Let $(M, \omega)$ be a symplectic manifold (not necessary compact) and $J$ be almost complex structure which is compatible with $\omega$, then $(TM, J)$ can be regarded as a complex vector bundle over $M$. We can define the first Chern class $c_1(M, J)$ of $(TM, J)$.
My question: Does $c_1(M, J)$ depend on $J$?