Gauge theoretical methods in the classification of non-Kaehlerian surfaces

TL;DR

This paper explores a novel approach using gauge theory techniques to address the challenging classification problem of class VII complex surfaces, aiming to prove the existence of certain curves and advance the classification.

Contribution

It introduces a new gauge-theoretic method inspired by Donaldson theory to establish the existence of curves on class VII surfaces, potentially resolving a major open problem.

Findings

New gauge-theoretic approach to classify class VII surfaces

Recent results supporting the existence of curves on these surfaces

Potential progress towards the classification conjecture

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 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.