# almost holomorphic line bundles

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?

I think you actually have to define the triple $(J_L,J_M,\nabla)$ to be an almost holomorphic structure on $L\to M$ if there are local nonzero holomorphic sections everywhere, since the definition of 'holomorphic section' depends on $\nabla$ as well as the choice of $J_L$ and $J_M$.
Also, your choice of $J_L$ on the 'symplectic' bundle $L\to M$ reduces its structure to an $\mathrm{SO}(2)$-structure, so the $2$-plane bundle $L$ with this structure is just an oriented Euclidean $2$-plane bundle. Any two reductions of structure of an oriented $2$-plane bundle from $\mathrm{GL}_+(2,\mathbb{R})$ to $\mathrm{SO}(2)$ are equivalent, so there's really no information in that. It's the $\mathrm{SO}(2)$-connection $\nabla$ and the almost complex structure $J_M$ that carry all of the geometry, so you really should be defining your 'almost holomorphic structure in terms of the pair $(J_M,\nabla)$.
Now, when the dimension of $M$ is greater than $2$ and $J_M$ is generic, then there are no nonconstant local holomorphic functions on $M$. This means that, if you have two nonvanishing 'holomorphic' sections of $(L,\nabla)$, then, on overlaps, the ratio of the two (which must be a local holomorphic function), will have to be constant, and hence the bundle $L$ will have a flat connection. This will imply that the Euler class of the $2$-plane bundle is zero, so that $L$ will be a trivial bundle if $M$ is simply connected.
In special cases, when $(M, J_M)$ admits nonconstant local holomorphic functions, you can get nontrivial bundles $L$, but that's rare.
• Actually, no. The Chern class of $L$ is only a cohomology class in $H^2(M)$; even a choice of $J_L$ does not yield a well-defined $2$-form until one chooses a compatible connection and computes its curvature. Without a more precise definition of what you mean by 'good', I don't think that your question is really answerable. Mar 10, 2014 at 14:48