the first stage respects the submanifold. Carleman Similarity Principle is used to take advantage of properties of solutions of Cauchy-Riemann equations that are preserved by a large class of perturbations of the equations. In a forthcoming paper [@divisorGWs], we will apply similar geometric principles to study relative GW-invariants and the absolute/relative correspondence in genus $0$ with applications to birational geometry in the spirit of Hu-Li-Ruan ([@HuLtR], [@HuR], [@LtR]) and McDuff ([@Mc]).\
The author would like to thank R. Pandharipande for bringing the “Fano case" of the Gopakumar-Vafa prediction to the author’s attention, D. McDuff for detailed comments and suggestions on an earlier version of this paper, and T. Graber, T.-J. Li, D. Maulik, and Y. Ruan for related discussions.

A comparison theorem for GW-invariants {#AbsGW_subs}
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We will denote by $\bar\Z^+$ the set of non-negative integers. Let $(X,\om)$ be a compact symplectic manifold. If $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, and $J$ is an $\om$-tame[^2] almost complex structure on $X$, denote by $\ov\M_{g,S}(X,\be;J)$ the moduli space of equivalence classes of stable $S$-marked genus $g$ degree $\be$ $J$-holomorphic maps to $X$. For each $j\!\in\!S$, there is a well-defined evaluation map \_j: \_[g,S]{}(X,;J)X.As standard in GW-theory, we will denote by $$\psi_j\in 