GW-invariants of a Kahler surface $X$ of general type localize to a canonical divisor. The situation is particularly beautiful if $X$ admits a smooth canonical divisor $\K_X$. If $X$ is minimal, the GW-invariants of $X$ in degrees other than multiples of $\K_X$ vanish. The GW-invariants of $X$ in degrees $\K_X$ and $2\K_X$ are computed in [@KLi] via an algebraic reformulation of [@LeP] and shown to satisfy a conjecture of [@MaP]. In the next paragraph we review the relevant statements from [@LeP].\
Let $(X,J_0)$ be a minimal Kahler surface of general type and $\al$ the real part of a non-zero holomorphic $(2,0)$-form such that $Y\!\equiv\!\al^{-1}(0)$ is smooth (and reduced). Since $X$ is minimal, $Y$ is connected. With $\lr{\cdot,\cdot}$ denoting the Riemannian metric on $X$, define $$\begin{gathered}
K_{\al}\in \Ga\big(X;\Hom_{\R}(TX,TX)\big),  \quad
R_{\al}\in \Ga\big(Y;\Hom_{\R}(TY\!\otimes\!_{\C}NY,NY)\big),
\qquad\hbox{by}\notag\\
\label{Rdfn_e}\begin{split}
\lr{v_1,K_{\al}v_2}&=\al(v_1,v_2)~~\forall\,v_1,v_2\!\in\!T_xX,\,x\!\in\!X;\\
R_{\al}(v_1,v_2)&= J_0\big\{\na_{v_2}K_{\al}\big\}(v_1)+T_xY
~~\forall\,v_1\!\in\!T_xY,\,v_2\!\in\!T_xX,\,x\!\in\!X.
\end{split}\end{gathered}$$ By [@LeP Lemmas 2.1,8.2], $R_{\al}$ is well-defined. The almost complex structure $J_{\al}$ on $X$ described in [@LeP Section 2] agrees with $J_0$ along the smooth complex curve $Y$. By [@LeP Lemma 2.3], every non-constant $J_{\al}$-holomorphic map $u\!:\Si_u\!\lra\!X$ is in fact a $J_0$-holomorphic map to $Y$ and so lies in the homology class $dY$ for some $d\!\in\!\Z^+$. By [@LeP Section 8], the operator on 