DOCSLIB.ORG

Topological Quantum Field Theory for Character Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topological Quantum Field Theory for Character Varieties Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Teor´ıasTopol´ogicasde Campos Cu´anticos para Variedades de Caracteres Angel´ Gonzalez´ Prieto Directores Prof. Marina Logares Jim´enez Prof. Vicente Mu~nozVel´azquez Tesis presentada para optar al grado de Doctor en Investigaci´onMatem´atica Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Angel´ Gonzalez´ Prieto Advisors: Prof. Marina Logares and Prof. Vicente Mu~noz Thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Research Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 \It's not who I am underneath, but what I do that defines me." Batman ABSTRACT Topological Quantum Field Theory for Character Varieties by Angel´ Gonzalez´ Prieto Abstract en Espa~nol La presente tesis doctoral est´adedicada al estudio de las estructuras de Hodge de un tipo especial de variedades algebraicas complejas que reciben el nombre de variedades de caracteres. Para este fin, proponemos utilizar una poderosa herramienta de naturaleza ´algebro-geom´etrica,proveniente de la f´ısicate´orica,conocida como Teor´ıaTopol´ogicade Campos Cu´anticos (TQFT por sus siglas en ingl´es). Con esta idea, en la presente tesis desarrollamos un formalismo que nos permite construir TQFTs a partir de dos piezas m´assencillas de informaci´on: una teor´ıade campos (informaci´ongeom´etrica)y una cuantizaci´on(informaci´onalgebraica). Como aplicaci´on,construimos una TQFT que calcula las estructuras de Hodge de variedades de representaciones y la usamos para calcular expl´ıcitamente los polinomios de Deligne-Hodge de SL2(C)-variedades de caracteres parab´olicas. Palabras clave: TQFT, variedades de caracteres, estructuras de Hodge, espacios de m´oduli, GIT. Abstract in English This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro- geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs from two simpler pieces of data: a field theory (geometric data) and a quantisation (algebraic data). As an application, we construct a TQFT computing Hodge structures on representation varieties and we use it for computing explicity the Deligne-Hodge polynomials of parabolic SL2(C)-character varieties. Key words: TQFT, character varieties, Hodge structures, moduli spaces, GIT. 2010 Mathematics Subject Classification. Primary: 57R56. Secondary: 14C30, 14D21, 14L24. Acknowledgements El camino hacia la consecuci´ondel Doctorado en Investigaci´onMatem´aticaque culmina con esta tesis ha sido largo, complejo y lleno de curvas. Sin embargo, de ninguna manera puedo decir que haya sido una senda dura o desagradable. Durante los tres a~nosen los que he realizado este doctorado he sido, simple y llanamente, feliz. Me he podido dedicar en cuerpo y alma a lo que siempre hab´ıa deseado y, lejos de defraudarme, ha sido una experiencia enormemente enriquecedora y, por qu´eno decirlo, divertida. Sin embargo, la causa ´ultimade toda esta felicidad no es sino la gente que me ha rodeado a lo largo de todo este proceso y, sin cuya ayuda y apoyo, esta tesis no habr´ıasido posible. Normalmente en este tipo de escritos se suele argumentar el manido \no mencionar´enombres para evitar omisiones". Yo creo que me arriesgar´e. En primer lugar, quiero dedicar esta tesis a mis padres. Durante toda mi vida, ellos han sido un gran apoyo, sustento y acicate para crecer y progresar. Si decid´ıhacerme cient´ıficofue gracias a ellos. De mis padres aprend´ıque la esencia del ser humano es cuestionarse cosas y que la cultura nos hace libres. Ellos me ense~naronque la oportunidad de aprender es la mayor fortuna que se puede tener. No