The Nahm transform maps irreducible Hermitian-Yang-Mills connections on a K3 surface to Hermitian-Yang-Mills connections on its Mukai dual, and the inverse transform recovers the original connection, giving a differential geometric proof of Fourier-Mukai duality.
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Mukai duality on K3 surfaces from the differential geometric perspective
The Nahm transform maps irreducible Hermitian-Yang-Mills connections on a K3 surface to Hermitian-Yang-Mills connections on its Mukai dual, and the inverse transform recovers the original connection, giving a differential geometric proof of Fourier-Mukai duality.