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Geometry of Moduli Spaces and Representation Theory [Hardback]

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  • Formāts: Hardback, 436 pages, height x width: 254x178 mm, weight: 915 g
  • Sērija : IAS/Park City Mathematics Series
  • Izdošanas datums: 30-Dec-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470435748
  • ISBN-13: 9781470435745
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Geometry of Moduli Spaces and Representation Theory
  • Formāts: Hardback, 436 pages, height x width: 254x178 mm, weight: 915 g
  • Sērija : IAS/Park City Mathematics Series
  • Izdošanas datums: 30-Dec-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470435748
  • ISBN-13: 9781470435745
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470435745.html
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This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program ``Geometry of moduli spaces and representation theory'', and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory.

Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan-Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry.

Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections.

The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.
Preface vii
Introduction ix
Roman Bezrukavnikov
Alexander Braverman
Zhiwei Yun
Perverse sheaves and the topology of algebraic varieties 1(58)
Mark Andrea de Cataldo
Lecture 1 The decomposition theorem
3(14)
1.1 Deligne's theorem in cohomology
3(1)
1.2 The global invariant cycle theorem
4(1)
1.3 Cohomological decomposition theorem
5(1)
1.4 The local invariant cycle theorem
6(1)
1.5 Deligne's theorem
7(1)
1.6 The decomposition theorem
8(2)
1.7 Exercises for Lecture 1
10(7)
Lecture 2 The category of perverse sheaves P(Y)
17(11)
2.1 Three "Whys", and a brief history of perverse sheaves
17(2)
2.2 The constructible derived category D (Y)
19(2)
2.3 Definition of perverse sheaves
21(1)
2.4 Artin vanishing and Lefschetz hyperplane theorems
22(2)
2.5 The perverse t-structure
24(1)
2.6 Intersection complexes
25(1)
2.7 Exercises for Lecture 2
26(2)
Lecture 3 Semi-small maps
28(9)
3.1 Semi-small maps
29(2)
3.2 The decomposition theorem for semi-small maps
31(1)
3.3 Hilbert schemes of points on surfaces and Heisenberg algebras
31(2)
3.4 The endomorphism algebra End(f*Qx)
33(1)
3.5 Geometric realization of the representations of the Weyl group
34(1)
3.6 Exercises for Lecture 3
35(2)
Lecture 4 Symmetries: VD, RHL, Jc splits off
37(9)
4.1 Verdier duality and the decomposition theorem
37(1)
4.2 Verdier duality and the decomposition theorem with large fibers
38(1)
4.3 The relative hard Lefschetz theorem
39(1)
4.4 Application of RHL: Stanley's theorem
40(1)
4.5 Intersection cohomology of the target as a direct summand
41(2)
4.6 Pure Hodge structure on intersection cohomology groups
43(1)
4.7 Exercises for Lecture 4
44(2)
Lecture 5 The perverse filtration
46(10)
5.1 The perverse spectral sequence and the perverse filtration
47(1)
5.2 Geometric description of the perverse filtration
48(2)
5.3 Hodge-theoretic consequences
50(1)
5.4 Character variety and Higgs moduli: P = W
50(3)
5.5 Let us conclude with a motivic question
53(1)
5.6 Exercises for Lecture 5
54(2)
References
56(3)
An introduction to affine Grassmannians and the geometric Satake equivalence 59(96)
Xinwen Zhu
Abstract. We introduce various affine Grassmannian varieties, study their geometric properties, and give some applications. We also discuss the geometric Satake equivalence. These are the expanded lecture notes for a mini-course in the 2015 PCMI summer school.
Introduction
60(6)
0.1 Some motivations
60(1)
0.2 Scope and contents
61(2)
0.3 Conventions and notations
63(3)
0.4 Acknowledgement
66(1)
Lecture 1 Affine Grassmannians and their first properties
66(15)
1.1 The affine Grassmannian of GLn
66(2)
1.2 Affine Grassmannians of general groups
68(3)
1.3 Groups attached to the punctured disc
71(3)
1.4 The Beauville-Laszlo theorem
74(2)
1.5 The determinant line bundle
76(3)
1.6 Affine Grassmannians over the complex numbers
79(1)
1.7 Affine Grassmannians for p-adic groups
80(1)
Lecture 2 More on the geometry of affine Grassmannians
81(15)
2.1 Schubert varieties
81(6)
2.2 Digression: Some sub-ind-schemes in Gr
87(1)
2.3 Opposite Schubert "varieties" and transversal slices
88(4)
2.4 The Picard group
92(2)
2.5 Central extensions and affine Kac-Moody algebras
94(2)
Lecture 3 Beilinson-Drinfeld Grassmannians and factorisation structures
96(12)
3.1 Beilinson-Drinfeld Grassmannians
96(7)
3.2 Factorisation property
103(1)
3.3 The Ran space
104(1)
3.4 Rigidified line bundles on GrRan
105(3)
Lecture 4 Applications to the moduli of G-bundles
108(8)
4.1 One point uniformization
108(2)
4.2 Line bundles and conformal blocks
110(1)
4.3 Adelic uniformization
