Finite group actions on the moduli space of self-dual connections. I.

*(English)*Zbl 0724.57013Let M be a simply-connected, smooth 4-manifold with a positive definite intersection form and \({\mathcal M}\) be the moduli space of SU(2) instantons over M with Chern number 1. It is known that for generic Riemannian metrics on M the subspace \({\mathcal M}^*\subset {\mathcal M}\) representing irreducible connections is a smooth 5-manifold, and itself is obtained by adding a finite number of singular points, around which \({\mathcal M}\) is modelled on a cone over \(CP^ 2\). The author extends the theory to the case when a finite group G acts on M. For a G-invariant metric on M there is a natural induced G action on \({\mathcal M}\), with fixed point set \({\mathcal M}^ G\) representing invariant connections. It is shown first that for generic G-invariant metrics \({\mathcal M}^ G\) is a manifold except for a finite number of cone singularities. The G-index theorem is used to compute the dimension of \({\mathcal M}^ G\). The other main topic is to seek G-invariant perturbations of the instanton equations which deform \({\mathcal M}\) into a G-space with standard singularities. It is shown that these exist if a certain topological obstruction in \(H^ 3({\mathcal M}^*,{\mathcal M}^*_ 0)\) vanishes, where \({\mathcal M}^*_ 0\) is the complement of a large compact set in \({\mathcal M}^*\).

Reviewer: S.Donaldson (Oxford)

##### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

##### Keywords:

4-manifold; intersection form; moduli space; instantons; Chern number; generic Riemannian metrics; connections; invariant metric; singularities
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\textit{Y. S. Cho}, Trans. Am. Math. Soc. 323, No. 1, 233--261 (1991; Zbl 0724.57013)

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