An Introduction to Intersection Homology Theory (Hardcover, 2)

INTRODUCTIONPoincare dualityMorse theory for siningular spac esde Rham cohomology and L2-c -cohomol ology The cohomology of pr projective vari ties REVIEW OF HOMOLOGY AND COHOMOLOGY Simplicial homologySingular homology Homology with close closed supportConclusion Further reading REVIEW OF SHEAF COHOMOLOGY AND DERIVED CATEGORIES Sheaves Cech cohomology of sheavesHypercohomologyFunctors and exactnessResolution of sheaves of complexes Cohomology and hypercohomology via derived functors Derived categories. Right derived functorsFurther reading. THE DEFINITION OF INTERSECTION HOMOLOGY Stratified spaces and pseudomanifolds Simplicial intersection homologySingular intersection homologySimple examples of intersection homologyNormalisati ons Relative groups and the Mayer-Vietoris sequence. The intersection homology of a cone Functoriality of intersection homologyHomology with local coefficientsQuasi-projec tive complex varieties Further readingWITT SPACES AND DUALITY Generalised Poincare duality. Witt spacesSignatures of Witt spacesThe Witt-bordism groups Further readingL2- COHOMOLOGY AND INTERSECTION ON COHOMOLOGY L2-cohomology and Hodge theory The L2-cohomology of a punctured cone Varieties with isolated conical singularities Locally symmetric varietiesFurther reading. SHEAF-THEORETIC INTERSECTION HOMOLOGYSheaves of singular chainsConstructibility and an axiomatic characterisationThe topological invariance of intersection homologyDuality in the derived categoryFurther readingPERVERSE SHEAVESPerverse sheavesPerverse sheaves on varietiesNearby and vanishing cyclesThe decomposition theoremFurther readingTHE INTERSECTION COHOMOLOGY OF FANSAffine toric varietiesToric varieties from fansMaps and torus actionsProjective toric varieties and convex polytopesStratifications of toric varietiesSubdivisions and desingularisationsEquivariant intersection cohomologyThe intersection cohomology of fansStanley's conjecturesFurther readingCHARACTERISTIC p AND THE WEIL CONJECTURESStatement of the Weil conjecturesBasic properties of ,-adic cohomologyEtale topology and cohomologyThe Weil conjectures for singular varietiesFurther readingD-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCEThe Riemann-Hilbert problemDifferential systems over CnDx-modules and intersection homologyThe characteristic variety of a Dx-moduleHolonomic differential systemsExamples of characteristic varietiesLeft and right Dx-modulesRestriction of Dx-modulesRegular singularitiesThe Riemann-Hilbert correspondenceFurther readingTHE KAZHDAN-LUSZTIG CONJECTUREVerma modulesD-modules over flag manifoldsCharacteristic pHecke algebras and the Kazhdan-Lusztig polynomialsFurther readingBibliographyIndex