$\lr{c_1(TX),\be}\!<\!0$, all degree $\be$ GW-invariants are zero, since the moduli space of unmarked maps has negative expected dimension. This leaves the “Calabi-Yau" case, $\lr{c_1(TX),\be}\!=\!0$, and the “Fano" case, $\lr{c_1(TX),\be}\!>\!0$. If $g,h\!\in\!\bar\Z^+$, define $C_{h,\be}^X(g)\!\in\!\Q$ by \_[g=0]{}\^[i]{}C\_[h,]{}\^X(g)t\^[2g]{} =()\^[2h-2+]{}.

\[FanoGV\_thm\] Suppose $(X,\om)$ is a compact symplectic $6$-fold, $\be\!\in\!H_2(X;\Z)$, $g\!\in\!\bar\Z^+$, $S$ is a finite set, and $\ka_j\!\in\!H_*(X;\Z)$ for $j\!\in\!S$ are such that \_ref[dimcond\_e]{} is satisfied with $a_j\!=\!0$. If $\lr{c_1(TX),\be}\!>\!0$,

1.  there exists a dense open subset $\cJ_{\reg}(g,\be)$ of the space of smooth $\om$-tame almost complex structures on $X$ such that for all $h\!\le\!g$:

    -   the moduli space $\M_{h,S}^*(X,\be;J)$ consists of regular maps;

    -   for a generic choice of pseudocycle representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, $\M_{h,\f}^*(X,\be;J)$ is a finite set of regular pairs $([u],(w_j)_{j\in S})$ such that $u$ is an embedding;

2.  the numbers $E_{h,\be}^X(\f,J)$, with $h\!\le\!g$, are independent of the choice of $J\!\in\!\cJ_{\reg}(g,\be)$ and $f_j$ and can thus be denoted $E_{h,\be}^X((\ka_j)_{j\in S})$;

3.  if $C_{g,\be}^X(h)$ is defined by \_ref[Cdfn\_e]{}, ((\_j)\_[jS]{})\_[g,]{}\^X= \_[h=0]{}\^[h=g]{} C\_[h,]{}\^X(g-h) E\_[h,]{}\^X((\_j)\_[jS]{}).\

For $g\!=\!0,1$, \_ref[FanoGV\_e]{} gives

((\_j)\_[jS]{})\_[0,]{}\^X&=E\_[0,]{}\^X((\_j)\_[jS]{}),\
((\_j)\_[jS]{})\_[1,]{}\^X&=E\_[1,]{}\^X((\_j)\_[jS]{}) +E\_[0,]{}\^X((\_j)\_[jS]{}).

The first identity expresses the well-known fact that the genus $0$ GW-invariants of a Fano manifold are enumerative. The second identity in \_ref[lowgenusGV\_e]{} is the $n\!=\!3$ case of the relation between the standard genus $1$ 