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Constructions of symplectic forms on 4-manifolds

Peter Lambert-Cole

Abstract

Given a symplectic 4-manifold $(X,ω)$ with rational symplectic form, Auroux constructed branched coverings to $(CP^2,ω_{FS})$. 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 $(X,ω)$ 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.

Constructions of symplectic forms on 4-manifolds

Abstract

Given a symplectic 4-manifold with rational symplectic form, Auroux constructed branched coverings to . By modifying a previous construction of Lambert-Cole--Meier--Starkston, we prove that the branch locus in can be assumed holomorphic in a neighborhood of the spine of the standard trisection of . Consequently, the symplectic 4-manifold admits a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of . 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.
Paper Structure (37 sections, 39 theorems, 85 equations, 3 figures)

This paper contains 37 sections, 39 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Construction
  4. Holomorphic trisections and Kähler-like forms
  5. Symplectic cone
  6. Symplectic 1-cocycles
  7. Comparison
  8. Acknowledgements
  9. Trisections
  10. Trisections
  11. Trisections of spheres
  12. Connected sum and stabilization
  13. Relative trisections
  14. Bridge position for surfaces
  15. Bridge perturbation
  16. ...and 22 more sections

Key Result

Corollary 1.2

Let $(X,\omega)$ be a closed, symplectic 4-manifold with rational symplectic form. There exists an almost-Kahler structure $(X,\omega',J)$ with $[\omega'] = [\omega] \in H^2_{DR}(X)$ and a smooth handle decomposition of $X$ such that $J$ is integrable over the 2-skeleton.

Figures (3)

  • Figure 1: The decomposition of Theorem \ref{['thrm:main-kahler']} applied to the moment polytope of $\mathbb{CP}^2$
  • Figure 2: Component pieces of a torus diagram for the branch locus $\mathcal{R}$
  • Figure 3: Local fundamental group calculation

Theorems & Definitions (91)

  • Conjecture 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 81 more