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Let us consider a map from a $\Sigma_g \longrightarrow N$, where $N$ is a symplectic manifold.

Then we define the moduli space as

$M= \{ f | f \mbox{ is a pseudoholomorphic map } \Sigma_g \to N, f_* ([\Sigma]) = [A]\}$, where $A \in N$.

We assume the dimension of $N$ is $2n$. We can cauculate the dimension of the moduli space $M$ is $n(2-2g)+2c_1(A)$

Questions:

  1. Is the moduli space non-empty for any Riemman surface and any $N$ and any $[A] \in H_2(N)$ ?

  2. Is there any simple way to judge whether the moduli space is not empty? What about this problem when N is a four dimension manifold?

The case I want to know most is when $N$ is a four-dimensional symplectic manifold with infinite fundamental group.

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    $\begingroup$ This is not an answer, just a comment. Even when $N$ is a complex manifold these are difficult questions. Most (positive) existence results I know of are either by explicit construction (not too hard, for instance, if $N$ is a low degree hypersurface in $\mathbb{C}P^n$), or by Mori's Bend-and-Break result. Mori's technique works by passing to algebraic varieties over positive characteristic fields, so it only applies in the algebraic case. $\endgroup$
    – Jason Starr
    Commented Jul 15, 2012 at 11:08
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    $\begingroup$ Two comments: For question 1, the answer is trivially yes if you take $A = 0$ and constant holomorphic curves... is that a reasonable interpretation of the question? There is also a typo in the index formula. If you want the real dimension of $M$, the formula is $n(2-2g) + 2c_1(A)$. $\endgroup$
    – Sam Lisi
    Commented Jul 15, 2012 at 14:28
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    $\begingroup$ The simplest way to get existence for all choices of $\Sigma_g$, $g$ positive, is to first prove existence for a genus $0$ Riemann surface. Since $\Sigma_g$ has positive degree maps to the Riemann sphere, the composition will give maps of $\Sigma_g$ to $N$. $\endgroup$
    – Jason Starr
    Commented Jul 15, 2012 at 16:26
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    $\begingroup$ I would say that without more information you can't say whether there is a curve or not. Concerning your question in a comment, even assuming that N is Kaehler isn't much use: a generic K3 surface has no curves because curves represent integral (1,1)-classes in cohomology and you can generically assume that the subspace $H^{1,1}$ is irrational (though I guess by Lefschetz's theorem on (1,1)-classes this is the only obstruction for Kaehler surfaces). Do you have a specific example in mind where you want to find a holomorphic curve? $\endgroup$
    – Jonny Evans
    Commented Jul 15, 2012 at 17:32
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    $\begingroup$ Is this a new question, or have I misunderstood the original questions? If you want a sequence of manifolds $N_i$ for which there exist non-constant pseudoholomorphic curves of genus $g$, then take your favourite sequence of symplectic manifolds $X_i$ and let $N_i=X_i \times \Sigma_g$. On this, you can take a split almost complex structure, $A= [pt]\otimes [\Sigma_g]$ and the pseudoholomorphic curve will be $\{pt\} \times \Sigma_g$. $\endgroup$
    – Sam Lisi
    Commented Jul 16, 2012 at 10:43

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I am trying to assemble the answers to the question(s) that were hashed out in the comments (and also in a separate discussion with Jonny Evans). This answer is community wiki since it is the outcome of collaborative discussion. Please feel free to edit this.

Rephrasing of question: does there exist a non-constant holomorphic curve in any symplectic manifold? (presumably, for generic choice of compatible J) Can we say more in dimension 4? Are there conditions we can put on the symplectic manifold so that there exist curves?

  • A generic K3 surface (which is a symplectic 4-manifold) does not have any curves, so the answer to the question in complete generality is "no". We therefore reinterpret the question to be about finding a large class of 4-manifolds for which we can say something.
  • If we drop the non-constant condition, there are the trivial (constant) holomorphic curves. This is why we require non-constant holomorphic curves.
  • If we allow ourselves to find a $J$-holomorphic curve for a very special (not generic!) almost complex structure $J$, it suffices to find an embedded symplectic surface and then construct $J$ to make this surface $J$-holomorphic. In dimension 4, we can find a symplectic surface by finding a Donaldson divisor.
  • If there exists a $J$-holomorphic sphere in $N$, then there exist $J$-holomorphic maps from domains of all genus, by composing with a branched cover.
  • There are two obvious infinite families of examples for which we can find non-constant holomorphic curves. The first are products of symplectic manifolds with surfaces. The second family of examples is obtained by blowing up a symplectic 4-manifold.
  • Another family of examples come from 4-manifolds $(N, \omega)$ for which the Gromov-Taubes invariant is non-vanishing. For instance, if $c_1(TN) \ne 0$.
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