Embedded surfaces in 4-manifolds. (English) Zbl 0746.53041

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 529-539 (1991).
[For the entire collection see Zbl 0741.00019.]
The purpose of the article is to outline a “proof” of the following conjecture due to J. W. Morgan: If \(X\) is a simply-connected, oriented 4- manifold for which Donaldson’s polynomial invariants are defined and non- zero, and if \(\Sigma\) is a smoothly embedded, oriented 2-manifold with positive self-intersection, then the genus \(g\) of \(\Sigma\) satisfies \(2g- 2\geq \Sigma\cdot\Sigma\). The conjecture is known to hold for \(g=0\) or \(=1\). The author’s argument uses gauge theory, but with a modification; he considers connections in some auxiliary \(SU(2)\) or \(SO(3)\) bundle over \(X\backslash \Sigma\), which have non-trivial holonomy around the small linking circles of \(\Sigma\). As for “branched instantons” some essential facts are missing, the author has to fill the gap by two other conjectures.


53C40 Global submanifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)


Zbl 0741.00019