# Relative Gromov-Witten Invariants

A central issue in defining relative GW-invariants on a symplectic manifold is the possibility that a sequence of relative pseudoholomorphic curves can degenerate in such a manner, that components lie completely in the relative hypersurface V.

This issue is addressed directly by Ionel-Parker in their paper on relative GW-invarianst and also by Li-Ruan in their virtual neighborhood construction. Both come to the conclusion, that after a generic choice of almost complex structure, such behavior can be ruled out.

However, in a preprint (arXiv:0809.3534 ) by McDuff the issue arises again, and she produces an example where is appears that such a degenerate curve has non-expected dimension. Moreover, she states a criterion for the exclusion of such higher level curves in genus 0.

Is McDuff attempting to illustrate an issue not related to the general defintion of relative GW-invariants, or is there an oversight in the papers defining relative GW-invariants?

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You say:

"This issue is addressed directly by Ionel-Parker in their paper on relative GW-invarianst and also by Li-Ruan in their virtual neighborhood construction. Both come to the conclusion, that after a generic choice of almost complex structure, such behavior can be ruled out."

but I don't think that you have correctly characterized their solution to the problem. You do not get to dismiss the behavior --- instead you deal with it in a way that maintains transversality by caveat. In order to explain, it is helpful to compare the way the problem is handled in algebraic geometry and then compare it with what is done in symplectic geometry.

In algebraic geometry, if you want to have a proper moduli space of relative stable maps to $(X,D)$ and maintain the condition that components of the map are not mapped into the divisor $D$, you are forced to allow the target $(X,D)$ to degenerate a singular target $(X,D)\cup\_D F$ consisting of the union of $X$ and $F=\mathbb{P}(\mathcal{O}\oplus N\_{D/X})$ glued along the zero section of $F$ and more generally, you need to allow a finite chain of copies of $F$ glued successively along the $0$ and $\infty$ sections: $X\cup_D F\cup_D\cdots\cup_D F$.

Although it is perhaps a little hidden, this same phenomenon is forced on you in the analytic setting of Li-Ruan and Parker-Ionel. Instead of the singular target considered above, they use a target of $X/D$ with a metric that looks like an infinite very, very long tube in the neighborhood of $D$. When they analyze the limiting behavior of maps to such a space, they see a similar phenomenon cropping up: the energy of a map to the tube partitions itself into sections so that in the limit, you can view it as given by a collection of maps to a bunch of long tubes, each of which give you a map to $F$ (the point being that the infinite tube is conformally equivalent to $F$ minus the 0 and $\infty$ sections).

There is a picture of this on the top of page 5 of this paper of Parker-Ionel:

http://arxiv.org/pdf/math/0010217

They are actually illustrating the gluing of relative invariants (so that manifolds appear on both sides of the tube instead of just one side), but you can see the appearance of multiple copies of $F$.

So to answer the question: there is no oversight in those Parker-Ionel, Li-Ruan papers. The phenomenon that McDuff is illustrating is a fact of life for relative invariants, even if it is manifested a little differently in the analytic setting.

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