  Given a symplectic 4-manifold $(X,\omega)$ with rational symplectic form,
Auroux constructed branched coverings to $(CP^2,\omega_{FS})$. By modifying a
previous construction of Lambert-Cole--Meier--Starkston, we prove that the
branch locus in $CP^2$ can be assumed holomorphic in a neighborhood of the
spine of the standard trisection of $CP^2$. Consequently, the symplectic
4-manifold $(X,\omega)$ admits a cohomologous symplectic form that is K\"ahler
in a neighborhood of the 2-skeleton of $X$. We define the Picard group of
holomorphic line bundles over the holomorphic 2-skeleton. We then investigate
Hodge theory and apply harmonic spinors to construct holomorphic sections over
the K\"ahler subset.