pod´ıafaltar aqu´ıuna menci´ona todos mis amigos que, de una u otra manera, me han acompa~nado y empujado a lo largo de este viaje. Gracias a Adri, Alberto, Nico y Nuria por esas fant´asticascervezas en el Ferm´ınenvueltas en risas y seguidas de una (mala) pel´ıcula de terror. Tambi´enquiero dar las gracias a mis amigos murcianos (de coraz´on,solo dos de ellos de nacimiento) Germ´an, Jaime, Luis, Miguel y Sara. Y a mi cu~nadaCristina y a mis suegros, Juanjo y Mar´ıa,porque siempre me han apoyado y me han hecho sentir uno m´asde la familia. Y, por supuesto, no me puedo olvidar de Elena, Kike y Tania por todas esas tardes en Principe P´ıo,probablemente en el Burger King, en las que las horas se nos van volarnos sin darnos ni cuenta. Pero no solo he contado con un gran apoyo fuera de la universidad, sino tambi´endentro de ella. Por eso, tambi´enquiero dedicar esta tesis a Angel´ Ramos, Antonio Br´u,Benjamin Ivorra, David G´omez- Castro y Susana G´omezpor todas esas interesantes charlas, buenos consejos y risas en tantas y tantas comidas. Muchas gracias de todo coraz´on. Gracias a vosotros, ir a la universidad cada d´ıaha sido fant´astico. Tambi´enquiero dedicar unas l´ıneas de estos agradecimientos a mis compa~nerosde departamento Giovanni, Juan Angel´ y Luc´ıa,por tantas largas conversaciones matem´aticasy por siempre ayudarme a responder todas mis preguntas sobre spinores, ´algebras de Clifford y operadores de Dirac. Vosotros me hab´eisense~nadoque la matem´aticano debe ser una actividad solitaria y que la verdadera comprensi´on solo se alcanza cuando se discute sobre ella. Y, por supuesto, tambi´enquer´ıadedicar esta tesis a Antonio Vald´espor todos estos a~nosque hemos compartido mientras le ayudaba en la docencia. Quiero darte las gracias sinceramente porque siempre me has escuchado, valorado y tenido muy en cuenta mi opini´on. A tu lado nunca me he sentido un mero ayudante, sino un profesor m´asy eso me ha permitido encontrar en la docencia una verdadera vocaci´on.Nuestros tes con porra y nuestras vi comidas de fin de cuatrimestre son siempre un aut´entico placer y en ti he encontrado no solo un colega sino tambi´en un amigo. I would like to thank the reviews of this thesis, David Ben-Zvi and Peter Newstead, for their careful reading of the manuscript. Their very helpful comments and constructive suggestions have improved the exposition of this thesis. I would like to specially thank Thomas Wasserman. I am greatly indebted for your invaluable help through the development of this thesis. It was you who taught me all the secrets of TQFTs and who gave me new tools and perspectives for addressing the problems I had found. I really enjoy all our talks about maths, physics and life in general. This thesis would have been impossible without your help and support. Por descontado, tambi´enme gustar´ıaagradecer a mis directores, Marina Logares y Vicente Mu~noz, todo el apoyo, cari~noy confianza que han depositado en m´ıdurante todos estos a~nos. Sinceramente, creo que no podr´ıahaber tenido mejores gu´ıaspara esta ruta. Gracias por haber escuchado con gran atenci´ontodas y cada una de mis ideas, por alocadas que fuesen, y por siempre haber sido capaces de extraer algo bueno de ellas. Nunca me hab´eisdesalentado y siempre me hab´eisanimado a no desistir de mis intentos por atacar un problema. Creo sinceramente que eso es lo que distingue a un mero supervisor de un buen director. Ojal´apueda estar a la altura cuando me llegue el momento de ser yo el director. Cuando os conoc´ı,podr´ıasaber m´aso menos matem´aticas(m´asbien menos), pero vosotros me hab´eisense~nadoa ser Matem´atico,con may´usculas.Sois mucho m´asque mis directores, sois mis maestros. Finalmente, mis ´ultimasl´ıneasde