perverse sheaves

perverse sheaves
Name	Perverse sheaves
Field	Algebraic geometry; Harvard University; École Normale Supérieure
Introduced	1980s
Key people	Alexander Grothendieck, Pierre Deligne, Maxim Kontsevich, Luc Illusie, Masaki Kashiwara, Joseph Bernstein, Wilfried Schmid, Robert MacPherson, Jean-Pierre Serre

perverse sheaves are an advanced class of constructible objects in the derived category of sheaves on a topological space or algebraic variety introduced to capture delicate cohomological phenomena. They encode local and global data simultaneously and provide a unifying language linking singularity theory, representation theory, and Hodge theory. Originating in the work of several mathematicians in the 1970s and 1980s, they have become central in modern research influenced by institutions such as Institut des Hautes Études Scientifiques and Princeton University.

Introduction

Perverse sheaves arise in the setting of derived categories associated to spaces studied at University of Paris VII and Harvard University by researchers building on ideas from Grothendieck's Éléments de géométrie algébrique, Deligne's Weil II, and the microlocal analysis developed in the school of Mikio Sato; they simultaneously refine constructs appearing in works of Wilfried Schmid on variations of Hodge structure and in the study of the Kazhdan–Lusztig conjectures by Joseph Bernstein and Masaki Kashiwara. The formalism connects to sheaf-theoretic methods popularized at Columbia University, Stanford University, and University of California, Berkeley and finds expression through functors akin to those used in the proof of the Riemann–Hilbert correspondence.

Definitions and Basic Properties

A perverse sheaf is defined by a set of cohomological vanishing and support conditions relative to a fixed stratification introduced in the style of Thom's isotopy theorem and stratified spaces considered by John Milnor; this uses truncation functors in a derived category following frameworks inspired by Alexander Beilinson and Joseph Bernstein. Key properties include self-duality under Verdier duality related to ideas of Jean Leray and control of cohomology stalks modeled on phenomena studied by René Thom and Hironaka's resolution of singularities as refined by Heisuke Hironaka. Perverse sheaves are abelian hearts of t-structures analogous to the t-structures considered in works at Institut Fourier and admit simple objects classified in many settings similarly to classification problems handled at Moscow State University.

Examples and Constructions

Standard examples include intersection cohomology complexes constructed in the spirit of Goresky–MacPherson intersection homology developed by Mark Goresky and Robert MacPherson, and nearby and vanishing cycle complexes that echo constructions used by Pierre Deligne in his study of monodromy in Étale cohomology. Other constructions appear in the study of Schubert varieties in flag varieties central to research at University of Chicago and Massachusetts Institute of Technology, relating to the representation-theoretic apparatus employed in proofs involving Kazhdan–Lusztig polynomials and work by David Kazhdan and George Lusztig. Examples also arise from local systems tied to monodromy problems investigated at Princeton University and University of Bonn.

Operations and Functoriality

Perverse sheaves are stable under standard derived functors such as pushforward and pullback when properness and smoothness hypotheses akin to those in theorems from École Polytechnique and University of Göttingen are satisfied; these operations are controlled by perverse t-structure considerations similar to methods by Masaki Kashiwara and Pierre Schapira. Nearby cycles and vanishing cycles provide microlocal operations that mirror microlocal analysis methods developed by Mikio Sato and applied in contexts explored at Nagoya University. The category admits convolution operations used extensively in geometric representation theory at University of Cambridge and Institut des Hautes Études Scientifiques, with coherence properties related to coherence results studied at École Normale Supérieure.

Relation to Topology and Hodge Theory

Perverse sheaves realize topological invariants such as intersection cohomology which generalized Poincaré duality phenomena studied by Henri Poincaré and refined in settings examined at University of Oxford; they bridge to Hodge theory through mixed Hodge modules developed in the program led by Wilfried Schmid and Saito's theory from Kyoto University and University of Tokyo. The interaction with Hodge structures parallels the use of spectral sequences in analyses emanating from École Normale Supérieure seminars and techniques from the study of variations of Hodge structure explored at Institute for Advanced Study.

Applications and Connections

Applications include proofs and formulations in geometric representation theory, notably in the context of the geometric Langlands program pursued at Institut des Hautes Études Scientifiques and University of Cambridge, and in the analysis of singularities relevant to mirror symmetry research at University of California, Berkeley and Kavli Institute for Theoretical Physics. They play a role in categorification projects linked to work by Maxim Kontsevich and Edward Witten and appear in descriptions of modular representation theory problems investigated at University of Chicago and Stanford University. Connections extend to arithmetic geometry themes central to research at CNRS and Princeton University.

Historical Development and Foundational Results

Foundational results crystallized in the late 1970s and 1980s through collaborations and parallel developments involving researchers affiliated with Institut des Hautes Études Scientifiques, Princeton University, École Normale Supérieure, and University of Bonn; key milestones include the formulation of perverse t-structures, intersection cohomology theorems by Goresky–MacPherson, and the Riemann–Hilbert correspondence extended by efforts of Masaki Kashiwara and Pierre Deligne. Subsequent advances tying perverse sheaves to representation theory and Hodge theory were advanced in seminars and programs at Institute for Advanced Study, Mathematical Sciences Research Institute, and Clay Mathematics Institute, influencing contemporary research across global mathematical centers such as University of Tokyo, University of Cambridge, and Harvard University.

Category:Algebraic geometry