identity \_ref[FanoGV\_e]{} follows from \_ref[P2\_e]{}.\
Theorem \[FanoGV\_thm\] confirms (a stronger version of) the Fano case of [@P2 Conjecture 2(i)], i.e. that the numbers $E_{h,\be}^X((\ka_j)_{j\in S})$ [*defined from GW-invariants by \_ref[FanoGV\_e]{}*]{} are integers. The Calabi-Yau case is fundamentally more difficult as it involves multiple covers of curves.[^8] On the other hand, it might be possible to approach [@P2 Conjecture 2(ii)], i.e. that $E_{h,\be}^X((\ka_j)_{j\in S})\!=\!0$ for a fixed $\be$ and all sufficiently large $g$ if $X$ is projective, by studying possible limits of $J_t$-holomorphic curves with $J_t\!\in\!\cJ_{\reg}(g,\be)$ as $J_t$ approaches the standard complex structure on $X\!\subset\!\P^n$ and using the Castelnuovo bound [@ACGH p116].\
An algebro-geometric approach to Theorem \[FanoGV\_thm\] has recently been proposed in [@KKO], at least in the usual, more narrow, meaning of [*Fano*]{} in algebraic geometry. The stable-map style invariants of smooth projective varieties defined in [@KKO] are a priori integers in the case of Fano varieties, just like the numbers $E_{h,\be}^X((\ka_j)_{j\in S})$. In addition, in this Fano case, they are non-negative integers and satisfy the vanishing prediction of [@P2 Conjecture 2(ii)]. However, it remains to be shown that they are related to the GW-invariants in the required way, i.e. as in \_ref[FanoGV\_e]{}.

Analytic Preliminaries {#analysis_sec}
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In this section, 