Constructions of symplectic forms on 4-manifolds

Given a symplectic 4-manifold $(X,\omega )$ with rational symplectic form, Auroux constructed branched coverings to $(C{P}^2,{\omega }_{FS})$. By modifying a previous construction of Lambert-Cole–Meier–Starkston, we prove that the branch locus in $C{P}^2$ can be assumed holomorphic in a neighborhood of the spine of the standard trisection of $C{P}^2$. Consequently, the symplectic 4-manifold $(X,\omega )$ admits a cohomologous symplectic form that is Kähler 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 Kähler subset.