DGLA controlling deformation of holomorphic curves Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = R\Gamma(C,df:T_C\to f^*T_X)$. Explicitly, this complex may be realized using the Dolbeault resolution of $T_C$ and $f^*T_X$. In this realization, there are three terms in this complex: 
$L^0 = \Omega^0(C,T_C)$, $L^1 = \Omega^{0,1}(C,T_C)\oplus\Omega^0(C,f^*T_X)$ and $L^2 = \Omega^{0,1}(C,f^*T_X)$ with the differentials $L^0\to L^1$ and $L^1\to L^2$ given by a sum of pushforward by $df$ and the canonical $\bar\partial$ operator on a holomorphic vector bundle.
By some general philosophy (for example in the deformation theory book by Kontsevich-Soibelman), $L^\bullet$ should carry the structure of a differential graded Lie algebra (DGLA) such that the deformations of $f$ over a local Artin ring $(A,\mathfrak m)$ with residue field $\mathbb C$ can be seen as solutions $\omega\in L^1\otimes\mathfrak m$ to the Maurer-Cartan equation $d\omega + \frac12[\omega,\omega] = 0$ modulo the gauge action of $\exp(L^0\otimes\mathfrak m)$.
Can we realize the DGLA structure in this case explicitly? In particular, what is the explicit expression for the bracket $[\cdot,\cdot]:L^1\otimes L^1\to L^2$? I am able to see that the degree zero bracket $L^0\otimes L^0\to L^0$ should be simply the usual commutator Lie bracket of vector fields. 
 A: Firstly, I assume you mean deformations of $C$ over $X$ ("deformations of $f$" is ambiguous, as it could mean fixing neither or both of $C$  and $X$).
The DGLA philosophy is then that there should exist some DGLA quasi-isomorphic to the explicit realisation of the complex $L$ you wrote down. It doesn't guarantee a DGLA structure on $L$ itself, though it will transfer a non-canonical $L_{\infty}$ structure.
In this case, you can reinterpret the problem as trying to deform $\mathcal{O}_C$ as a sheaf of $f^{-1}\mathcal{O}_X$-algebra. The DGLA you want should then be an explicit model for $\mathbf{R}\Gamma(C,\mathbf{R}\mathrm{Der}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C))$. At this point, you encounter the problem that free algebra resolutions and flabby sheaf resolutions don't interact well. 
One explicit model is given by first forming the Harrison complex (or a natural analogue for holomorphic functions) $\mathrm{Harr}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C)$ (a sheaf of DGLAs), then take a nice open cover $\mathfrak{U}$ of $C$ and form a Cech complex $\check{C}(\mathfrak{U},\mathrm{Harr}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C))$, giving a cosimplicial DGLA. Then apply Thom-Whitney cochains to give a DGLA.
You'll find various related constructions in several works by Iacono, Manetti and Fiorenza, as well as Ciocan-Fontanine's derived Hilbert schemes and some of my early papers.  
