:
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

This article follows the ideas of my talk “A duality theorem for instanton moduli spaces on class VII surfaces” given at the workshop “INdAM Meeting: Complex and Symplectic Geometry” which took place in June 2016 in Cortona; it gives a survey on my recent results concerning the existence of a cycle of curves on class VII surfaces with small b2. The main problem in the classification of class VII surfaces is the existence of holomorphic curves. My approach uses a combination of techniques coming from complex geometry and gauge theory. The main object used in the proofs is a moduli space \(\mathcal{M}\) of polystable holomorphic bundles on the considered surface. This moduli space is identified with an instanton moduli space via the Kobayashi-Hitchin correspondence. The existence (non-existence) of curves on the base surface is related to geometric properties of the corresponding moduli space.

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Notes

Acknowledgements

The author thanks Daniele Angella, Paolo De Bartolomeis, Costantino Medori, and Adriano Tomassini, the organisers of the “INdAM Meeting: Complex and Symplectic Geometry”, for the invitation to give a talk and to submit an article for the proceedings of the conference. The author is grateful to the referee for careful reading of the paper, and for his useful suggestions and comments.

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© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Aix Marseille Université, CNRS, Centrale MarseilleMarseilleFrance

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