  Let $X$ be a smooth complex algebraic variety and let $\operatorname{Coh}
(X)$ denote its Abelian category of coherent sheaves. By the work of W. Lowen
and M. Van den Bergh, it is known that the deformation theory of
$\operatorname{Coh} (X)$ as an Abelian category can be seen to be controlled by
the Gerstenhaber-Schack complex associated to the restriction of the structure
sheaf $\mathcal O_X \vert_{\mathfrak U}$ to a cover of affine open sets. We
construct an explicit $L_\infty$ algebra structure on the Gerstenhaber-Schack
complex controlling the higher deformation theory of $\mathcal O_X
\vert_{\mathfrak U}$ in case $X$ can be covered by two acyclic open sets,
giving an explicit deformation-obstruction calculus for such deformations.
Deformations of complex structures and deformation quantizations of $X$ are
recovered as degenerate cases, as is shown by means of concrete examples.
