Let E G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over C. We show that E G is semistable if and only if it admits approximate Hermitian-Einstein structures.

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Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, φ), consisting of a flat vector bundle E over M and a flat nonzero section φ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on Kähler manifolds to the situation of flat pairs on affine manifolds.

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