Picard–Lefschetz theory

E912804

mathematical theory theory in algebraic geometry theory in symplectic geometry

Picard–Lefschetz theory is a branch of algebraic and symplectic geometry that studies how the topology of complex algebraic varieties changes under deformation, particularly via vanishing cycles and monodromy around singularities.

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Statements (46)

Predicate	Object
instanceOf	mathematical theory ⓘ theory in algebraic geometry ⓘ theory in symplectic geometry ⓘ
appliesTo	Lefschetz fibrations in symplectic geometry ⓘ Milnor fibers NERFINISHED ⓘ complex algebraic varieties ⓘ complex hypersurfaces ⓘ
describes	behavior of cycles near critical values of a map ⓘ change of homology basis under analytic continuation ⓘ
developedIn	20th century ⓘ
field	algebraic geometry ⓘ symplectic geometry ⓘ
focusesOn	Lefschetz singularities NERFINISHED ⓘ non-degenerate critical points ⓘ
formalism	cohomology theory ⓘ homology theory ⓘ local systems ⓘ
hasKeyResult	Picard–Lefschetz formula for monodromy on homology NERFINISHED ⓘ computation of intersection forms via vanishing cycles ⓘ description of vanishing cycles in terms of critical points ⓘ
historicalOrigin	work of Solomon Lefschetz ⓘ work of Émile Picard ⓘ
relatedTo	Gauss–Manin connection NERFINISHED ⓘ Hodge theory NERFINISHED ⓘ mirror symmetry ⓘ monodromy representation ⓘ singularity theory ⓘ variation of Hodge structure ⓘ
studies	Lefschetz fibrations NERFINISHED ⓘ Lefschetz pencils NERFINISHED ⓘ monodromy around singularities ⓘ singularities of complex hypersurfaces ⓘ topology of complex algebraic varieties ⓘ vanishing cycles ⓘ variation of topology under deformation ⓘ
usedIn	algebraic geometry ⓘ complex geometry ⓘ mathematical physics ⓘ mirror symmetry research ⓘ symplectic topology ⓘ
usesConcept	Morse theory NERFINISHED ⓘ Picard–Lefschetz formula NERFINISHED ⓘ intersection form ⓘ middle-dimensional homology ⓘ monodromy operator ⓘ vanishing cycle ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Milnor fibration → relatedTo → Picard–Lefschetz theory ⓘ

Lefschetz pencil → usedFor → Picard–Lefschetz theory ⓘ