Gromov-Witten invariants counting curves passing through two points - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:46:27Z http://mathoverflow.net/feeds/question/49971 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49971/gromov-witten-invariants-counting-curves-passing-through-two-points Gromov-Witten invariants counting curves passing through two points Mike Usher 2010-12-20T16:05:49Z 2011-01-27T05:10:17Z <p>Let us say that a closed symplectic manifold $X$ is <strong>$GW_g$-connected</strong> if there is a nonvanishing <a href="http://en.wikipedia.org/wiki/Gromov-Witten_invariant" rel="nofollow">Gromov-Witten invariant</a> of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in other words a nonvanishing invariant that formally counts (pseudo-)holomorphic curves of genus g passing through two generic points and satisfying some other constraints $\beta$ coming from $\bar{M}_{g,n}$ and $\alpha_i$ coming from other incidence conditions in $X$.</p> <p>When $g=0$ this is something like saying that $X$ is rationally connected in the algebro-geometric sense, and there's been recent work (such as <a href="http://arxiv.org/abs/1006.2486" rel="nofollow">arXiv:1006.2486</a>) relating to the question of whether the notions are the same. But in higher genus an analogous statement should fail--for instance in the product X of two elliptic curves there's a (reducible) genus two curve passing through any two points, but X is certainly not $GW_2$-connected.</p> <blockquote> <p><strong>Question</strong>: For which symplectic four-manifolds (or Kahler surfaces) $X$ does there exist g such that $X$ is $GW_g$-connected?</p> </blockquote> <p>My personal motivation for this question comes from the fact that if $X$ is $GW_g$-connected for some g then by a result <a href="http://arxiv.org/abs/math/0103195" rel="nofollow">of Lu</a> $X$ has finite Hofer-Zehnder capacity; however the question seems reasonably interesting aside from that. I restrict to dimension four here only because I expect doing so to make the question more tractable; insights into higher-dimensional cases would also be welcome.</p> <hr> <p>Here are some preliminary observations in the direction of an answer:</p> <p>It's an easy consequence of a result <a href="http://aif.cedram.org/aif-bin/item?id=AIF_1992__42_1-2_369_0" rel="nofollow">of McDuff</a> that the only symplectic four-manifolds that are $GW_0$-connected are the rational ones (i.e. those related to $\mathbb{C}P^2$ by blowups and blowdowns). </p> <p>For larger g, I've convinced myself that it's likely that any ruled surface over a curve of genus g ought to be $GW_g$-connected, though I haven't written down a careful proof--if someone knows where one can be found or knows that I'm wrong about this I'd be glad to hear about it.</p> <p>I'd expect that symplectic four-manifolds with $b^+>1$ (for complex surfaces this means $p_g>0$) should rarely if ever have this property, since they typically don't have GW invariants counting curves with nontrivial incidence constraints. In fact for Kahler surfaces with $p_g>0$ this follows from a result <a href="http://front.math.ucdavis.edu/0610.5570" rel="nofollow">of Lee and Parker</a>.</p> <p>For symplectic manifolds with $b^+=1$ which are not rational or ruled I'm not really sure what to expect. These usually have a decent supply of nontrivial Gromov-Witten invariants (as can be seen from Taubes-Seiberg-Witten theory), but it's not clear to me in general whether one should expect a nonvanishing invariant with two point constraints.</p> <p>EDIT: Since originally posting this question I looked a little more carefully at the literature on four-manifolds with $b^+=1$, and found that work of Li, Liu, and others based on Taubes-Seiberg-Witten theory is enough to show that any closed symplectic four-manifold with $b^+=1$ is $GW_g$-connected for some $g$. I've provided details of the argument in the appendix of <a href="http://arxiv.org/abs/1101.4986" rel="nofollow">this</a> preprint.</p> <p>So it seems likely that the answer to the original question is that a closed symplectic four-manifold is $GW_g$-connected iff it has $b^+=1$: the backward implication is always true (by Li-Liu), and the forward implication is definitely true if one restricts to Kahler surfaces (by Lee-Parker), and there are also many non-Kahler examples for which it can be checked. It seems a good deal harder to say anything about higher-dimensional cases. </p> http://mathoverflow.net/questions/49971/gromov-witten-invariants-counting-curves-passing-through-two-points/49992#49992 Answer by Zhiyu for Gromov-Witten invariants counting curves passing through two points Zhiyu 2010-12-20T19:32:18Z 2010-12-20T19:32:18Z <p>I am not familiar with symplectic geometry so let's assume everything here is at least K\'ahler.</p> <p>If $g=1$, then the condition $\langle [pt], [pt], \ldots \rangle^X_{1, [C]}\neq 0$ implies that the variety is uniruled, which is equivalent to $\langle [pt], \ldots \rangle^X_{0, {C}}$.</p> <p>I hope it is true that for a rationally connected fibration over a curve of any genus, your condition $\langle \beta, [pt], [pt], \ldots \rangle^X_{g, [C]}\neq 0$ is always true. And it is true when the fiber dimension is at most $2$. Basically as long as you know that there is a section which gives non-zero GW invariant, you can glue this section with curves in a general fiber which is minimal among all curves with non-vanishing GW invariant $\langle [pt], [pt], \ldots \rangle$.</p> <p>For ruled surface, what you said is true. The methods used in the paper <a href="http://arxiv.org/abs/1006.2486" rel="nofollow">here</a> certainly work.</p>