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Full Description
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.
Contents
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.
