intersection cohomology

intersection cohomology
Name	Intersection cohomology
Field	Algebraic topology, Algebraic geometry
Introduced by	Mark Goresky and Robert MacPherson
Introduced in	1980
Keywords	perverse sheaf, stratification, Poincaré duality

intersection cohomology.

Intersection cohomology is a refinement of cohomology theory developed to extend Poincaré duality and related invariants to singular spaces such as complex algebraic varieties, analytic spaces, and stratified pseudomanifolds. It was introduced to resolve failures of classical Poincaré duality on singular spaces and has deep ties to sheaf theory, representation theory, and mathematical physics, influencing research in Hodge theory, Langlands program, and the study of singularities.

Introduction

Intersection cohomology provides groups, typically denoted IH^*(X), that recover ordinary cohomology for smooth manifolds while producing corrected invariants for singular varieties such as algebraic variety, Schubert variety, or complex analytic spaces encountered in settings like the Riemann–Hilbert correspondence and the Decomposition Theorem. The theory is formulated using stratifications inspired by work in Morse theory, Whitney stratification, and Thom–Mather theory, and it interfaces with categorical concepts from derived category approaches used in Algebraic geometry and Representation theory.

Historical development and motivation

Intersection cohomology was developed by Mark Goresky and Robert MacPherson in the late 1970s and published in 1980 to address pathologies discovered in examples like singular algebraic curves and Schubert cell closures in flag varietys. Early motivations drew on the failure of classical invariants exemplified in work by Poincaré, Lefschetz, and later studies by Hironaka on resolution of singularities and Deligne on Weil conjectures and l-adic cohomology. The theory quickly connected to the geometry of permutation group actions on flag manifolds, the topology of nilpotent cones studied by George Lusztig, and the categorical frameworks later formalized by Pierre Deligne and Jean-Pierre Serre.

Basic definitions and examples

Intersection cohomology is defined for a stratified space X with a fixed stratification X = ⋃ S_i; one fixes a perversity function p (often middle perversity) and imposes chain conditions on allowable chains meeting strata of codimension k. Classical examples where intersection cohomology yields new information include singular projective varietys, cones over manifolds such as the cone on a complex projective space or the affine cone over a Schubert variety, and quotient spaces like orbifolds arising in Calabi–Yau degenerations studied in mirror symmetry. Computations on simple examples illustrate how IH^* corrects Betti number anomalies that appear in singular cohomology for spaces encountered in moduli space constructions such as moduli of vector bundles and compactified spaces in Geometric Invariant Theory.

Intersection homology and duality

Intersection homology groups IH_* provide a homology theory dual to intersection cohomology, restoring forms of Poincaré duality for oriented pseudomanifolds. The duality pairs IH_k(X) with IH^{n-k}(X) for an n-dimensional oriented space under suitable conditions, generalizing classical results used by Hodge in smooth contexts. The duality and nondegenerate pairings play key roles in proofs of the Decomposition Theorem for proper maps between varieties, which was central to breakthroughs by Beilinson, Bernstein, Deligne, and Gabber in the theory of perverse sheaves and their applications to Representation theory of reductive groups such as GL_n and SL_n.

Perverse sheaves and sheaf-theoretic formulation

A sheaf-theoretic formulation identifies intersection cohomology as hypercohomology of the intersection complex IC_X, an object in the derived category of constructible sheaves that is a prototype of a perverse sheaf. The language of perverse sheaves situates IH^* inside the formalism developed by Alexandre Beilinson, Joseph Bernstein, and Pierre Deligne and connects to the Riemann–Hilbert correspondence via work by Masaki Kashiwara and Zoghman Mebkhout. This approach links intersection cohomology to categorical phenomena in Geometric Langlands program contexts and to equivalences exploited in the study of D-modules on flag varietys and singular supports studied by Microlocal analysis experts.

Computation techniques and invariants

Computational tools include Mayer–Vietoris arguments adapted to stratifications, spectral sequences such as the Deligne spectral sequence, and the use of decomposition results for proper maps like the Decomposition Theorem. Intersection cohomology groups are often computed using stratified Morse theory, resolution of singularities techniques by Hironaka, and algorithmic approaches in Schubert calculus where combinatorial invariants like Kazhdan–Lusztig polynomials (studied by David Kazhdan and George Lusztig) encode dimensions of intersection homology groups for Schubert varieties. Other invariants interacting with IH^* include mixed Hodge structures due to Pierre Deligne and L^2-cohomology methods developed in analytic settings by Jeff Cheeger and Mark Stern.

Applications and connections to other areas

Intersection cohomology appears in diverse applications: in Representation theory via the proof of the Kazhdan–Lusztig conjecture relating characters of Bernstein–Gelfand–Gelfand category O to intersection cohomology of Schubert varieties; in Algebraic geometry through the Decomposition Theorem used in the study of perverse filtrations on cohomology of maps like the Hitchin fibration studied by Nigel Hitchin; in Number theory via intersection cohomology of arithmetic quotients related to automorphic forms and the cohomology of Shimura varietys; and in mathematical physics through applications to mirror symmetry and topological field theories influenced by work of Edward Witten and Maxim Kontsevich.

Category:Algebraic topology Category:Algebraic geometry Category:Representation theory