Constructions of symplectic forms on 4-manifolds

2024/03/08 by Lambert-Cole, Peter
#FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2403.05512

Abstract

Given a symplectic 4-manifold (X,ω) with rational symplectic form, Auroux constructed branched coverings to (CP²,ω_FS). 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 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.

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