esta secci´onson, como no podr´ıaser de otro modo, para Laura. Gracias. Gracias por todo tu amor, tu apoyo, tu cari~noy tu comprensi´ondurante toda la realizaci´onde esta tesis (y antes). T´uhas sido mi energ´ıa,mi motor y mi fuerza para seguir estudiando, trabajando y avanzando durante todo el doctorado. T´usiempre confiaste en m´ı,incluso cuando ni yo mismo lo hac´ıay los teoremas no sal´ıan. Siempre cre´ısteen m´ı. Simplemente, gracias por existir y por estar a mi lado. Contents Abstract iv Acknowledgements vi Table of Contents viii Introduction xii Introducci´on xxii 1 Topological Quantum Field Theories1 1.1 Review of category theory and TQFTs...........................1 1.1.1 Monoidal categories.................................4 1.1.2 Duality properties and low dimensional classification of TQFTs.........6 1.1.3 Lax monoidality and 2-categories.......................... 10 1.2 Lax monoidal Topological Quantum Field Theories.................... 14 1.2.1 C-algebras....................................... 14 1.2.2 Quantisation..................................... 19 1.2.3 Field theory...................................... 21 1.3 Other versions of Topological Quantum Field Theories.................. 26 1.3.1 TQFTs over a sheaf................................. 26 1.3.2 Almost-TQFTs and the category of tubes..................... 32 1.3.3 Reduction of a TQFT................................ 34 1.3.4 Soft TQFTs...................................... 37 2 Hodge Theory 41 2.1 A panoramic view of K-theory............................... 41 2.1.1 Review of derived categories............................. 43 2.1.2 Grothendieck group................................. 46 2.2 Hodge structures....................................... 50 2.2.1 Pure Hodge structures................................ 50 2.2.2 Mixed Hodge structures............................... 53 2.3 Mixed Hodge modules.................................... 57 2.3.1 Riemann-Hilbert correspondence.......................... 57 2.3.2 Saito's theory of mixed Hodge modules...................... 64 2.4 Monodromy as mixed Hodge module...........................
Load more

Recommended publications
  • Perverse Sheaves
    Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see Kleiman's paper entitled \The development/history of intersection homology theory". On manifolds, the idea is that you can intersect cycles via Poincar´eduality|we want to be able to do this on singular spces, not just manifolds. Deligne figured out how to compute intersection homology via sheaf cohomology, and does not use anything about cycles|only pullbacks and truncations of complexes of sheaves. In any derived category you can do this|even in characteristic p. The basic summary is that we define an abelian subcategory that lives inside the derived category of constructible sheaves, which we call the category of perverse sheaves. We want to get to what is called the decomposition theorem. Outline of Course 1. Derived categories, t-structures 2. Six Functors 3. Perverse sheaves—definition, some properties 4. Statement of decomposition theorem|\yoga of weights" 5. Application 1: Beilinson, et al., \there are enough perverse sheaves", they generate the derived category of constructible sheaves 6. Application 2: Radon transforms. Use to understand monodromy of hyperplane sections. 7. Some geometric ideas to prove the decomposition theorem. If you want to understand everything in the course you need a lot of background. We will assume Hartshorne- level algebraic geometry. We also need constructible sheaves|look at Sheaves in Topology. Problem sets will be given, but not collected; will be on the webpage. There are more references than BBD; they will be online.
