We study infinitesimal deformations of pairs (X, D) with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential-graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X, D) in many cases, for example, whenever (X, D) is a log Calabi–Yau pair, in the case of a smooth divisor D in a Calabi–Yau variety X and when D is a smooth divisor in | − mKX|, for some positive integer m.

In this article we investigate diffeomorphism classes of Calabi-Yau threefolds. In particular, we focus on those embedded in toric Fano manifolds. Along the way, we give various examples and conclude with a curious remark regarding mirror symmetry.

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf ${\mathcal F}$ are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf $\Eps nd^*(\Eps^\cdot)$, where $\Eps^\cdot$ is any locally free resolution of ${\mathcal F}$. In particular, one recovers the well known fact that the tangent space to $\Def_{\mathcal F}$ is $\Ext^1({\mathcal F},{\mathcal F})$, and obstructions are contained in $\Ext^2({\mathcal F},{\mathcal F})$. \par The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA ${\mathfrak g}^\Delta$, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor $H^1_{ m sc}(\exp {\mathfrak g}^\Delta)$.

We describe an abstract version of the Theorem of Bogomolov-Tian-Todorov, whose underlying idea is already contained in various papers by Bandiera, Fiorenza, Iacono, Manetti. More explicitly, we prove an algebraic criterion for a differential graded Lie algebra to be homo- topy abelian. Then, we collect together many examples and applications in deformation theory and other settings.