to the number \_ref[GWdfn\_e]{} is given by \_[g,]{}(Y,\_Y)= e((D\_J\^[NY]{}))\_[jS]{}\_j\^[a\_j]{}, \^[vir]{}.\

\[CY\_eg\] Suppose $(X,J)$ is a Calabi-Yau $3$-fold and $Y\!\subset\!X$ is a smooth isolated rational curve with $NY\!\approx\!\O(-1)\!\oplus\!\O(-1)$. We can then apply Theorem \[main\_thm\] with $S\!=\!\eset$, $g\!=\!0$, and $\be\!=\!d\io_{Y*}([Y])$ for any $d\!\in\!\Z^+$. The assumption on the normal bundle implies that $\ker(D_{J;u}^{NY})$ is trivial and thus Condition (b) is satisfied. The right-hand side of \_ref[mainthm\_e2]{} is then the famous multiple-cover contribution of $1/d^3$ ([@AsMo], [@MirSym Section 27.5], [@Voisin]).

\[inY\_eg\] If the image of each map $f_j$ in Theorem 1.2 lies in $Y$, the second part of Condition (a) is automatically satisfied. Condition (b) is equivalent to the homomorphisms $$\bigoplus_{j\in S}\ti\ev_j\!:
\ker(D_{J;u}^{NY})\lra \bigoplus_{j\in S}N_{z_j(u)}Y,
\qquad ([u],w)\in \ov\M_{g,\f}(X,\be;J),$$ being isomorphisms. For example, this is the case if $X\!=\!\P^n$, $Y\!=\!\P^1\!\subset\!X$, $S\!=\!\{1,2\}$, $g\!=\!0$, $\be\!=\!\io_{Y*}([Y])$ is the homology class of a line, $a_1,a_2\!=\!0$, and $f_1,f_2\!:pt\!\lra\!Y$ are maps to two distinct points. In this particular case, $$\ov\M_{0,\f}(X,\be;J)=\ov\M_{0,\f}(Y,\be_Y;J),$$ where $\be_Y\!=\![Y]$, and $\cok(D_J^{NY})$ is the zero vector bundle. Thus, $$\big(pt,pt\big)_{0,\be}^{\P^n}=
\big((\tau_{a_j}\ka_j)_{j\in S}\big)_{0,\be}^X
=\bC_{0,\f}(Y,\be_Y)
=\,^{\pm}\big|\ov\M_{g,\f}(Y,\be_Y;J)\big|
=\big(pt,pt\big)_{0,\be_Y}^{\P^1}
=1,$$ as expected.[^6]

\[transversetoY\_eg\] If each map $f_j$ in Theorem 1.2 is transverse to $Y$, the second part of Condition (a) is again automatically satisfied. Condition (b) is equivalent to the injectivity of the operators 