    [Show full text]
  • MORSE THEORY and TILTING SHEAVES 1. Introduction This
    MORSE THEORY AND TILTING SHEAVES DAVID BEN-ZVI AND DAVID NADLER 1. Introduction This paper is a result of our trying to understand what a tilting perverse sheaf is. (See [1] for definitions and background material.) Our basic example is that of the flag variety B of a complex reductive group G and perverse sheaves on the Schubert stratification S. In this setting, there are many characterizations of tilting sheaves involving both their geometry and categorical structure. In this paper, we propose a new geometric construction of such tilting sheaves. Namely, we show that they naturally arise via the Morse theory of sheaves on the opposite Schubert stratification Sopp. In the context of a flag variety B, our construction takes the following form. Given a sheaf F on B, a point p ∈ B, and a germ F of a function at p, we have the ∗ corresponding local Morse group (or vanishing cycles) Mp,F (F) which measures how F changes at p as we move in the family defined by F . If we restrict to sheaves F constructible with respect to Sopp, then it is possible to find a sheaf ∗ Mp,F on a neighborhood of p which represents the functor Mp,F (though Mp,F is not constructible with respect to Sopp.) To be precise, for any such F, we have a natural isomorphism ∗ ∗ Mp,F (F) ' RHomD(B)(Mp,F , F). Since this may be thought of as an integral identity, we call the sheaf Mp,F a Morse kernel. The local Morse groups play a central role in the theory of perverse sheaves.
    [Show full text]
  • Refined Invariants in Geometry, Topology and String Theory Jun 2
    Refined invariants in geometry, topology and string theory Jun 2 - Jun 7, 2013 MEALS Breakfast (Bu↵et): 7:00–9:30 am, Sally Borden Building, Monday–Friday Lunch (Bu↵et): 11:30 am–1:30 pm, Sally Borden Building, Monday–Friday Dinner (Bu↵et): 5:30–7:30 pm, Sally Borden Building, Sunday–Thursday Co↵ee Breaks: As per daily schedule, in the foyer of the TransCanada Pipeline Pavilion (TCPL) Please remember to scan your meal card at the host/hostess station in the dining room for each meal. MEETING ROOMS All lectures will be held in the lecture theater in the TransCanada Pipelines Pavilion (TCPL). An LCD projector, a laptop, a document camera, and blackboards are available for presentations. SCHEDULE Sunday 16:00 Check-in begins @ Front Desk, Professional Development Centre 17:30–19:30 Bu↵et Dinner, Sally Borden Building Monday 7:00–8:45 Breakfast 8:45–9:00 Introduction and Welcome by BIRS Station Manager 9:00–10:00 Andrei Okounkov: M-theory and DT-theory Co↵ee 10.30–11.30 Sheldon Katz: Equivariant stable pair invariants and refined BPS indices 11:30–13:00 Lunch 13:00–14:00 Guided Tour of The Ban↵Centre; meet in the 2nd floor lounge, Corbett Hall 14:00 Group Photo; meet in foyer of TCPL 14:10–15:10 Dominic Joyce: Categorification of Donaldson–Thomas theory using perverse sheaves Co↵ee 15:45–16.45 Zheng Hua: Spin structure on moduli space of sheaves on CY 3-folds 17:00-17.30 Sven Meinhardt: Motivic DT-invariants of (-2)-curves 17:30–19:30 Dinner Tuesday 7:00–9:00 Breakfast 9:00–10:00 Alexei Oblomkov: Topology of planar curves, knot homology and representation
    [Show full text]
  • Intersection Spaces, Perverse Sheaves and Type Iib String Theory
    INTERSECTION SPACES, PERVERSE SHEAVES AND TYPE IIB STRING THEORY MARKUS BANAGL, NERO BUDUR, AND LAURENT¸IU MAXIM Abstract. The method of intersection spaces associates rational Poincar´ecomplexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA the- ory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the inter- section space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar´eduality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities. Contents 1. Introduction 2 2. Prerequisites 5 2.1. Isolated hypersurface singularities 5 2.2. Intersection space (co)homology of projective hypersurfaces 6 2.3. Perverse sheaves 7 2.4. Nearby and vanishing cycles 8 2.5.
