Does there exist any theorem claiming that if a variety with negative first Chern class has no rational curves then every GW invariant is zero?

$\begingroup$ I fixed the grammar; I hope I have interpreted your question correctly. $\endgroup$ – Kevin H. Lin Dec 29 '09 at 15:20

1$\begingroup$ Kevin's edit is unambiguous, but I don't know whether it's what the questioner meant. Alexander? I read it this way: does a variety with negative first Chern class necessarily lack rational curves (and therefore have vanishing genuszero GW invariants)? Dmitri addresses both questions. $\endgroup$ – Tim Perutz Dec 29 '09 at 22:34

$\begingroup$ Thanks Kevin!!. Tim, Kevin interpreted my question correctly. Thanks for your answers, Dmitri. $\endgroup$ – Alexander Cruz Dec 30 '09 at 9:33
EDITED.
The first version of this answer contained correct information, but was answering not the question that was asked:))
So here is the new version of the answer.
1) If a variety with ample canonical bundle does not have rational curves then, indeed, its RATIONAL GW invariants vanish. Because GW invariants can be counted via algebraic geometry and the corresponding moduli spaces will be empty.
But this will not imply that all GW invariants vanish. Indeed, you can take minimal complex surfaces of general type with $b_+>1$. Then the GW invariant is nonzero, the canonical curve gives a nonzero contribution. But of course it has a nonzero genus.
The following is a discussion of vanishing and nonvanishing of GW invariants of varieties of general type. This was my previous answer. This is not immediately related to the question, but I decided to leave it here because it may be helpfull (for somebody).
Hypersurfaces of degree $2n1$ and less in $CP^n$ always contain lines. And for $2n1>n+1$ this will be an example of a manifold with ample canonical bundle. I think it should be possible to show that for these hypersurfaces their rational GW invariants are nonzero, though I am not aware if such a calculation has been done for all these examples.
There was a related discussion here:
Why is a variety of general type hyperbolic?
Nethertheless "morally" it is ture that for a large part of varieties of general type of dimesnion at least 4 with ample canonical bundle GW invarinats vanish. For example, it is conjectured that a "generic" hypersurfaces of degree $d\ge 2n+1$ in $CP^n$ does not contain rational curves (you need generic here becasue for every d there will be a hypersurface that contains a line). This is a hard conjecture. But it implies that rational GW invariant of such varieties vanishes.
A different thing that can be said about varieties of general type is that they are not projectively unirulled and this can be seen as vanishing of certain rational GW invariants. This follows from a result of Kollar and Ruan. You can see the discussion on page 4 of the following paper:
SYMPLECTIC BIRATIONAL GEOMETRY