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What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation of perfect complexes is most welcome.

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    $\begingroup$ One possibility is that (under appropriate hypotheses) equivalences of derived categories of coherent sheaves are always given by a Fourier-Mukai transform. The kernel of this transform will not necessarily be a coherent sheaf, but it will be a perfect complex. Understanding the deformation theory of it can sometimes be useful. $\endgroup$
    – Matt
    Commented Sep 14, 2013 at 4:42

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Here a reference which gives technical details on deformation of bounded complexes on a smooth projective variety (so they are all perfect, by Serre's Theorem) : http://arxiv.org/abs/0805.3527.

One application of such a theory is the construction of a deformation invariant virtual cycle for moduli spaces of objects in the derived category of a given threefold (this is explained in the paper).

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