@Tim Yes, this is exactly my question!

@Misha My question was about the symplectic submanifolds of Kahler surface. I think I made this very clear in my post.

@inkspot Thanks. Very good point! No, I assume the torus $T$ is a symplectic submanifold of $X$.

@Misha Sorry, it is not clear to me where you used the fact that $X$ is a ball quotient when you relaize $\eta$ as a composition. Consider the following example: Let $X = E(1)$, an elliptic surface obtained by blowing up a pencil of cubics in $CP^2$ at $9$ base points of the pencil. Note that the class of the fiber torus is non-zero, and the loops of the fiber torus are nullhomotopic. $X$ is not a ball quotient in this case.

Thanks Misha. How you define the composition?

Yes, BS that is what exactly I am asking. THANK YOU! According to the page 389 of Gompf and Stipsicz an embedded surface $\Sigma$ is a symplectic submanifold if and only if there is a compatiable almost complex structure $J$ such that $\Sigma$ is pseudo-holomorphic. I know there are examples of Kahler surfaces such as $\mathbb{T} \times \mathbb{T}$ which contain a symplectic but non-complex connected curves. For example, there is a connected symplectic surface which represent the homology class $n(\mathbb{T} \times {pt})$ (for any $n \geq 2$), but there exist no such holomorphic curve.

@Jason I am using the definition on page 389 of R. Gompf and A. Stipsicz "4-Manifolds and Kirby calculus". It defines pseudo-holomorphic submanifold on the almost-complex 4-manifold $X$ as a real 2-dimensional submanifold $\Sigma$ such that if $J$ maps the tangent bundle of $\Sigma$ into itself. If you look the same page, it remarks that if $X(\omega, J, g)$ is an almost-Kahler manifold then a pseudo-holomorphic submanifold $\Sigma$ is alwyas a symplectic submanifold. I didn't know that pseudo-holomorphic should be holomorphic as you remark (in Kahler case).

Yes, I assume $X$ is compact. The torus $T$ is a submanifold of $X$, and I consider the subgroup of $\pi_{1}(X)$ generated by the loops on $T$.