    [Show full text]
  • Arxiv:1804.05014V3 [Math.AG] 25 Nov 2020 N Oooyo Ope Leri Aite.Frisac,Tedeco the U Instance, for for Varieties
    PERVERSE SHEAVES ON SEMI-ABELIAN VARIETIES YONGQIANG LIU, LAURENTIU MAXIM, AND BOTONG WANG Abstract. We give a complete (global) characterization of C-perverse sheaves on semi- abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell’s work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser’s results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties. 1. Introduction Perverse sheaves are fundamental objects at the crossroads of topology, algebraic geometry, analysis and differential equations, with important applications in number theory, algebra and representation theory. They provide an essential tool for understanding the geometry and topology of complex algebraic varieties. For instance, the decomposition theorem [3], a far-reaching generalization of the Hard Lefschetz theorem of Hodge theory with a wealth of topological applications, requires the use of perverse sheaves. Furthermore, perverse sheaves are an integral part of Saito’s theory of mixed Hodge module [31, 32]. Perverse sheaves have also seen spectacular applications in representation theory, such as the proof of the Kazhdan-Lusztig conjecture, the proof of the geometrization of the Satake isomorphism, or the proof of the fundamental lemma in the Langlands program (e.g., see [9] for a beautiful survey). A proof of the Weil conjectures using perverse sheaves was given in [20]. However, despite their fundamental importance, perverse sheaves remain rather mysterious objects. In his 1983 ICM lecture, MacPherson [27] stated the following: The category of perverse sheaves is important because of its applications.
    [Show full text]
  • Perverse Sheaves in Algebraic Geometry
    Perverse sheaves in Algebraic Geometry Mark de Cataldo Lecture notes by Tony Feng Contents 1 The Decomposition Theorem 2 1.1 The smooth case . .2 1.2 Generalization to singular maps . .3 1.3 The derived category . .4 2 Perverse Sheaves 6 2.1 The constructible category . .6 2.2 Perverse sheaves . .7 2.3 Proof of Lefschetz Hyperplane Theorem . .8 2.4 Perverse t-structure . .9 2.5 Intersection cohomology . 10 3 Semi-small Maps 12 3.1 Examples and properties . 12 3.2 Lefschetz Hyperplane and Hard Lefschetz . 13 3.3 Decomposition Theorem . 16 4 The Decomposition Theorem 19 4.1 Symmetries from Poincaré-Verdier Duality . 19 4.2 Hard Lefschetz . 20 4.3 An Important Theorem . 22 5 The Perverse (Leray) Filtration 25 5.1 The classical Leray filtration. 25 5.2 The perverse filtration . 25 5.3 Hodge-theoretic implications . 28 1 1 THE DECOMPOSITION THEOREM 1 The Decomposition Theorem Perhaps the most successful application of perverse sheaves, and the motivation for their introduction, is the Decomposition Theorem. That is the subject of this section. The decomposition theorem is a generalization of a 1968 theorem of Deligne’s, from a smooth projective morphism to an arbitrary proper morphism. 1.1 The smooth case Let X = Y × F. Here and throughout, we use Q-coefficients in all our cohomology theories. Theorem 1.1 (Künneth formula). We have an isomorphism M H•(X) H•−q(Y) ⊗ Hq(F): q≥0 In particular, this implies that the pullback map H•(X) H•(F) is surjective, which is already rare for fibrations that are not products.
    [Show full text]
  • The Decomposition Theorem, Perverse Sheaves and the Topology Of
    The decomposition theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo and Luca Migliorini∗ Abstract We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples. Contents 1 Overview 3 1.1 The topology of complex projective manifolds: Lefschetz and Hodge theorems 4 1.2 Families of smooth projective varieties . ........ 5 1.3 Singular algebraic varieties . ..... 7 1.4 Decomposition and hard Lefschetz in intersection cohomology . 8 1.5 Crash course on sheaves and derived categories . ........ 9 1.6 Decomposition, semisimplicity and relative hard Lefschetz theorems . 13 1.7 InvariantCycletheorems . 15 1.8 Afewexamples.................................. 16 1.9 The decomposition theorem and mixed Hodge structures . ......... 17 1.10 Historicalandotherremarks . 18 arXiv:0712.0349v2 [math.AG] 16 Apr 2009 2 Perverse sheaves 20 2.1 Intersection cohomology . 21 2.2 Examples of intersection cohomology . ...... 22 2.3 Definition and first properties of perverse sheaves . .......... 24 2.4 Theperversefiltration . .. .. .. .. .. .. .. 28 2.5 Perversecohomology .............................. 28 2.6 t-exactness and the Lefschetz hyperplane theorem . ...... 30 2.7 Intermediateextensions . 31 ∗Partially supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie” 1 3 Three approaches to the decomposition theorem 33 3.1 The proof of Beilinson, Bernstein, Deligne and Gabber .
