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arxiv: 2509.24348 · v3 · submitted 2025-09-29 · 🧮 math.AG · math.CO

Motivic Classes of isotropic degeneracy loci and symmetric orbit closures

Minyoung Jeon This is my paper

Pith reviewed 2026-05-18 12:57 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords motivic Chern classesHirzebruch classesdegeneracy lociisotropic Grassmannianssymplectic orbit closuresorthogonal Grassmanniansflag varieties
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The pith

Explicit formulas compute motivic Chern and Hirzebruch classes of degeneracy loci in symplectic and odd orthogonal Grassmannians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops explicit formulas for the motivic Chern and Hirzebruch classes of degeneracy loci, with emphasis on those arising from symplectic and odd orthogonal Grassmannians. These classes specialize to give Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. The formulas extend earlier work on ordinary Grassmannian degeneracy loci to the isotropic and maximal even orthogonal settings. Applications include the same classes for orthogonal and symplectic orbit closures inside flag varieties. A reader cares because the formulas make refined invariants computable for a wider family of varieties in algebraic geometry.

Core claim

We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our results are inspired by, and partially extends, those of Anderson--Chen--Tarasca in the case of ordinary Grassmannian degeneracy loci to isotropic and odd orthogonal Grassmannians as well as maximal even orthogonal Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag vari

What carries the argument

Explicit formulas for motivic Chern and Hirzebruch classes adapted from ordinary Grassmannians to isotropic degeneracy loci and symmetric orbit closures.

If this is right

  • Specializations of the formulas immediately give Chern-Schwartz-MacPherson classes for the same isotropic loci.
  • K-theory classes and Cappell-Shaneson L-classes become available by the same specialization process.
  • Motivic classes are now obtained for orthogonal and symplectic orbit closures in flag varieties.
  • The same explicit expressions apply to maximal even orthogonal Grassmannians.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formulas may support algorithmic implementations that calculate these classes for larger examples in existing algebraic geometry software.
  • Similar adaptation techniques could apply to degeneracy loci in other homogeneous spaces beyond Grassmannians.
  • The refined classes might reveal new relations among invariants of isotropic subvarieties in flag varieties.

Load-bearing premise

The methods and results developed for ordinary Grassmannian degeneracy loci can be adapted and extended to the isotropic, odd orthogonal, and maximal even orthogonal Grassmannian settings.

What would settle it

A direct computation of the motivic Chern class for a concrete low-dimensional isotropic degeneracy locus, such as a rank-1 locus inside a small symplectic Grassmannian, that fails to match the proposed formula.

read the original abstract

We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our results are inspired by, and partially extends, those of Anderson--Chen--Tarasca in the case of ordinary Grassmannian degeneracy loci to isotropic and odd orthogonal Grassmannians as well as maximal even orthogonal Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript provides explicit formulas for the motivic Chern and Hirzebruch classes of degeneracy loci arising from symplectic and odd orthogonal Grassmannians (and maximal even orthogonal cases). These extend the Anderson--Chen--Tarasca results for ordinary Grassmannians via modifications to Chern class computations that respect the symplectic and orthogonal forms. Specializations yield Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. Applications identify symmetric orbit closures in flag varieties with appropriate degeneracy conditions to obtain their motivic classes.

Significance. If the stated formulas are verified, the work supplies a concrete extension of motivic class computations to isotropic settings, which are central to the geometry of classical groups and symmetric varieties. The direct adaptation of prior methods, the explicit formulas, and the identification of orbit closures with degeneracy loci provide usable tools for calculations in these contexts, building on the cited Anderson--Chen--Tarasca framework without introducing circular steps.

minor comments (2)
  1. The introduction and abstract state that the results 'partially extend' Anderson--Chen--Tarasca; a brief explicit comparison of which formulas are carried over verbatim versus which require new modifications for the isotropic case would improve readability.
  2. In the sections presenting the formulas for motivic Chern and Hirzebruch classes, confirm that the specializations to CSM, K-theory, and L-classes are accompanied by the precise substitution rules or references to standard results used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments or points of criticism were raised in the report. We are pleased that the extension of the Anderson--Chen--Tarasca framework to isotropic and orthogonal settings, along with the applications to symmetric orbit closures, was viewed as providing usable tools without circularity.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation adapts Anderson--Chen--Tarasca methods for ordinary Grassmannians to isotropic, odd orthogonal, and maximal even orthogonal cases via explicit modifications to Chern class computations. Motivic Chern and Hirzebruch class formulas are stated directly as results of these adaptations, with CSM, K-theory, and L-class specializations obtained by standard substitutions. Orbit closure applications follow from identifying loci with degeneracy conditions in flag varieties. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain; the work is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract mentions no free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that prior methods adapt to the new geometric settings.

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Mihalcea,Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Compos

    Paolo Aluffi and Leonardo C. Mihalcea,Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Compos. Math.152(2016), no. 12, 2603–2625

    work page 2016

  2. [2]

    Mihalcea, J¨ org Sch¨ urmann, and Changjian Su,From motivic chern classes of schubert cells to their hirzebruch and csm class, preprint, arxiv:2212.12509 (2022)

    Paolo Aluffi, Leonardo C. Mihalcea, J¨ org Sch¨ urmann, and Changjian Su,From motivic chern classes of schubert cells to their hirzebruch and csm class, preprint, arxiv:2212.12509 (2022)

    work page arXiv 2022

  3. [3]

    J.172(2023), no

    ,Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells, Duke Math. J.172(2023), no. 17, 3257–3320

    work page 2023

  4. [4]

    ,Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem, Ann. Sci. ´Ec. Norm. Sup´ er. (4)57(2024), no. 1, 87–141

    work page 2024

  5. [5]

