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.