## Geometry & Topology

### Chern–Schwartz–MacPherson classes of degeneracy loci

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3575-3622.

Dates
Revised: 29 January 2018
Accepted: 5 March 2018
First available in Project Euclid: 29 September 2018

https://projecteuclid.org/euclid.gt/1538186743

Digital Object Identifier
doi:10.2140/gt.2018.22.3575

Mathematical Reviews number (MathSciNet)
MR3858770

Zentralblatt MATH identifier
06945132

#### Citation

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

#### References

• P Aluffi, Characteristic classes of singular varieties, from “Topics in cohomological studies of algebraic varieties” (P Pragacz, editor), Birkhäuser, Basel (2005) 1–32
• P Aluffi, Classes de Chern des variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris 342 (2006) 405–410
• P Aluffi, L C Mihalcea, Chern classes of Schubert cells and varieties, J. Algebraic Geom. 18 (2009) 63–100
• P Aluffi, L C Mihalcea, Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, Compos. Math. 152 (2016) 2603–2625
• P Aluffi, L C Mihalcea, J Schuermann, C Su, Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells, preprint (2017)
• G Bérczi, A Szenes, Thom polynomials of Morin singularities, Ann. of Math. 175 (2012) 567–629
• A S Buch, Quiver coefficients of Dynkin type, Michigan Math. J. 57 (2008) 93–120
• A S Buch, R Rimányi, A formula for non-equioriented quiver orbits of type $A$, J. Algebraic Geom. 16 (2007) 531–546
• G S Call, D J Velleman, Pascal's matrices, Amer. Math. Monthly 100 (1993) 372–376
• D Edidin, W Graham, Equivariant intersection theory, Invent. Math. 131 (1998) 595–634
• L M Fehér, R Rimányi, Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces, Cent. Eur. J. Math. 1 (2003) 418–434
• L M Fehér, R Rimányi, Calculation of Thom polynomials and other cohomological obstructions for group actions, from “Real and complex singularities” (T Gaffney, M A S Ruas, editors), Contemp. Math. 354, Amer. Math. Soc., Providence, RI (2004) 69–93
• L M Fehér, R Rimányi, On the structure of Thom polynomials of singularities, Bull. Lond. Math. Soc. 39 (2007) 541–549
• L M Fehér, R Rimányi, Thom series of contact singularities, Ann. of Math. 176 (2012) 1381–1426
• L M Fehér, R Rimányi, A Weber, Motivic Chern classes and K-theoretic stable envelopes, preprint (2018)
• W Fulton, Intersection theory, 2nd edition, Ergeb. Math. Grenzgeb. 2, Springer (1998)
• W Fulton, P Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics 1689, Springer (1998)
• V Gorbounov, R Rimányi, V Tarasov, A Varchenko, Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra, J. Geom. Phys. 74 (2013) 56–86
• J Huh, Positivity of Chern classes of Schubert cells and varieties, J. Algebraic Geom. 25 (2016) 177–199
• V Y Kaloshin, A geometric proof of the existence of Whitney stratifications, Mosc. Math. J. 5 (2005) 125–133
• M E Kazarian, Non-associative Hilbert scheme and Thom polynomials, preprint (2009)
• B Kőm\Huves, Equivariant Chern–Schwartz–MacPherson classes of coincident root loci, in preparation
• A Knutson, E Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. 161 (2005) 1245–1318
• D Laksov, A Lascoux, A Thorup, On Giambelli's theorem on complete correlations, Acta Math. 162 (1989) 143–199
• B Lindström, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973) 85–90
• R D MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974) 423–432
• D Maulik, A Okounkov, Quantum groups and quantum cohomology, preprint (2012)
• T Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc. 140 (2006) 115–134
• T Ohmoto, Singularities of maps and characteristic classes, from “School on real and complex singularities” (R N Araújo dos Santos, V H Jorge Pérez, T Nishimura, O Saeki, editors), Adv. Stud. Pure Math. 68, Math. Soc. Japan, Tokyo (2016) 191–265
• A Okounkov, Lectures on K–theoretic computations in enumerative geometry, from “Geometry of moduli spaces and representation theory” (R Bezrukavnikov, A Braverman, Z Yun, editors), IAS/Park City Math. Ser. 24, Amer. Math. Soc., Providence, RI (2017) 251–380
• A Parusiński, P Pragacz, Chern–Schwartz–MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc. 8 (1995) 793–817
• I R Porteous, Simple singularities of maps, from “Proceedings of Liverpool singularities symposium, I” (C T C Wall, editor), Lecture Notes in Math. 192, Springer (1971) 286–307
• P Pragacz, A Weber, Positivity of Schur function expansions of Thom polynomials, Fund. Math. 195 (2007) 85–95
• S Promtapan, Characteristic classes of symmetric and skew-symmetric degeneracy loci, PhD thesis in preparation, University of North Carolina at Chapel Hill
• R Rimányi, Thom polynomials, symmetries and incidences of singularities, Invent. Math. 143 (2001) 499–521
• R Rimányi, Quiver polynomials in iterated residue form, J. Algebraic Combin. 40 (2014) 527–542
• R Rimányi, V Tarasov, A Varchenko, Partial flag varieties, stable envelopes, and weight functions, Quantum Topol. 6 (2015) 333–364
• R Rimányi, V Tarasov, A Varchenko, Elliptic and K–theoretic stable envelopes and Newton polytopes, preprint (2017)
• R Rimányi, A Varchenko, Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae, from “Schubert varieties, equivariant cohomology and characteristic classes”, IMPANGA 15, Eur. Math. Soc. (2018) 225–235
• O Straser, Algebraic stratifications of $G$–varieties, MathOverflow post (2013) Available at \setbox0\makeatletter\@url http://mathoverflow.net/questions/129218 {\unhbox0
• V Tarasov, A Varchenko, Geometry of $q$–hypergeometric functions as a bridge between Yangians and quantum affine algebras, Invent. Math. 128 (1997) 501–588
• A Weber, Equivariant Chern classes and localization theorem, J. Singul. 5 (2012) 153–176
• X Zhang, Chern classes and characteristic cycles of determinantal varieties, J. Algebra 497 (2018) 55–91