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Let $(X,g)$ be a Riemannian 4-manifold and let $G \to X$ be a principal $G$ bundle. The ASD equations $F^+=-F$ are satisfied for $F=dA$ the curvature two-form of the $G$-equivariant connection $A$.

My question is the following: How can one determine if Abelian instantons exist for generic metric $g$? What are the conditions that have to be met and why most generic metrics do not admits Abelian instanton solutions?

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