Computation of Gromov-Witten invariants for symplectic manifolds

Question

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?

Answer 1

There are some very good reasons why the majority of calculations are done for algebraic manifolds. Maybe the most naive reason is as follows: it is harder to solve PDEs than to draw lines through two points in a space. Somehow everyone knows that for two points in $\mathbb CP^n$ there is exactly one line that passes through them, and so a certain GW invariant equals $1$. From the point of view of symplectic geometry this is a completely non-trivial result. Indeed, if you take the standard symplectic structure $w$ on $\mathbb CP^n$ (coming from the Fubini-Study metric), take $J$ that is tamed by $w$ and try to make the calculation, you will first need to know that almost complex curves exist for $J$ locally (which is already non-trivial), then using different compactness arguments will need to homothopy $J$ to the standard complex structure and prove that GW invariant does not change during homothopy, and finally once you get the standard complex structure on $\mathbb CP^n$ perform the above elementary calculation to get $1$ line through any two points. Gromov non-squeezing http://en.wikipedia.org/wiki/Nonsqueezing_theorem is also based on this idea, so all the above is a completely non-trivial task...

A different reason that majority of calculations are done in algebraic case is that up to recently majority of compact symplectic manifolds we constructed from algebraic pieces, by taking Gompf's sum. In this case again on can try first to calculate GW invariants for these pieces and then glue them. Also in small dimensions symplectic manifolds with sufficiently non-trivial GW invariants tend to be algebraic. For example, in dimension 4 it is conjectured that if a GW invariant of rational curves passing trough one point is non-zero then the 4-manifold is a unirulled or rational surface (in particular, it is algebraic).

If you want an explicit calculation for manifolds that are not Kahler, check the following article of McDuff "Hamiltonian S^1 manifolds are uniruled" http://arxiv.org/abs/0706.0675 . She proves non-vanishing of certain GW invariants for manifolds admitting Hamiltonian $S^1$ action and it is not hard to construct such non-Kahler manifolds.

More generally, Tien-Jun Li and Ruan argue in "Symplectic Birational Geometry" http://arxiv.org/abs/0906.3265 that rational GW invariants are responsible for birational geometry of symplectic manifolds.

Answer 2

There are lots of computations of the Seiberg-Witten invariants for non-complex symplectic 4-manfiolds. Using Taubes' SW=GW theorem, these give many, many computations of GW invariants of symplectic (non-complex) manifolds. Moreover, the techniques often heavily use the symplectic category (for example Fintushel and Stern's rational blow-down operation is often applied in situations where the requisite configuration of curves exists symplectically, but not algebraically).

Nevertheless, it is true that the majority of the computations in the GW theory are in the algebraic category. My philosophy on this (not incompatible with the previous answer) is that flexible categories (like symplectic manifolds) and rigid categories (like complex algebraic varieties) have different and complimentary strengths and weaknesses: in a flexible category there may be more freedom to perturb your problem to a "generic" situation with good properties, whereas in a rigid category, there might be less freedom to perturb to a generic situation, but more tools to deal with non-generic situations. The former is usually better for proving rather general theorems, whereas the later is usually better for computation of specific examples.

Answer 3

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).