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Zbl 0625.53031
Itoh, Mitsuhiro
Yang-Mills connections over a complex surface and harmonic curvature. (English)
[J] Compos. Math. 62, 95-106 (1987). ISSN 0010-437X; ISSN 1570-5846/e

The Yang-Mills equations for a connection A on a bundle P over a Riemannian manifold M are $d\sp*\sb A F=0$. Here $F\in \Omega\sp 2({\cal F}\sb P)$ is the curvature of A. If M is a Kähler manifold, of real dimension 4, the equations can be re-expressed using the complex structure as: $\partial\sp*\sb A F\sp{2,0}=-i\partial\sb A F\sp 0$ where $F=F\sp{2,0}+F\sp{\perp}+F\sp 0\omega +F\sp{0,2}$ is the full decomposition of the curvature. ($\omega$ is the Kähler form and $F\sp 0\in \Omega\sp 0({\cal F}\sb P).)$ \par In this paper the author introduces the notion of a ``harmonic connection'', defined to be one with $\partial\sp*\sb A F\sp{2,0}=0$. He proves the following results: \par Theorem 1. If M has positive scalar curvature S then any harmonic Yang- Mills connection is anti-self-dual. \par Theorem 2. There is a $c=c(M)>0$, such that if A is a Yang-Mills connection with $\Vert F\Vert\sb{L\sp 2}<c$ then A is anti-self-dual. (The anti-self-duality condition is: $F\sp{2,0}=F\sp 0=0.)$ \par For irreducible SU(2) connections on a bundle with $c\sb 2(P)=k$ he proves: Theorem 3. If $[F\sp{2,0}\wedge F\sp{0,2}]=0$, then either: (i) A is anti-self-dual, (ii) A is contained in a family of harmonic Yang-Mills connections of real dimension at least: $2(1+4k-3P\sb a(M))$. These results are proved using Weitzenböck formulae. For example Theorem 2 follows from an argument of Min-Oo applied to the formula: $$ \nabla\sp*\sb A \nabla\sb A F\sp{2,0}-3i[F\sp 0,F\sp{2,0}]+2sF\sp{2,0}=0 $$ valid for any Yang-Mills connection A.
[S.K.Donaldson]

MSC 2000:: *53C05 Connections, general theory
53C55 Complex differential geometry (global)

Keywords: anti-self dual connection; Yang-Mills equations; harmonic connection; positive scalar curvature; Yang-Mills connection

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