Equivariant Cohomology in Algebraic Geometry

A graduate-level introduction to the core notions of equivariant cohomology, an indispensable tool in several areas of modern mathematics.

David Anderson (Author), William Fulton (Author)

9781009349970, Cambridge University Press

Paperback / softback, published 15 May 2025

464 pages
22.9 x 15.2 x 2.4 cm, 0.666 kg

'Equivariant Cohomology in Algebraic Geometry by David Anderson and William Fulton offers a comprehensive, accessible exploration of the development, standard examples, and recent contributions in this fascinating field. The authors have successfully struck a balance between rigor and approachability, making it an excellent resource for young researchers in the field. The book's real strength lies in its application to toric varieties and Schubert varieties across various settings, including Grassmannians, flag varieties, degeneracy loci, and extensions to other classical types and Kac–Moody groups. The authors' treatment of Bott-Samelson desingularizations of Schubert varieties is particularly noteworthy, displaying elegance and coherence within the context of the book's material. With over 450 pages of content, Equivariant Cohomology in Algebraic Geometry offers a comprehensive resource for researchers and scholars. It is poised to become a standard reference in the field, leaving a lasting impact on the flourishing area of research for years to come.' Sara Billey, University of Washington

Equivariant cohomology has become an indispensable tool in algebraic geometry and in related areas including representation theory, combinatorial and enumerative geometry, and algebraic combinatorics. This text introduces the main ideas of the subject for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics. The first six chapters cover the basics: definitions via finite-dimensional approximation spaces, computations in projective space, and the localization theorem. The rest of the text focuses on examples – toric varieties, Grassmannians, and homogeneous spaces – along with applications to Schubert calculus and degeneracy loci. Prerequisites are kept to a minimum, so that one-semester graduate-level courses in algebraic geometry and topology should be sufficient preparation. Featuring numerous exercises, examples, and material that has not previously appeared in textbook form, this book will be a must-have reference and resource for both students and researchers for years to come.

1. Preview
2. Defining equivariant cohomology
3. Basic properties
4. Grassmannians and flag varieties
5. Localization I
6. Conics
7. Localization II
8. Toric varieties
9. Schubert calculus on Grassmannians
10. Flag varieties and Schubert polynomials
11. Degeneracy loci
12. Infinite-dimensional flag varieties
13. Symplectic flag varieties
14. Symplectic Schubert polynomials
15. Homogeneous varieties
16. The algebra of divided difference operators
17. Equivariant homology
18. Bott–_Samelson varieties and Schubert varieties
19. Structure constants
A. Algebraic topology
B. Specialization in equivariant Borel–_Moore homology
C. Pfaffians and Q-polynomials
D. Conventions for Schubert varieties
E. Characteristic classes and equivariant cohomology
References
Notation index
Subject index.

Subject Areas: Geometry [PBM]