Gauge theoretical methods in the classification of non-Kaehlerian surfaces

Details Gauge theoretical methods in the classification of non-Kaehlerian surfaces Gauge theoretical methods in the classification of non-Kaehlerian surfaces

2008-04-03

arxiv.org/abs/0804.0557v1

Mathematics

Complex Variables

12 pages, 2 figures, talk given at the Postnikov Memorial Conference, Bedlewo (Poland) June 2007, to appear in Banach Center P

Scientific paper

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $b_2>0$ has $b_2$ curves. By the results of Kato, Nakamura and Dloussky/Oeljeklaus/Toma, this conjecture (if true) would solve this classification problem completely. We explain a new approach (based on techniques from Donaldson theory) to prove existence of curves on class VII surfaces, and we present recent results obtained using this approach.

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