Involutions of varieties and Rost’s degree formula
-
Olivier Haution
Abstract
To an algebraic variety equipped with an involution, we associate a cycle class in the modulo two Chow group of its fixed locus. This association is functorial with respect to proper morphisms having a degree and preserving the involutions. Specialising to the exchange involution of the square of a complete variety, we obtain Rost’s degree formula in arbitrary characteristic (this formula was proved by Rost and Merkurjev in characteristic not two).
Acknowledgements
I would like to acknowledge the crucial influence of the ideas of Markus Rost, and especially those appearing in the introduction of the preprint [24]. In particular, his proof that the Segre number is even in characteristic two was the starting point of this work. I also drew inspiration from the construction of Steenrod squares given in [25, §2] and [3, Chapter XI] (related constructions are performed in [26, §5]). I thank the referee for his/her careful reading of the paper, and his/her constructive suggestions.
References
[1] P. Aluffi, Shadows of blow-up algebras, Tohoku Math. J. (2) 56 (2004), no. 4, 593–619. 10.2748/tmj/1113246753Search in Google Scholar
[2] N. Bourbaki, Éléments de mathématique. Algèbre commutative. Chapitre 10, Springer, Berlin 2007. 10.1007/978-3-540-34395-0Search in Google Scholar
[3] R. Elman, N. A. Karpenko and A. S. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society, Providence 2008. 10.1090/coll/056Search in Google Scholar
[4] W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. 10.1007/978-1-4612-1700-8Search in Google Scholar
[5] A. Grothendieck, Éléments de géométrie algébrique. II: Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961), 1–222. 10.1007/BF02699291Search in Google Scholar
[6] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Séconde partie), Publ. Math. Inst. Hautes Études Sci. 24 (1965), 1–231. 10.1007/BF02684322Search in Google Scholar
[7] A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960–61. Revêtements étales et groupe fondamental (SGA 1), Doc. Math. (Paris) 3, Société Mathématique de France, Paris 2003. Search in Google Scholar
[8] O. Haution, Integrality of the Chern character in small codimension, Adv. Math. 231 (2012), no. 2, 855–878. 10.1016/j.aim.2012.04.030Search in Google Scholar
[9] O. Haution, Reduced Steenrod operations and resolution of singularities, J. K-Theory 9 (2012), no. 2, 269–290. 10.1017/is011006030jkt162Search in Google Scholar
[10] O. Haution, Degree formula for the Euler characteristic, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1863–1869. 10.1090/S0002-9939-2012-11450-9Search in Google Scholar
[11] O. Haution, Invariants of upper motives, Doc. Math. 18 (2013), 1555–1572. 10.4171/dm/436Search in Google Scholar
[12] O. Haution, Detection by regular schemes in degree two, Algebr. Geom. 2 (2015), no. 1, 44–61. 10.14231/AG-2015-003Search in Google Scholar
[13] D. W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995), no. 3, 461–476. 10.1007/BF02572626Search in Google Scholar
[14] D. W. Hoffmann and A. Laghribi, Isotropy of quadratic forms over the function field of a quadric in characteristic 2, J. Algebra 295 (2006), no. 2, 362–386. 10.1016/j.jalgebra.2004.02.038Search in Google Scholar
[15] O. T. Izhboldin, Motivic equivalence of quadratic forms. II, Manuscripta Math. 102 (2000), no. 1, 41–52. 10.1007/s002291020041Search in Google Scholar
[16] N. A. Karpenko, Canonical dimension, Proceedings of the international congress of mathematicians. II: Invited lectures (ICM 2010), World Scientific, Hackensack (2010), 146–161. 10.1142/9789814324359_0044Search in Google Scholar
[17] S. Kimura, Fractional intersection and bivariant theory, Comm. Algebra 20 (1992), no. 1, 285–302. 10.1080/00927879208824340Search in Google Scholar
[18] M.-A. Knus, A. S. Merkurjev, M. Rost and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence 1998. 10.1090/coll/044Search in Google Scholar
[19] M. Levine and F. Morel, Algebraic cobordism, Springer Monogr. Math., Springer, Berlin 2007. Search in Google Scholar
[20] A. S. Merkurjev, Rost’s degree formula, Notes of mini-course in Lens (2001), www.math.ucla.edu/~merkurev/papers/lens.dvi. Search in Google Scholar
[21] A. S. Merkurjev, Algebraic oriented cohomology theories, Algebraic number theory and algebraic geometry, Contemp. Math. 300, American Mathematical Society, Providence (2002), 171–193. 10.1090/conm/300/05148Search in Google Scholar
[22] A. S. Merkurjev, Steenrod operations and degree formulas, J. reine angew. Math. 565 (2003), 13–26. 10.1515/crll.2003.098Search in Google Scholar
[23] M. Rost, Norm varieties and algebraic cobordism, Proceedings of the international congress of mathematicians. II: Invited lectures (ICM 2002), Higher Education Press, Beijing (2002), 77–85. Search in Google Scholar
[24]
M. Rost,
On Frobenius,
[25] A. Vishik, Symmetric operations (in Russian), Tr. Mat. Inst. Steklova 246 (2004), 92–105; translation in Tr. Mat. Inst. Steklova 246 (2004), 79–92. Search in Google Scholar
[26] A. Vishik, Symmetric operations in algebraic cobordism, Adv. Math. 213 (2007), no. 2, 489–552. 10.1016/j.aim.2006.12.012Search in Google Scholar
[27] V. Voevodsky, The Milnor conjecture, preprint (1996), www.math.uiuc.edu/K-theory/0170/. Search in Google Scholar
[28] M. Zhykhovich, Décompositions motiviques des variétés de Severi–Brauer généralisées et isotropie des involutions unitaires, Ph.D. thesis, Université Pierre et Marie Curie, Paris 2012. Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Equivariant basic cohomology of Riemannian foliations
- Motivic decomposition of compactifications of certain group varieties
- A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2
- Associated forms and hypersurface singularities: The binary case
- Stratified-algebraic vector bundles
- Best possible rates of distribution of dense lattice orbits in homogeneous spaces
- K-homological finiteness and hyperbolic groups
- Involutions of varieties and Rost’s degree formula
- Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
- Addendum to “Singular equivariant asymptotics and Weyl’s law”
Articles in the same Issue
- Frontmatter
- Equivariant basic cohomology of Riemannian foliations
- Motivic decomposition of compactifications of certain group varieties
- A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2
- Associated forms and hypersurface singularities: The binary case
- Stratified-algebraic vector bundles
- Best possible rates of distribution of dense lattice orbits in homogeneous spaces
- K-homological finiteness and hyperbolic groups
- Involutions of varieties and Rost’s degree formula
- Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
- Addendum to “Singular equivariant asymptotics and Weyl’s law”