    Dave Anderson, Linda Chen, and Nicola Tarasca,Motivic classes of degeneracy loci and pointed Brill- Noether varieties, J. Lond. Math. Soc. (2)105(2022), no. 3, 1787–1822

    work page 2022

  6. [6]

    Math.350(2019), 440–485

    David Anderson,K-theoretic Chern class formulas for vexillary degeneracy loci, Adv. Math.350(2019), 440–485

    work page 2019

  7. [7]

    Math.154(2018), no

    David Anderson and William Fulton,Chern class formulas for classical-type degeneracy loci, Compos. Math.154(2018), no. 8, 1746–1774

    work page 2018

  8. [8]

    Kazuhiko Aomoto,On some double coset decompositions of complex semisimple Lie groups, J. Math. Soc. Japan18(1966), 1–44

    work page 1966

  9. [9]

    Jean-Paul Brasselet, J¨ org Sch¨ urmann, and Shoji Yokura,Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal.2(2010), no. 1, 1–55

    work page 2010

  10. [10]

    William Fulton,Intersection theory, Second, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998

    work page 1998

  11. [11]

    I. M. Gessel and X. G. Viennot,Determinants, paths, and plane partitions, Preprint (1989), 23

    work page 1989

  12. [12]

    William Graham, Minyoung Jeon, and Scott Larson,Irreducible characteristic cycles for orbit closures of a symmetric subgroup, In preparation

    work page

  13. [13]

    Math.320(2017), 115–156

    Thomas Hudson, Takeshi Ikeda, Tomoo Matsumura, and Hiroshi Naruse,Degeneracy loci classes inK- theory—determinantal and Pfaffian formula, Adv. Math.320(2017), 115–156

    work page 2017

  14. [14]

    Combin.70(2018), 190–201

    Thomas Hudson and Tomoo Matsumura,Vexillary degeneracy loci classes inK-theory and algebraic cobordism, European J. Combin.70(2018), 190–201

    work page 2018

  15. [15]

    Algebraic Geom.25(2016), no

    June Huh,Positivity of Chern classes of Schubert cells and varieties, J. Algebraic Geom.25(2016), no. 1, 177–199

    work page 2016

  16. [16]

    Mihalcea, and Hiroshi Naruse,FactorialP- andQ-Schur functions represent equivariant quantum Schubert classes, Osaka J

    Takeshi Ikeda, Leonardo C. Mihalcea, and Hiroshi Naruse,FactorialP- andQ-Schur functions represent equivariant quantum Schubert classes, Osaka J. Math.53(2016), no. 3, 591–619

    work page 2016

  17. [17]

    Minyoung Jeon,Mather classes of Schubert varieties via small resolutions, Math. Res. Lett.30(2023), no. 2, 463–507

    work page 2023

  18. [18]

    Jones,Singular Chern classes of Schubert varieties via small resolution, Int

    Benjamin F. Jones,Singular Chern classes of Schubert varieties via small resolution, Int. Math. Res. Not. IMRN8(2010), 1371–1416

    work page 2010

  19. [19]

    Math.372(2020), 107299, 43

    Eric Marberg and Brendan Pawlowski,K-theory formulas for orthogonal and symplectic orbit closures, Adv. Math.372(2020), 107299, 43. 29

    work page 2020

  20. [20]

    Toshihiko Matsuki,The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan31(1979), no. 2, 331–357

    work page 1979

  21. [21]

    Adam Parusi´ nski and Piotr Pragacz,Chern-Schwartz-MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc.8(1995), no. 4, 793–817

    work page 1995

  22. [22]

    Groups26(2021), no

    Brendan Pawlowski,Universal graph Schubert varieties, Transform. Groups26(2021), no. 4, 1417–1461

    work page 2021

  23. [23]

    Sutipoj Promtapan and Rich´ ard Rim´ anyi,Characteristic classes of symmetric and skew-symmetric de- generacy loci, Facets of algebraic geometry. Vol. II, 2022, pp. 254–283

    work page 2022

  24. [24]

    R. W. Richardson and T. A. Springer,The Bruhat order on symmetric varieties, Geom. Dedicata35 (1990), no. 1-3, 389–436

    work page 1990

  25. [25]

    Reine Angew

    Harry Tamvakis,Giambelli, Pieri, and tableau formulas via raising operators, J. Reine Angew. Math.652 (2011), 207–244

    work page 2011

  26. [26]

    Toshiyuki Tanisaki,Holonomic systems on a flag variety associated to Harish-Chandra modules and repre- sentations of a Weyl group, Algebraic groups and related topics (Kyoto/Nagoya, 1983), 1985, pp. 139–154

    work page 1983

  27. [27]

    Wolf,The action of a real semisimple group on a complex flag manifold

    Joseph A. Wolf,The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc.75(1969), 1121–1237

    work page 1969

  28. [28]

    Wyser and A

    B. Wyser and A. Yong,Polynomials for symmetric orbit closures in the flag variety, Transform. Groups 22(2017), no. 1, 267–290

    work page 2017

  29. [29]

    Wyser,K-orbit closures onG/Bas universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form, Transform

    Benjamin J. Wyser,K-orbit closures onG/Bas universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form, Transform. Groups18(2013), no. 2, 557–594

    work page 2013

  30. [30]

    Shoji Yokura,A singular Riemann-Roch for Hirzebruch characteristics, Singularities Symposium— lojasiewicz 70 (Krak´ ow, 1996; Warsaw, 1996), 1998, pp. 257–268

    work page 1996

  31. [31]

    Algebra497(2018), 55–91

    Xiping Zhang,Chern classes and characteristic cycles of determinantal varieties, J. Algebra497(2018), 55–91. Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon, 34126, Republic of Korea Email address:minyoungjeon@ibs.re.kr

    work page 2018