TITLE: Computation of Gromov-Witten invariants for symplectic manifolds
QUESTION [12 upvotes]: According to references, Gromov-Witten invariants were first defined for symplectic manifolds and later for projective varieties algebraically, and they coincide on the overlap. Because I thought Gromov-Witten invariants are important invariants for symplectic manifolds, I looked up papers about the computation. But recent results I found were about algebraic ones. Also simple examples calculated in textbooks don't seem to use symplectic structures.
I wonder why people don't do computations in symplectic category. Is it because of the difficulty, or are there some other reasons? Maybe I am just ignorant about it. I am more familiar with symplectic geometry, so I want to know methods to compute the invariant symplectically. In particular, I want to know if explicit computation was done for some non-K\"ahler manifolds. And is it recommended to study albebraic techniques of computation even if what I am interested in is symplectic manifold?

REPLY [4 votes]: I noticed this question a few weeks ago but waited to post an answer before I could say: here's a new computation (http://arxiv.org/abs/1106.3959) for a class of non-Kaehler manifolds.
Psychologically one reason that the algebraic category is easier is that gluing works more easily there. For instance, Robbin-Ruan-Salamon have a paper (http://www.math.ethz.ch/~salamon/PREPRINTS/smUW.pdf) that puts a smooth orbifold structure on the (compactified) moduli space of regular stable maps into an integrable almost complex manifold (without using any of the standard gluing techniques). One feels happier about using e.g. equivariant localisation there than in the non-Kaehler case where the smooth structure on any given moduli space (and its obstruction bundle) is maybe not so canonical.
Of course, as the other posters point out, the main reason there are few computations is that it's not so easy to find semipositive (e.g. monotone) non-Kaehler manifolds (for which one can use the "standard" definitions of GW invariants due to Ruan-Tian/McDuff-Salamon).