Constructions of symplectic forms on 4-manifolds

Given a symplectic 4-manifold (X,ω) with rational symplectic form, Auroux constructed branched coverings to (CP²,ωFS). By modifying a previous construction of Lambert-Cole--Meier--Starkston, we prove that the branch locus in CP² can be assumed holomorphic in a neighborhood of the spine of the standard trisection of CP². Consequently, the symplectic 4-manifold (X,ω) 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.