Equivariant Cohomology in Algebraic Geometry

Contents
Preface
1 Preview
	1.1 The Borel construction
	1.2 Fiber bundles
	1.3 The localization package
	1.4 Schubert calculus and Schubert polynomials
	Notes
2 Defining Equivariant Cohomology
	2.1 Principal bundles
	2.2 Definitions
	2.3 Chern classes and fundamental classes
	2.4 The general linear group
	2.5 Some other groups
	2.6 Projective space
	Notes
	Hints for Exercises
3 Basic Properties
	3.1 Tori
	3.2 Functoriality
	3.3 Invariance
	3.4 Free and trivial actions
	3.5 Exact sequences
	3.6 Gysin homomorphisms
	3.7 Poincaré duality
	Hints for Exercises
4 Grassmannians and flag varieties
	4.1 Schur polynomials
	4.2 Flag bundles
	4.3 Projective space
	4.4 Complete flags
	4.5 Grassmannians and partial flag varieties
	4.6 Poincaré dual bases
	4.7 Bases and duality from subvarieties
	Notes
	Hints for Exercises
5 Localization I
	5.1 The main localization theorem (first approach)
	5.2 Integration formula
	5.3 Equivariant formality
	Notes
	Hints for Exercises
6 Conics
	6.1 Steiner’s problem
	6.2 Cohomology of a blowup
	6.3 Complete conics
	Notes
	Hints for Exercises
7 Localization II
	7.1 The general localization theorem
	7.2 Invariant curves
	7.3 Image of the restriction map
	7.4 The image theorem for nonsingular varieties
	Notes
	Hints for Exercises
8 Toric Varieties
	8.1 Equivariant geometry of toric varieties
	8.2 Cohomology rings
	8.3 The Stanley–Reisner ring
	8.4 Other presentations
	Notes
	Hints for Exercises
9 Schubert Calculus on Grassmannians
	9.1 Schubert cells and Schubert varieties
	9.2 Schubert classes and the Kempf–Laksov formula
	9.3 Tangent spaces and normal spaces
	9.4 Double Schur polynomials
	9.5 Poincaré duality
	9.6 Multiplication
	9.7 Grassmann duality
	9.8 Littlewood–Richardson rules
	Notes
	Hints for Exercises
10 Flag Varieties and Schubert Polynomials
	10.1 Rank functions and Schubert varieties
	10.2 Neighborhoods and tangent weights
	10.3 Invariant curves in the flag variety
	10.4 Bruhat order for the symmetric group
	10.5 Opposite Schubert varieties and Poincaréduality
	10.6 Schubert polynomials
	10.7 Multiplying Schubert classes
	10.8 Partial flag varieties
	10.9 Stability
	10.10 Properties of Schubert polynomials
	Notes
	Hints for Exercises
11 Degeneracy Loci
	11.1 The Cayley–Giambelli–Thom–Porteous formula
	11.2 Flagged degeneracy loci
	11.3 Irreducibility
	11.4 The class of a degeneracy locus
	11.5 Essential sets
	11.6 Degeneracy loci for maps of vector bundles
	11.7 Universal properties of Schubert polynomials
	11.8 Further properties of Schubert polynomials
	Notes
	Hints for Exercises
12 Infinite-Dimensional Flag Varieties
	12.1 Stability revisited
	12.2 Infinite Grassmannians and flag varieties
	12.3 Schubert varieties and Schubert polynomials
	12.4 Degeneracy loci
	Notes
	Hints for Exercises
13 Symplectic Flag Varieties
	13.1 Degeneracy loci for symmetric maps
	13.2 Isotropic subspaces
	13.3 Symplectic flag bundles
	13.4 Lagrangian Grassmannians
	13.5 Cohomology rings
	Notes
	Hints for Exercises
14 Symplectic Schubert Polynomials
	14.1 Schubert varieties
	14.2 Double Q-polynomials and Lagrangian Schubert classes
	14.3 Symplectic degeneracy loci
	14.4 Type C Schubert polynomials
	14.5 Properties of type C Schubert polynomials
	Notes
	Hints for Exercises
15 Homogeneous Varieties
	15.1 Linear algebraic groups
	15.2 Flag varieties
	15.3 Parabolic subgroups and partial flag varieties
	15.4 Invariant curves
	15.5 Compact groups
	15.6 Borel presentation and equivariant line bundles
	Notes
	Hints for Exercises
16 The Algebra of Divided Difference Operators
	16.1 Push-Pull operators
	16.2 Restriction to fixed points
	16.3 Difference operators and line bundles
	16.4 The right W-action
	16.5 Left-Handed actions and operators
	16.6 The convolution algebra
	Notes
	Hints for Exercises
17 Equivariant Homology
	17.1 Equivariant Borel–Moore homology and Chow groups
	17.2 Segre classes
	17.3 Localization
	17.4 Equivariant multiplicities
	Notes
	Hints for Exercises
18 Bott–Samelson Varieties and Schubert Varieties
	18.1 Definitions, fixed points, and tangent spaces
	18.2 Desingularizations of Schubert varieties
	18.3 Poincaré duality and restriction to fixed points
	18.4 A presentation for the cohomology ring
	18.5 A restriction formula for Schubert varieties
	18.6 Duality
	18.7 A nonsingularity criterion
	Notes
	Hints for Exercises
19 Structure Constants
	19.1 Chevalley’s formula
	19.2 Characterization of structure constants
	19.3 Positivity via transversality
	19.4 Positivity via degeneration
	Notes
	Hints for Exercises
Appendix A. Algebraic Topology
	A.1 Homology and cohomology
	A.2 Borel–Moore homology
	A.3 Class of a subvariety
	A.4 Leray–Hirsch theorem
	A.5 Chern classes
	A.6 Gysin homomorphisms
	A.7 The complement of a variety in affine space
	A.8 Limits
Appendix B. Specialization in Equivariant Borel–Moore Homology
Appendix C. Pfaffians and Q-Polynomials
	C.1 Pfaffians
	C.2 Schur Q-polynomials
	C.3 Double Q-polynomials and interpolation
	Hints for Exercises
Appendix D. Conventions for Schubert Varieties
	D.1 Grassmannians
	D.2 Flag varieties
	D.3 General G/P
Appendix E. Characteristic Classes and Equivariant Cohomology
References
Notation Index
Subject Index