the normal bundle $NY$ of $Y$ induced by the linearization of the $\dbar_{J_{\al}}$-operator for maps to $X$ at such a map $u$ is given by D\_[J\_;u]{}\^[NY]{}=\_[u\^\*NY]{}+R\_(du,): L\^p\_1(\_u;u\^\*NY) L\^p(\_u;T\^\*\_u\^[0,1]{}\_u\^\*NY),where $\dbar_{u^*NY}$ is the $\dbar$-operator in the holomorphic bundle $u^*(NY,J_0)\!\lra\!\Si_u$. By [@LeP Proposition 8.6], $D_{J_{\al};u}^{NY}$ is injective. In light of Theorem \[main\_thm\], Corollary \[LeP\_crl\] below is thus simply a re-formulation of the main conclusion of [@LeP].

\[LeP\_crl\] Suppose $(X,J_0)$ is a minimal Kahler surface of general type, $\al$ is the real part of a non-zero holomorphic $(2,0)$-form such that $Y\!\equiv\!\al^{-1}(0)$ is smooth, $g\!\in\!\bar\Z^+$, $d\!\in\!\Z^+$, $S$ is a finite set, $S_2\!\subset\!S$, $a_j\!\in\!\bar\Z^+$ for each $j\!\in\!S$, and $\ka_j\!\in\!H_2(X;\Z)$ for each $j\!\in\!S_2$. If $R_{\al}$ is defined by \_ref[Rdfn\_e]{}, then the cokernels of the operators \_ref[DSurf\_e]{} form a natural oriented vector orbi-bundle $$\cok\big(D_{\al}^{NY}\big) \lra \ov\M_{g,S}(Y,dY)$$ and $$\begin{split}
&\big((\tau_{a_j}\ka_j)_{j\in S_2},(\tau_{a_j}1)_{j\in S-S_2}\big)_{g,d\K_X}^X\\
&\qquad
=\bigg(\prod_{j\in S_2}\lr{c_1(T^*X),\ka_j}\bigg)
\bigg\lan e\big(\cok(D_{\al}^{NY})\big)\!\prod_{j\in S_2}\!\!\!\big(\ev_j^*PD_Y(pt)\big)
\prod_{j\in S}\!\psi_j^{a_j},
\big[\ov\M_{g,S}(Y,dY)\big]^{vir}\bigg\ran.
\end{split}$$

The Fano case of the Gopakumar-Vafa prediction {#FanoGV_subs}
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GW-invariants are generally not integers. On the other hand, at least in the case of projective $3$-folds (symplectic $6$-manifolds), certain combinations of them are believed to be integers. Ideally these combinations would be precisely counts of curves of fixed genus and degree and passing through appropriate constraints. A 