    [Show full text]
  • Geometric Structures and Varieties of Representations
    GEOMETRIC STRUCTURES AND VARIETIES OF REPRESENTATIONS WILLIAM M. GOLDMAN Abstract. Many interesting geometric structures on manifolds can be interpreted as structures locally modelled on homogeneous spaces. Given a homogeneous space (X; G) and a manifold M, there is a deformation space of structures on M locally modelled on the geometry of X invariant under G. Such a geometric struc- ture on a manifold M determines a representation (unique up to inner automorphism) of the fundamental group ¼ of M in G. The deformation space for such structures is \locally modelled" on the space Hom(¼; G)=G of equivalence classes of representations of ¼ ! G. A strong interplay exists between the local and global structure of the variety of representations and the corresponding geometric structures. The lecture in Boulder surveyed some as- pects of this correspondence, focusing on: (1) the \Deformation Theorem" relating deformation spaces of geometric structures to the space of representations; (2) representations of surface groups in SL(2; R), hyperbolic structures on surfaces (with singularities), Fenchel-Nielsen coordinates on TeichmullerÄ space; (3) convex real projective structures on surfaces; (4) representations of Schwarz triangle groups in SL(3; C). This paper represents an expanded version of the lecture. Introduction According to Felix Klein's Erlanger program of 1872, geometry is the study of the properties of a space which are invariant under a group of transformations. A geometry in Klein's sense is thus a pair (X; G) where X is a manifold and G is a Lie group acting (transitively) on X. If (X; G) is Euclidean geometry (so X = Rn and G is its group of isometries), then one might \in¯nitesimally model" a manifold M on Euclidean space by giving each tangent space a Euclidean geometry, which varies from point to point; the resulting in¯nitesimally Euclidean geometry is a Riemannian metric.
    [Show full text]
  • Intersection Homology & Perverse Sheaves with Applications To
    LAUREN¸TIU G. MAXIM INTERSECTIONHOMOLOGY & PERVERSESHEAVES WITHAPPLICATIONSTOSINGULARITIES i Contents Preface ix 1 Topology of singular spaces: motivation, overview 1 1.1 Poincaré Duality 1 1.2 Topology of projective manifolds: Kähler package 4 Hodge Decomposition 5 Lefschetz hyperplane section theorem 6 Hard Lefschetz theorem 7 2 Intersection Homology: definition, properties 11 2.1 Topological pseudomanifolds 11 2.2 Borel-Moore homology 13 2.3 Intersection homology via chains 15 2.4 Normalization 25 2.5 Intersection homology of an open cone 28 2.6 Poincaré duality for pseudomanifolds 30 2.7 Signature of pseudomanifolds 32 3 L-classes of stratified spaces 37 3.1 Multiplicative characteristic classes of vector bundles. Examples 38 3.2 Characteristic classes of manifolds: tangential approach 40 ii 3.3 L-classes of manifolds: Pontrjagin-Thom construction 44 Construction 45 Coincidence with Hirzebruch L-classes 47 Removing the dimension restriction 48 3.4 Goresky-MacPherson L-classes 49 4 Brief introduction to sheaf theory 53 4.1 Sheaves 53 4.2 Local systems 58 Homology with local coefficients 60 Intersection homology with local coefficients 62 4.3 Sheaf cohomology 62 4.4 Complexes of sheaves 66 4.5 Homotopy category 70 4.6 Derived category 72 4.7 Derived functors 74 5 Poincaré-Verdier Duality 79 5.1 Direct image with proper support 79 5.2 Inverse image with compact support 82 5.3 Dualizing functor 83 5.4 Verdier dual via the Universal Coefficients Theorem 85 5.5 Poincaré and Alexander Duality on manifolds 86 5.6 Attaching triangles. Hypercohomology long exact sequences of pairs 87 6 Intersection homology after Deligne 89 6.1 Introduction 89 6.2 Intersection cohomology complex 90 6.3 Deligne’s construction of intersection homology 94 6.4 Generalized Poincaré duality 98 6.5 Topological invariance of intersection homology 101 6.6 Rational homology manifolds 103 6.7 Intersection homology Betti numbers, I 106 iii 7 Constructibility in algebraic geometry 111 7.1 Definition.