111(5)
Lecture 5 The geometric Satake equivalence
116(29)
5.1 The Satake category Sate
117(2)
5.2 Sate as a Tannakian category
119(3)
5.3 Langlands dual group
122(8)
5.4 Fusion product
130(4)
5.5 Bootstraps
134(8)
5.6 From the geometric Satake to the classical Satake
142(3)
A Complements on sheaf theory
145(11)
A.1 Equivariant category of perverse sheaves
145(4)
A.2 Universally local acyclicity
149(6)
Lectures on Springer theories and orbital integrals 155(96)
Zhiwei Yun
Abstract. These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in
2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.
0 Introduction
156(1)
0.1 Topics of these notes
156(1)
0.2 What we assume from the reader
157(1)
Lecture 1 Springer fibers
157(13)
1.1 The setup
157(1)
1.2 Springer fibers
158(1)
1.3 Examples of Springer fibers
159(3)
1.4 Geometric Properties of Springer fibers
162(1)
1.5 The Springer correspondence
163(4)
1.6 Comments and generalizations
167(1)
1.7 Exercises
168(2)
Lecture 2 Affine Springer fibers
170(20)
2.1 Loop group, parahoric subgroups and the affine flag variety
170(4)
2.2 Affine Springer fibers
174(2)
2.3 Symmetry on affine Springer fibers
176(2)
2.4 Further examples of affine Springer fibers
178(3)
2.5 Geometric Properties of affine Springer fibers
181(5)
2.6 Affine Springer representations
186(2)
2.7 Comments and generalizations
188(1)
2.8 Exercises
189(1)
Lecture 3 Orbital integrals
190(13)
3.1 Integration on a p-adic group
190(1)
3.2 Orbital integrals
191(3)
3.3 Relation with affine Springer fibers
194(1)
3.4 Stable orbital integrals
195(3)
3.5 Examples in SL2
198(3)
3.6 Remarks on the Fundamental Lemma
201(1)
3.7 Exercises
202(1)
Lecture 4 Hitchin fibration
203(49)
4.1 The Hitchin moduli stack
204(2)
4.2 Hitchin fibration
206(2)
4.3 Hitchin fibers
208(2)
4.4 Relation with affine Springer fibers
210(2)
4.5 A global version of the Springer action
212(1)
4.6 Exercises
213(38)
Lectures on K-theoretic computations in enumerative geometry 251(130)
Andrei Okounkov
Lecture 1 Aims & Scope
252(11)
1.1 K-theoretic enumerative geometry
252(2)
1.2 Quantum K-theory of Nakajima varieties
254(3)
1.3 K-theoretic Donaldson-Thomas theory
257(4)
1.4 Old vs. new
261(1)
1.5 Acknowledgements
262(1)
Lecture 2 Before we begin
263(10)
2.1 Symmetric and exterior algebras
263(3)
2.2 KG (X) and KG (X)
266(3)
2.3 Localization
269(3)
2.4 Rigidity
272(1)
Lecture 3 The Hilbert scheme of points of 3-folds
273(25)
3.1 Our very first DT moduli space
273(2)
3.2 Ovir and avir
275(4)
3.3 Nekrasov's formula
279(5)
3.4 Tangent bundle and localization
284(7)
3.5 Proof of Nekrasov's formula
291(7)
Lecture 4 Nakajima varieties
298(6)
4.1 Algebraic symplectic reduction
298(1)
4.2 Nakajima quiver varieties [ 35,61,62]
299(2)
4.3 Quasimaps to Nakajima varieties
301(3)
Lecture 5 Symmetric powers
304(10)
5.1 PT theory for smooth curves
304(4)
5.2 Proof of Theorem 5.1.16
308(2)
5.3 Hilbert schemes of surfaces and threefolds
310(4)
Lecture 6 More on quasimaps
314(21)
6.1 Balanced classes and square roots
314(3)
6.2 Relative quasimaps in an example
317(5)
6.3 Stable reduction for relative quasimaps
322(5)
6.4 Moduli of relative quasimaps
327(4)
6.5 The degeneration formula and the gluing operator
331(4)
Lecture 7 Nuts and bolts
335(11)
7.1 The Tube
335(3)
7.2 The Vertex
338(3)
7.3 The index limit
341(1)
7.4 Cap and capping
342(1)
7.5 Large framing vanishing
343(3)
Lecture 8 Difference equations
346(9)
8.1 Shifts of Kahler variables
346(2)
8.2 Shifts of equivariant variables
348(4)
8.3 Difference equations for vertices
352(3)
Lecture 9 Stable envelopes and quantum groups
355(12)
9.1 K-theoretic stable envelopes
355(4)
9.2 Triangle lemma and braid relations
359(5)
9.3 Actions of quantum groups
364(3)
Lecture 10 Quantum Knizhnik-Zamolodchikov equations
367(14)
10.1 Commuting difference operators from R-matrices
367(2)
10.2 Minuscule shifts and qKZ
369(3)
10.3 The gluing operator in the stable basis
372(2)
10.4 Proof of Theorem 10.2.11
374(7)
Lectures on perverse sheaves on instanton moduli spaces 381
Hiraku Nakajima
Lecture 1 Introduction
381(6)
Lecture 2 Uhlenbeck partial compactification - in brief
387(3)
Lecture 3 Heisenberg algebra action on the Gieseker partial compactification
390(7)
Lecture 4 Stable envelopes
397(9)
Lecture 5 Sheaf theoretic analysis of stable envelopes
406(8)
Lecture 6 R-matrix for Gieseker partial compactification
414(6)
Lecture 7 Perverse sheaves on instanton moduli spaces
420(8)
Lecture 8 W-algebra representation on circle plusd IHT(UdG)
428(3)
Lecture 9 Concluding remarks
431
Roman Bezrukavnikov, Massachusetts Institute of Technology, Cambridge, MA.

Alexander Braverman, University of Toronto, ON, Canada, Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, and Skolkovo Institute for Science and Technology, Moscow, Russia.

Zhiwei Yun, Yale University, New Haven, CT.