Derived categories of complex manifolds, their DG-enhancement and Bott-Chern classes

I will describe the following series of results, partly obtained in collaboration with A. Rosly, about derived categories of coherent sheaves on complex manifolds. For a general complex torus $X, D^b(\mathrm{Coh}-X)$ has a semiorthogonal decomposition, and it is not equivalent to $D^b_{\mathrm{coh}}(\mathcal{O}_X-\mathrm{mod})$. There is a twist-closed DG-enhancement of the latter category by dbar-superconnections for any smooth compact complex manifold. This DG-enhancement allows us to define Bott-Chern cohomology for any object of $D^b_{\mathrm{coh}}(\mathcal{O}_X-\mathrm{mod})$, in particular, for a coherent sheaf. If time permits, I will describe the extension of the enhancement theorem to the case of non-compact complex manifolds and applications to constructing the moduli space of objects in the above category.