Intersection theory of punctured pseudoholomorphic curves

Abstract

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings. We then turn our attention to curves in the symplectization $\mathbb{R}\times M$ of a $3$-manifold $M$ admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the $3$-manifold, and we present conditions that will guarantee the projection of a curve to the $3$-manifold is an embedding. Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition, and an application concerning the existence of generalized pseudoholomorphic curves.

109 pages; updated so that section numbering matches that of published version

References in corpus (2)

• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Contact Homology of Orbit Complements and Implied Existence

Cited by in corpus (29)

• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Weak Symplectic Fillings and Holomorphic Curves
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• A dynamical characterization of universally tight lens spaces
• A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$
• Contact Homology of Orbit Complements and Implied Existence
• Higher symplectic capacities
• Holomorphic curves in the presence of holomorphic hypersurface foliations
• Proof of the simplicity conjecture
• Counting curves with local tangency constraints
• Algebraic Torsion in higher-dimensional contact manifolds
• Systems of transversal sections near critical energy levels of Hamiltonian systems in ${\mathbb{R}}^4$
• Symplectic capacities, unperturbed curves, and convex toric domains
• Lectures on Polyfolds and Symplectic Field Theory
• Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four
• Contact 3-manifolds, holomorphic curves and intersection theory
• Embedded contact homology of prequantization bundles
• Polyfold and SFT Notes I: A Primer on Polyfolds and Construction Tools
• Real holomorphic curves and invariant global surfaces of section
• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification
• Polyfold and SFT Notes II: Local-Local M-Polyfold Constructions
• A categorification of the Alexander polynomial in embedded contact homology
• Symplectic embeddings of polydisks
• Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves
• Global surfaces of section for Reeb flows in dimension three and beyond
• SFT computations and intersection theory in higher-dimensional contact manifolds
• Holomorphic curves in log-symplectic manifolds
• Hamiltonian Floer theory on surfaces
• Homological invariants of codimension 2 contact submanifolds