Instanton invariants and flat connections on the Kummer surface
The author shows how to calculate a certain integer-valued instanton invariant for the \(K3\) surface \(X\) and any homotopy \(K3\) surface \(X(p)\) obtained by performing a logarithmic transform of order \(p\) on \(X\). He reproves the result of Friedman and Morgan that the \(X(p)\) are not diffeomorphic to \(X\) and in fact represent infinitely many distinct diffeomorphism types. The calculations neither use Donaldson's translation to stable bundles nor the \(C\)-infinity techniques developed by Gompf and Mrowka (in order to compute the same invariant), but draw on the fact that for certain nonsimply connected 4-manifolds or orbifolds the moduli space of flat connections has a real meaning for the theory. This moduli space can be understood purely topologically, which is the reason for the simplifications coming from this approach. In this paper the orbifold is the 4-torus divided by the involution \(t\to -t\), which gives the \(K3\) surface \(X\) after resolving the 16 double points with (\(-2\))-curves. The main idea now is that, though the manifolds \(X(p)\) also contain 16 (\(-2\))-curves, the fundamental groups of the complements (w.r.t. the standard embedding) are different, and through the instanton moduli space this difference can be detected in the manifolds \(X\) resp. \(X(p)\).
instanton invariant for the \(K3\) surface
scientific article; zbMATH DE number 58545