I am reading the proof of the theorem by Donaldson that we can build a Lefschetz pencil on a given symplectic manifold. The idea is to try to mimic what happens in the Kähler case, when one uses holomorphic sections of a line bundle to define a projective embedding, and then cut the manifold there with appropriate hyperplane sections.
However, the lack of a complex structure on the symplectic manifold $(V,\omega)$ can be bypassed if we are able to say when a section $s$ of a complex line bundle $p:L\to V$ is "approximately holomorphic". The line bundle $L$ that we consider is determined by the fact that its first Chern class corresponds to an integral lift of the symplectic form $\omega$ (which is assumed to exist). A compatible choice of metric $g$ on V and almost-complex structure $J$ gives a way of writing the covariant derivative of $s$ (for a suitable connection on $L$ characterized by its curvature being $-i\omega$) as $$\nabla s=\partial s + \bar{\partial}s$$ via a splitting in the complex-linear and complex-anti-linear parts. The idea that "holomorphic sections have $\bar{\partial}s=0$" is translated in the symplectic case as "the norm of $\bar{\partial}s$ is very small". Now my question is: $\fbox{what should be the natural norm to consider? }$ Note that the section $s$ is a section of $L$, while $g$ is a metric for the bundle $TV$. How can we come up with an interesting bundle metric on $L$?
I am not very practical with tools coming from differential and complex geometry, so maybe this can sound just as an easy question. In the paper "Lefschetz pencils on symplectic manifolds" it is said that:
"We can define the norms $||s||$, $||\bar{\partial}s||$ in the usual way (???), using the metric g, and its Levi-Civita connection."
Thank you for any suggestion or reference!