L_infinity morphisms and semiregularity

The main topic of this thesis is the construction of canonical $L_{\infty}$ liftings of the components of the Buchweitz-Flenner semiregularity map for coherent sheaves on complex manifolds, using Chern-Simons classes for curved DG-pairs. As an application, we obtain that the Buchweitz-Flenner semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. We also introduce semiregularity maps for a Lie pair $(\mathcal{L}, \mathcal{A})$ and a locally free $\mathcal{A}$-module, and prove they annihilate all obstructions to deformations of the $\mathcal{A}$-module, provided that a certain spectral sequence degenerates at $E_1$.