Gauge theoretical methods in the classification of non-Kählerian surfaces

Abstract

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 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.

A vanishing result for a Casson-type instanton invariant

“…. , the cohomology groups of the second complex (31). The following proposition is a well known and easily proved folklore result at least for the untwisted case, and the proof for the twisted case is completely analogous.…”

“…From the point of view of investigating invariants of smooth four-manifold, our vanishing result is negative. However, it appears that this result has relevance to Teleman's classification programme on complex surfaces of class VII (see [29,31]), and in this perspective, it is rather a positive result. We discuss this briefly in Subsection 4.3.…”

On the torsion of the first direct image of a locally free sheaf

“…In our previous articles [Te2]- [Te5] we developed a program for proving the existence of curves on minimal class VII surfaces with b 2 > 0 using certain moduli spaces of polystable bundles on these surfaces: for a class VII surface X we consider a differentiable rank 2 bundle E on X with c 2 = 0 and det(E) = K X , where K X denotes the underlying differentiable line bundle of the canonical line bundle K X . The fundamental object intervening in our program is the moduli space M pst := M pst K (E) of polystable holomorphic structures on E which induce the canonical holomorphic structure K X on K X .…”

On Casson-Type Instanton Moduli Spaces Over Negative Definite 4-Manifolds

“…PSfrag replacements (12) (34) (14) (23) (24) Figure 6. An example of what a partial diagrammatic description of a suitable immersionL → S 4 may look like.…”

“…Recently Andrei Teleman considered moduli spaces of projectively anti-selfdual instantons in certain Hermitian rank-2 bundles over a closed oriented 4-manifold with negative definite intersection form [12]. These play a role in his classification program on Class VII surfaces [13] [14]. However, in certain situations the instanton moduli spaces involved consist of projectively flat connections and therefore have very interesting topological implications.…”