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.

MSC:

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

Zbl 0741.00019