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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 $n(2-2g)+c_1(A)$

Questions: 1 Is the moduli space is non empty for any riemman surface and any N and any $[A] \in H_2(N)$ ?

2 Is there any simple way to dudge wether the moduli space is not empty? Or what about this problem when N is a four dimension manifold?

Most case I want to know is N is a four-dimensional symplectic manifold with the fundamental group infinit.

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.

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 $n(2-2g)+c_1(A)$

Questions: 1 Is the moduli space is non empty for any riemman surface and any N and any $[A] \in H_2(N)$ ?

2 Is there any simple way to dudge wether the moduli space is not empty? Or what about this problem when N is a four dimension manifold?

Sorry,there is my first time to use Tex and I donn't konw how to add the back space. I beg your pardon.

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 $n(2-2g)+c_1(A)$

Questions: 1 Is the moduli space is non empty for any riemman surface and any N and any $[A] \in H_2(N)$ ?

2 Is there any simple way to dudge wether the moduli space is not empty? Or what about this problem when N is a four dimension manifold?

Most case I want to know is N is a four-dimensional symplectic manifold with the fundamental group infinit.

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

Then we define the moduli space as

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

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

Questions: 1 Is the moduli space is non empty for any riemman surface and any N and any $[A] \in H_2(N)$ ?

2 Is there any simple way to dudge wether the moduli space is not empty? Or what about this problem when N is a four dimension manifold?

Sorry,there is my first time to use Tex and I donn't konw how to add the back space. I beg your pardon.

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 $n(2-2g)+c_1(A)$

Questions: 1 Is the moduli space is non empty for any riemman surface and any N and any $[A] \in H_2(N)$ ?

2 Is there any simple way to dudge wether the moduli space is not empty? Or what about this problem when N is a four dimension manifold?

Sorry,there is my first time to use Tex and I donn't konw how to add the back space. I beg your pardon.