I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex associated and, more importantly, the differential graded Lie algebra structure that apparently appears in this problem.


I am interested in learning about this machinery in general more than only for the specific case of complex structures, so references about the general set-up are also welcome. In particular, my ultimate goal is to understand how $L\_{\infty}$-algebras are associated to deformation problems, and since differential graded Lie algebras are particular instances of $L\_{\infty}$-algebras that looks like a good place where to start.


Thanks.



References for the moduli space of complex structures
Concerning the deformation theory of complex manifolds, there are of course the seminal papers of Kodaira-Spencer. There are also some more recent notes of Manetti, *Lectures on deformations of complex manifolds*, which are available on arxiv and could be of interest for you.


The general principle relating deformation problems to DGLAs or more general $L\_{\infty}$-algebras has a very long history. The notion of deformation functor goes back to Schlessinger's paper *Functors of Artin rings*. The general principle that every deformation problem should arise as a deformation functor associated to a certain dgla emerged from unpublished work of Deligne and Drinfeld and found its first striking application in Goldman-Millson's paper *The deformation theory of representations of fundamental groups of compact Kähler manifolds*.
Briefly, to every dgla $g$ one can associate its set of Maurer-Cartan elements $MC(g)$, on which acts the so called gauge group exp(g^0) (here we have to add a pronilpotency condition on $g$ to have a well defined gauge group). The quotient is called the Maurer-Cartan moduli set, and can be made into a functor from artinian rings to sets which is the deformation functor associated to $g$. This is well explained in Manetti's *Deformation theory via differential graded Lie algebras* for instance (but there are certainly plenty of other references). Maurer-Cartan elements are the structures you want to study and the gauge group gives you an equivalence relation between such structures.
It turns out that the Maurer-Cartan moduli set is invariant under quasi-isomorphisms of dlgas and more generally under $L\_{\infty}$-quasi-isomorphisms, a propery which was crucial in Kontsevich's work *Deformation quantization of Poisson manifolds*.


Now let us go in a ""derived" world : one can define a simplicial Maurer-Cartan set whose 1-simplices gives a notion of homotopy between two Maurer-Cartan elements, which is equivalent to the action of the gauge group. The main advantage is that such homotopies can be defined for $L\_{\infty}$-algebras, for which there is no gauge groupe anymore.
The simplicial Maurer-Cartan set construction and several properties of it are well explained in Getzler's *Lie theory for nilpotent $L\_{\infty}$-algebras*.
Like the Maurer-Cartan moduli set, it is invariant (up to homotopy) under quasi-isomorphisms of $L\_{\infty}$-algebras, and even under $L\_{\infty}$-quasi-isomorphisms according to a recent paper of Dolgushev-Rogers *A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras*.


The "Deligne principle" of deformation theory was stated rigorously as a theorem only recently (several years ago) in the realm of derived algebraic geometry, as an equivalence of $\infty$-categories between dglas and moduli problems. This is due to Lurie in *Derived algebraic geometry X* and Pridham in *Unifying derived deformation theories*.


