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I have the following question. Any input would be appreciated. If you have a counterexample, I would be very interested to know.

Question. Let $(X, \omega, J)$ be a Kahler manifold of complex dimension $2$ of general type, with $\omega$ a symplectic form and $J$ an integrable complex structure compatible with $\omega$. Let $L$ be an ample line bundle on $X$. Assume the divisor $D$ corresponding to $L$ is smooth, connected, symplectic curve with a positive self-intersection. Is there is a surjection $\pi_{1}(D) \rightarrow \pi_{1}(X)$?

Remark: Note that here I don't assume that the divisor $D$ is holomorphic. In this case the answer to my question is yes by Nori's result. I am asking if Nori's result holds in symplectic setting.

I have the following question. Any input would be appreciated. If you have a counterexample, I would be very interested to know.

Question. Let $(X, \omega, J)$ be a Kahler manifold of complex dimension $2$ of general type, with $\omega$ a symplectic form and $J$ an integrable complex structure compatible with $\omega$. Let $L$ be an ample line bundle on $X$. Assume the divisor $D$ corresponding to $L$ is connected, symplectic curve with a positive self-intersection. Is there is a surjection $\pi_{1}(D) \rightarrow \pi_{1}(X)$?

Remark: Note that here I don't assume that the divisor $D$ is holomorphic. In this case the answer to my question is yes by Nori's result. I am asking if Nori's result holds for symplectic submanifold.

I have the following question. Any input would be appreciated. If you have a counterexample, I would be very interested to know.

Question. Let $(X, \omega, J)$ be a Kahler manifold of complex dimension $2$ of general type, with $\omega$ a symplectic form and $J$ an integrable complex structure compatible with $\omega$. Let $L$ be an ample line bundle on $X$. Assume the divisor $D$ corresponding to $L$ is smooth, connected, pseudo-holomorphic curve with a positive self-intersection. Is there is a surjection $\pi_{1}(D) \rightarrow \pi_{1}(X)$?

Remark: Note that here I don't assume that the divisor $D$ is holomorphic. In this case the answer to my question is yes by Nori's result. I am asking if Nori's result holds in pseudo-holomorphic setting.

I have the following question. Any input would be appreciated. If you have a counterexample, I would be very interested to know.

Question. Let $(X, \omega, J)$ be a Kahler manifold of complex dimension $2$ of general type, with $\omega$ a symplectic form and $J$ an integrable complex structure compatible with $\omega$. Let $L$ be an ample line bundle on $X$. Assume the divisor $D$ corresponding to $L$ is smooth, connected, symplectic curve with a positive self-intersection. Is there is a surjection $\pi_{1}(D) \rightarrow \pi_{1}(X)$?

Remark: Note that here I don't assume that the divisor $D$ is holomorphic. In this case the answer to my question is yes by Nori's result. I am asking if Nori's result holds in symplectic setting.