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:**

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

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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