McDuff–Salamon theory of J-holomorphic curves

E621122

framework for pseudoholomorphic curves mathematical theory theory in symplectic geometry

The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.

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All labels observed (4)

Label	Occurrences
J-holomorphic Curves and Symplectic Topology	3
Gromov–Witten theory	1
Gromov’s theory of pseudo-holomorphic curves	1
McDuff–Salamon theory of J-holomorphic curves canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6834194 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: McDuff–Salamon theory of J-holomorphic curves
Context triple: [Dusa McDuff, notableWork, McDuff–Salamon theory of J-holomorphic curves]

A. The geometry of four-manifolds
The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
B. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
C. Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
D. Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
E. Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: McDuff–Salamon theory of J-holomorphic curves
Target entity description: The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.

A. The geometry of four-manifolds
The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
B. Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
C. Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
D. Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
E. Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	framework for pseudoholomorphic curves ⓘ mathematical theory ⓘ theory in symplectic geometry ⓘ
appliesTo	symplectic manifolds ⓘ
associatedWith	Dietmar Salamon ⓘ Dusa McDuff ⓘ
buildsOn	Gromov theory of pseudoholomorphic curves ⓘ linked to: Gromov compactness theorem
clarifies	analytic foundations of pseudoholomorphic curve theory ⓘ compactness and bubbling phenomena ⓘ transversality and regularity issues ⓘ
contributesTo	Floer theory ⓘ construction of Gromov–Witten invariants ⓘ foundations of symplectic topology ⓘ linked to: Introduction to Symplectic Topology
develops	analysis of J-holomorphic curves ⓘ applications of J-holomorphic curves ⓘ topology of J-holomorphic curves ⓘ
documentedIn	Introduction to Symplectic Topology ⓘ J-holomorphic Curves and Symplectic Topology ⓘ linked to: McDuff–Salamon theory of J-holomorphic curves
emphasizes	Fredholm setup for Cauchy–Riemann operators ⓘ gluing constructions for curves ⓘ orientation of moduli spaces ⓘ
field	symplectic geometry ⓘ
focusesOn	J-holomorphic curves ⓘ pseudoholomorphic curves ⓘ
frameworkFor	systematic study of J-holomorphic curves in symplectic manifolds ⓘ
providesToolsFor	Gromov–Witten theory ⓘ linked to: Floer theory Hamiltonian dynamics ⓘ symplectic topology ⓘ
studies	holomorphic curves in almost complex manifolds ⓘ intersection theory in symplectic manifolds ⓘ moduli spaces of pseudoholomorphic curves ⓘ symplectic invariants ⓘ
usedFor	defining invariants of symplectic manifolds ⓘ proving symplectic non-squeezing results ⓘ studying Hamiltonian periodic orbits ⓘ studying symplectic embeddings ⓘ
usesConcept	Fredholm theory ⓘ linked to: Fredholm operator Gromov compactness ⓘ linked to: Gromov compactness theorem almost complex structures ⓘ bubbling analysis ⓘ compactness theorems ⓘ compatible almost complex structures ⓘ elliptic regularity ⓘ energy estimates ⓘ moduli spaces of curves ⓘ tame almost complex structures ⓘ transversality ⓘ

How these facts were elicited