I am trying to understand the construction of symplectic inflation and I am stuck in the following point.
Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \subset M$ is a 2 dimensional symplectic submanifold. Let $J$ be a tame (with respect to $\omega$) almost complex structure. Further we are given that $N$ is $J$-holomorphic for the above $J$.
Let $S(N)$ denote the symplectic normal bundle of $N$. Given a point $p$ on the the intersection of a fibre $F_p$ of $S(N)$ and the zero section are the following statements true?
1)The tangent space $T_p F_p$ is invariant under $J$.
2)Let $w=(u,v) \in T_pF_p \oplus T_p N$ and $w^\prime= (u^\prime,v^\prime)\in T_pF_p \oplus T_p N$. Then $\omega_p((u,v),(u^\prime,v^\prime)) = u^T J_p v + {u^\prime}^T J_p v^\prime$.
I can see that these statements should be true when $J$ is compatible with $\omega$, but I'm unable to show them for a tame almost complex structure.