    [Show full text]
  • Conditions of Smoothness of Moduli Spaces of Flat Connections and of Character Varieties
    This is a repository copy of Conditions of smoothness of moduli spaces of flat connections and of character varieties. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/149815/ Version: Accepted Version Article: Ho, Nankuo, Wilkin, Graeme Peter Desmond and Wu, Siye (2018) Conditions of smoothness of moduli spaces of flat connections and of character varieties. Mathematische Zeitschrift. ISSN 1432-1823 https://doi.org/10.1007/s00209-018-2158-2 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Conditions of smoothness of moduli spaces of flat connections and of character varieties Nan-Kuo Ho1,a, Graeme Wilkin2,b and Siye Wu1,c 1 Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan 2 Department of Mathematics, National University of Singapore, Singapore 119076 a E-mail address: [email protected] b E-mail address: [email protected] c E-mail address: [email protected] Abstract.
    [Show full text]
  • Representation Theory As Gauge Theory
    Representation theory Quantum Field Theory Gauge Theory Representation Theory as Gauge Theory David Ben-Zvi University of Texas at Austin Clay Research Conference Oxford, September 2016 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Outline Representation theory Quantum Field Theory Gauge Theory Representation theory Quantum Field Theory Gauge Theory Fourier Series G = U(1) acts on S1, hence on L2(S1): Fourier series 2 1 [M 2πinθ L (S ) ' Ce n2Z joint eigenspaces of rotation operators See last slide for all image credits Representation theory Quantum Field Theory Gauge Theory The dual Basic object in representation theory: describe the dual of a group G: Gb = f irreducible (unitary) representations of Gg e.g.
    [Show full text]
  • D-Modules, Perverse Sheaves, and Representation Theory
    Progress in Mathematics Volume 236 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Ryoshi Hotta Kiyoshi Takeuchi Toshiyuki Tanisaki D-Modules, Perverse Sheaves, and Representation Theory Translated by Kiyoshi Takeuchi Birkhauser¨ Boston • Basel • Berlin Ryoshi Hotta Kiyoshi Takeuchi Professor Emeritus of Tohoku University School of Mathematics Shirako 2-25-1-1106 Tsukuba University Wako 351-0101 Tenoudai 1-1-1 Japan Tsukuba 305-8571 [email protected] Japan [email protected] Toshiyuki Tanisaki Department of Mathematics Graduate School of Science Osaka City University 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 Japan [email protected] Mathematics Subject Classification (2000): Primary: 32C38, 20G05; Secondary: 32S35, 32S60, 17B10 Library of Congress Control Number: 2004059581 ISBN-13: 978-0-8176-4363-8 e-ISBN-13: 978-0-8176-4523-6 Printed on acid-free paper. c 2008, English Edition Birkhauser¨ Boston c 1995, Japanese Edition, Springer-Verlag Tokyo, D Kagun to Daisugun (D-Modules and Algebraic Groups) by R. Hotta and T. Tanisaki. All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]