Equivariant Cohomology in Algebraic Geometry

Erscheinungsjahr 2025
ISBN: 978-1-009-34997-0
Verlag: Cambridge University Press

Buch, Englisch, Band 210, 462 Seiten, Format (B × H): 229 mm x 151 mm, Gewicht: 678 g

Reihe: Cambridge Studies in Advanced Mathematics

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.

Autoren/Hrsg.

Anderson, David

Fulton, William

Inhaltsverzeichnis

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.

Über Autor(innen)

Anderson, David
David Anderson is Associate Professor at The Ohio State University. He works in combinatorial algebraic geometry and has written over three dozen papers on topics including Schubert calculus, Newton–Okounkov bodies, and equivariant K-theory. In 2020, he received a CAREER Award from the National Science Foundation.

Fulton, William
William Fulton is Oscar Zariski Distinguished University Professor Emeritus at the University of Michigan. He is an algebraic geometer, and author or co-author of approximately five dozen papers and a dozen books, including 'Intersection Theory,' which won a Steele Prize from the American Mathematical Society. Fulton is a member of the National Academy of Sciences, and a foreign member of the Royal Swedish Academy of Sciences.