Self-dual connections on 4-manifolds with indefinite intersection matrix.

*(English)*Zbl 0552.53011One of the fundamental starting points for the celebrated theorem of .S. K. Donaldson [ibid. 18, 279-315 (1983; Zbl 0507.57010)] was the existence theorem for self-dual connections on manifolds with positive definite intersection form, proved by the author [ibid. 17, 139-170 (1982; Zbl 0484.53026)]. The paper under review here presents an extension of those results to manifolds with indefinite intersection matrix, and is almost certainly a precursor to further analysis of the topology of 4-manifolds by using the tool of gauge theories. The situation is now, however, considerably more complicated. Whereas for manifolds of definite intersection form, self-dual SU(2) connections begin to exist when \(-c_ 2(P)>0\), in the more general case the results are metric dependent. For example, as the author proves, the complex projective plane with its standard metric (or any Kähler metric) has no anti-self-dual SU(2) connections with \(c_ 2(P)=1\) but does for a generic metric.

The most general theorem proved is the following: Let M be a smooth compact, oriented 4-manifold and Q its intersection matrix, with \(b_- =(rank Q\)- sign Q). Then if \(b_-\neq 0,1,3\) for a dense open set of metrics on M in the \(C^{2+r}\) topology, a principal SU(2) bundle P admits a smooth irreducible self-dual connection when \(-c_ 2(P)\geq b_-\). The analogous result for any metric requires \(-c_ 2(P)\geq 4b_-/3\) with special cases for \(b_-\leq 3\). The techniques involve an extremely subtle use of non-linear analysis and a sideways glance at Kuranishi’s theory of deformations of complex structures.

The most general theorem proved is the following: Let M be a smooth compact, oriented 4-manifold and Q its intersection matrix, with \(b_- =(rank Q\)- sign Q). Then if \(b_-\neq 0,1,3\) for a dense open set of metrics on M in the \(C^{2+r}\) topology, a principal SU(2) bundle P admits a smooth irreducible self-dual connection when \(-c_ 2(P)\geq b_-\). The analogous result for any metric requires \(-c_ 2(P)\geq 4b_-/3\) with special cases for \(b_-\leq 3\). The techniques involve an extremely subtle use of non-linear analysis and a sideways glance at Kuranishi’s theory of deformations of complex structures.

Reviewer: N.Hitchin

##### MSC:

53C05 | Connections, general theory |

58E30 | Variational principles in infinite-dimensional spaces |

55R10 | Fiber bundles in algebraic topology |