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We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier--Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive 4-dimensional space.
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W. Lowen, Algebroid prestacks and deformations of ringed spaces , Trans. Amer. Math. Soc. 360
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Y. Toda, Deformations and Fourier–Mukai transforms , J. Diff. Geom. 81
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