Poincaré duality for tautological Chern subrings of orthogonal grassmannians

Authors

  • Nikita A. Karpenko
  • Alexander S. Merkurjev

DOI:

https://doi.org/10.7146/math.scand.a-132376

Abstract

Let $X$ be an orthogonal grassmannian of a nondegenerate quadratic form $q$ over a field. Let $C$ be the subring in the Chow ring $\text {CH}(X)$ generated by the Chern classes of the tautological vector bundle on $X$. We prove Poincaré duality for $C$. For $q$ of odd dimension, the result was already known due to an identification between $C$ and the Chow ring of certain symplectic grassmannian. For $q$ of even dimension, such an identification is not available.

References

Borel, A., Linear algebraic groups, second ed., Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0941-6

Demazure, M., Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301. https://doi.org/10.1007/BF01418790

Devyatov, R. A., Karpenko, N. A., and Merkurjev, A. S., Maximal indexes of flag varieties for spin groups, Forum Math. Sigma 9 (2021), Paper No. e34. https://doi.org/10.1017/fms.2021.30

Edidin, D. and Graham, W., Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634. https://doi.org/10.1007/s002220050214

Elman, R., Karpenko, N., and Merkurjev, A., The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, American Mathematical Society, Providence, RI, 2008. https://doi.org/10.1090/coll/056

Fulton, W., Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenz­gebiete. 3. Folge. vol. 2, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-1-4612-1700-8

Karpenko, N. A., On generic flag varieties for odd spin groups, Preprint (4 May 2021, 11 pages). Available on authors' webpage.

Karpenko, N. A., Chow groups of some generically twisted flag varieties, Ann. $K$-Theory 2 (2017), no. 2, 341–356. https://doi.org/10.2140/akt.2017.2.341

Karpenko, N. A., On generic quadratic forms, Pacific J. Math. 297 (2018), no. 2, 367–380. https://doi.org/10.2140/pjm.2018.297.367

Karpenko, N. A., On generically split generic flag varieties, Bull. Lond. Math. Soc. 50 (2018), 496–508. https://doi.org/10.1112/blms.12161

Karpenko, N. A. and Merkurjev, A. S., Canonical $p$-dimension of algebraic groups, Adv. Math. 205 (2006), no. 2, 410–433. https://doi.org/10.1016/j.aim.2005.07.013

Matsumura, H., Commutative algebra, second ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.

Merkurjev, A., $R$-equivalence on three-dimensional tori and zero-cycles, Algebra Number Theory 2 (2008), no. 1, 69–89. https://doi.org/10.2140/ant.2008.2.69

Totaro, B., The Chow ring of a classifying space,Algebraic $K$-theory (Seattle, WA, 1997), 249–281, Proc. Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999. https://doi.org/10.1090/pspum/067/1743244

Totaro, B., The torsion index of the spin groups, Duke Math. J. 129 (2005), no. 2, 249–290. https://doi.org/10.1215/S0012-7094-05-12923-4

Vishik, A., Fields of $u$-invariant $2^r+1$, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 661–685, Progr. Math., 270, Birkhäuser Boston, Boston, MA, 2009. https://doi.org/10.1007/978-0-8176-4747-6_22

Published

2022-06-11

How to Cite

Karpenko, N. A., & Merkurjev, A. S. (2022). Poincaré duality for tautological Chern subrings of orthogonal grassmannians: Array. MATHEMATICA SCANDINAVICA, 128(2). https://doi.org/10.7146/math.scand.a-132376

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Articles