(see Section \[NRS\_subs\]), let $\H_u$ denote the space of deformations of the complex structure on $\Si_u$. The linearization of the $\dbar_J$-operator for maps to $X$, $$D_{J;u}^X\!:\H_u\oplus
L^p_1(\Si_u;u^*TX)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*TX),
\quad p\!>\!2,$$ induces a generalized Cauchy-Riemann operator $$D_{J;u}^{NY}\!:  L^p_1(\Si_u;u^*NY)\lra 
L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*NY).$$ For each $j\!\in\!S$, define $$\ti\ev_j\!:\ker D_{J;u}^{NY}\lra N_{z_j(u)}Y
\qquad\hbox{by}\quad \xi\lra \xi(z_j(u))+T_{z_j(u)}Y,$$ where $z_j(u)\!\in\!\Si_u$ is the $j$-th marked point; this homomorphism is the composition of the differential of the evaluation map \_ref[evmap\_e]{} with the projection to the normal bundle.

\[main\_thm\] Suppose $(X,\om)$ is a compact symplectic $2n$-manifold, $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, $a_j\!\in\!\bar\Z^+$ for each $j\!\in\!S$, and $f_j\!:M_j\!\lra\!X$ is a cobordism representative for $\ka_j\!\in\!H_*(X;\Z)$ for each $j\!\in\!S$. If $J$ is an $\om$-tame almost complex structure on $X$, $Y$ is a compact almost complex submanifold of $(X,J)$, and $\be_Y\!\in\!H_2(Y;\Z)$ are such that

1.  $\io_{Y*}(\be_Y)\!=\!\be$ and $f_j$ intersects $Y$ properly for each $j\!\in\!S$;

2.  for every $([u],(w_j)_{j\in S})\!\in\!\ov\M_{g,\f}(Y,\be_Y;J)$, the homomorphism (D\_[J;u]{}\^[NY]{})\_[jS]{}N\_[f\_j(w\_j)]{}\^[f\_j]{}Y, (\_j()+(d\_[w\_j]{}f\_j))\_[jS]{},is an isomorphism,

then

1.  the space $\ov\M_{g,\f}(Y,\be_Y;J)$ carries a natural VFC (dependent on the orientations of $f_j^{-1}(Y)$) with

    \^[vir]{} &=\_[g,S]{}(X,)-\_[jS]{}(2n-\_j)\
    &+\_[jS]{}N\^[f\_j]{}Y-2(+\_NY(1-g));

2.  the vector spaces $\cok(D_{J;u}^{NY})$ form a natural oriented vector orbi-bundle $$\cok\big(D_J^{NY}\big) \lra \ov\M_{g,\f}(Y,\be_Y;J)$$ with \_ (D\_[J;u]{}\^[NY]{})= \_[jS]{}N\^[f\_j]{}Y-2(+\_NY(1-g));

3.  $\ov\M_{g,\f}(Y,\be_Y;J)$ is a union of connected components of $\ov\M_{g,\f}(X,\be;J)$ and its contribution 