Geometry & Topology
- Geom. Topol.
- Volume 22, Number 6 (2018), 3575-3622.
Chern–Schwartz–MacPherson classes of degeneracy loci
The Chern–Schwartz–MacPherson class (CSM) and the Segre–Schwartz–MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method — inspired by recent work of Maulik and Okounkov and of Gorbounov, Rimányi, Tarasov and Varchenko — does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas — including residue generating sequences — for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (two-step) positivity properties for some relevant representations.
Geom. Topol., Volume 22, Number 6 (2018), 3575-3622.
Received: 11 July 2017
Revised: 29 January 2018
Accepted: 5 March 2018
First available in Project Euclid: 29 September 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 32S20: Global theory of singularities; cohomological properties [See also 14E15] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14N15: Classical problems, Schubert calculus 57R20: Characteristic classes and numbers
Fehér, László M; Rimányi, Richárd. Chern–Schwartz–MacPherson classes of degeneracy loci. Geom. Topol. 22 (2018), no. 6, 3575--3622. doi:10.2140/gt.2018.22.3575. https://projecteuclid.org/euclid.gt/1538186743