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

Kronheimer, Peter B.

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.

Reviewer: H.Reckziegel (Köln)

MSC:

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

Keywords:

Morgan’s conjecture; Donaldson’s polynomial; gauge theory; branched instantons

Citations:

Zbl 0741.00019