S})$ replaced by $E_{h,\be}^X(\f,J)$. In turn, this identity follows from Theorem \[main\_thm\] and the proof of [@P1 Theorem 3]. Let $Y$ be a degree $\be$ $J$-holomorphic curve of genus $h\!\le\!g$ meeting each $f_j$. By the above, the assumptions of Theorem \[main\_thm\] are satisfied. By definition (see Section \[CR\_subs2\]), the orbi-bundle $\cok(D_J^{NY})$ is dual to the bundle $\ker((D_J^{NY})^*)$ of kernels of the dual operators $(D_J^{NY})^*$. For each $$\big([u],(w_j)_{j\in S}\big)\in\ov\M_{g,\f}(Y,[Y];J)
\subset\ov\M_{g,\f}(X,\be;J),$$ the operator $(D_{J;u}^{NY})^*$ is the natural extension of the operator $\bigoplus_i (D_{J;u_i}^{NY})^*$ to $(1,0)$-forms on $\Si_u$ with poles at the nodes such that the residues at each node sum up to $0$. Since $(D_{J;u_{\eff}}^{NY})^*$ is injective by the regularity of $u_{\eff}$, the projection $$\eta\lra \bigoplus_{\Si_{u;i}\neq\Si_{u_{\eff}}}\eta|_{\Si_{u;i}}$$ to the contracted components is injective. Since $(D_{J;u_i}^{NY})^*\!=\!\dbar^*$ if $u_i$ is constant, the image of this homomorphism is determined by $\Si_u$ and is independent of $D_{J;u_{\eff}}^{NY}$ (as long as $D_{J;u_{\eff}}^{NY}$ is surjective). Thus, $\cok(D_J^{NY})$ is isomorphic to the restriction to $\ov\M_{g,\f}(Y,[Y];J)$ of the obstruction bundle in [@P1 Section 3], i.e. the bundle of cokernels of the operators $D_{J;u}^{NY}$ as above, but for a holomorphic vector bundle $NY$. Thus,

\_[g,]{}(Y,\_Y) &=e((D\_J\^[NY]{})), \^[vir]{}\
&=C\_[h,]{}\^X(g-h)(\[u\_\],(w\_j)\_[jS]{})

by \_ref[mainthm\_e2]{} and [@P1 Theorem 3]. Since $$\ov\M_{g,\f}(X,\be;J)
=\bigsqcup_{h=0}^{h=g}\bigsqcup_{([u],(w_j)_{j\in S})\in\M_{h,\f}^*(X,\be;J)}
\!\!\!\!\!\!  \ov\M_{g,\f}(\Im\,u,[\Im\, u];J),$$ the 