Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex structure ($J_L$ and $J_M$, respectively) such that $g_L=\omega_L(\cdot,J_L\cdot)$ and $g_M=\omega_M(\cdot,J_M\cdot)$ are hermitian metrics on the $L$ and $TM$ respectively.

Fora given pair $(J_L,J_M)$ consider a hermitian connection $\nabla$for $g_L$ (i.e. $\nabla g_L=0$, $\nabla J_L=0$), and for a local section $s\in \Gamma(L)$ define $$ \nabla^{0,1} s= \frac{\nabla s + J_L (\nabla s) \circ J_M}{2}, $$ i.e. $$ \nabla^{0,1}_v s= \frac{\nabla_v s + J_L \nabla_{J_M(v)} s}{2}. $$ We say $s$ is holomorphic if $\nabla^{0,1} s=0$.

We say $(J_L,J_M)$ is an almost holomorphic structure on $L\to M$, if non-zero local holomorphic sections exit everywhere.

Question: What are the obstructions on $L\to M$ toward the existence of an almost holomorphic structure?