Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). Suppose first that $C$ is a non-singular closed Riemann surface (say of genus $g$ equipped with $n$ marked points) and $f:C\to X$ is a $J$-holomorphic curve which is a stable map. Denote the complex structure on $C$ by $j$. Letting $f_*[C] = \beta\in H_2(X,\mathbb Z)$, we can ask if the moduli space $\overline{\mathcal M}_{g,n}(X,J,\beta)$ is transversely cut-out at the point $(C,f)$, i.e., if it is "unobstructed".

The answer to this is given by considering the following complex $L^\bullet$. Its terms are given by $L^0=\Omega^0(C,TC)$, $L^1 = \Omega^{0,1}(C,TC)\oplus\Omega^0(C,f^*TX)$ and $L^2 = \Omega^{0,1}(C,f^*TX)$, where the last vector space is defined using $J$. The map $L^0\to L^1$ is given by the difference of the canonical $\bar\partial$ operator on $TC$ and the map $df:TC\to f^*TX$. The map $L^1\to L^2$ is given by the sum of $df$ and the linearization $D_u\bar\partial_{J,j}$ (in the sense of McDuff-Salamon's book) of the $\bar\partial_{J,j}$-operator at the point $f$ (which is a real-linear Cauchy Riemann operator). Stability is equivalent to $H^0(L^\bullet)=0$. Further, we can show that $(C,f)$ is unobstructed iff $H^2(L^\bullet)=0$.

Now, suppose we replace $C$ by a nodal curve (of arithmetic genus $g$ and $n$ marked points) and suppose $f:C\to X$ is a stable holomorphic map. We can ask if $(C,f)$ is unobstructed as an element of the moduli space $\overline{\mathcal M}_{g,n}(X,J,\beta)$. My question is then the following:

Can we write down a complex which detects unobstructedness in the nodal case?

In the case when $(X,J)$ is a genuine complex manifold, I believe that the cochain complex $\text{RHom}_{C}(f^*\Omega^1_X\to\Omega^1_C,\mathcal O_C)$ answers this question (here, $\Omega^1_X,\Omega^1_C$ are the sheaves of Kähler differentials on $X,C$ respectively and the map $f^*\Omega^1_X\to\Omega^1_C$ is the "pull-back by $f$" map). However, in the case of a general symplectic manifold with a compatible almost complex structure, I think the answer might be subtle, since the usual route to investigating if $(C,f)$ is unobstructed in the nodal case seems to pass through the gluing theorem for holomorphic curves.

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). Suppose first that $C$ is a non-singular closed Riemann surface (say of genus $g$ equipped with $n$ marked points) and $f:C\to X$ is a $J$-holomorphic curve which is a stable map. Denote the complex structure on $C$ by $j$. Letting $f_*[C] = \beta\in H_2(X,\mathbb Z)$, we can ask if the moduli space $\overline{\mathcal M}_{g,n}(X,J,\beta)$ is transversely cut-out at the point $(C,f)$, i.e., if it is "unobstructed".

The answer to this is given by considering the following complex $L^\bullet$. Its terms are given by $L^0=\Omega^0(C,TC)$, $L^1 = \Omega^{0,1}(C,TC)\oplus\Omega^0(C,f^*TX)$ and $L^2 = \Omega^{0,1}(C,f^*TX)$, where the last vector space is defined using $J$. The map $L^0\to L^1$ is given by the difference of the canonical $\bar\partial$ operator on $TC$ and the map $df:TC\to f^*TX$. The map $L^1\to L^2$ is given by the sum of $df$ and the linearization $D_f\bar\partial_{J,j}$ (in the sense of McDuff-Salamon's book) of the $\bar\partial_{J,j}$-operator at the point $f$ (which is a real-linear Cauchy Riemann operator). Stability is equivalent to $H^0(L^\bullet)=0$. Further, we can show that $(C,f)$ is unobstructed iff $H^2(L^\bullet)=0$.

Now, suppose we replace $C$ by a nodal curve (of arithmetic genus $g$ and $n$ marked points) and suppose $f:C\to X$ is a stable holomorphic map. We can ask if $(C,f)$ is unobstructed as an element of the moduli space $\overline{\mathcal M}_{g,n}(X,J,\beta)$. My question is then the following:

Can we write down a complex which detects unobstructedness in the nodal case?

In the case when $(X,J)$ is a genuine complex manifold, I believe that the cochain complex $\text{RHom}_{C}(f^*\Omega^1_X\to\Omega^1_C,\mathcal O_C)$ answers this question (here, $\Omega^1_X,\Omega^1_C$ are the sheaves of Kähler differentials on $X,C$ respectively and the map $f^*\Omega^1_X\to\Omega^1_C$ is the "pull-back by $f$" map). However, in the case of a general symplectic manifold with a compatible almost complex structure, I think the answer might be subtle, since the usual route to investigating if $(C,f)$ is unobstructed in the nodal case seems to pass through the gluing theorem for holomorphic curves.