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Chern class formulas for classical-type degeneracy loci

Part of: Projective and enumerative geometry Special varieties Algebraic combinatorics

Published online by Cambridge University Press:  18 July 2018

David Anderson  and
William Fulton
David Anderson
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA email [email protected]
William Fulton
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email [email protected]
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Abstract

Employing a simple and direct geometric approach, we prove formulas for a large class of degeneracy loci in types B, C, and D, including those coming from all isotropic Grassmannians. The results unify and generalize previous Pfaffian and determinantal formulas. Specializing to the Grassmannian case, we recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. Our method yields streamlined proofs which proceed in parallel for all four classical types, substantially simplifying previous work on the subject. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.

MSC classification

Primary: 14N15: Classical problems, Schubert calculus
Secondary: 14M12: Determinantal varieties 14M15: Grassmannians, Schubert varieties, flag manifolds 05E05: Symmetric functions and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 8 , August 2018 , pp. 1746 - 1774
Copyright
© The Authors 2018 

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Footnotes

DA was partially supported by NSF grant DMS-1502201.

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