Gauge theoretical methods in the classification of non-Kählerian surfaces
Volume 85 / 2009
Banach Center Publications 85 (2009), 109-120
MSC: Primary 32Q57; Secondary 57R57, 53C07, 32Q55.
DOI: 10.4064/bc85-0-8
Abstract
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 [Ka1]–[Ka3], [Na1]–[Na3], [DOT], [OT] this conjecture (if true) would solve the 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.
