On the equivariant cohomology of Hilbert schemes of points in the plane
[Cohomologie équivariante des schémas de Hilbert de points dans le plan]
Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1201-1250.

Soit S le plan affine muni de sa structure de variété torique via l’action du tore T de dimension deux. Nous étudions l’anneau de Chow équivariant A K * (S [n] ) du schéma de Hilbert S [n] . Nous calculons les formules de changement de base entre les bases naturelles introduites par Nakakjima, Ellingsrud et Strømme, et la base classique associée aux points fixes. Nous calculons les relations de commutation quivariantes entre les opérateurs de création/destruction. Nous exprimons la classe de la petite diagonale de S [n] en fonction des classes de Chern équivariante du fibré tautologique. Nous montrons que le schéma de Hilbert imbriqué paramétrant les couples de schémas ponctuels imbriqués de degrés respectifs n et n+1 est irréductible.

Let S be the affine plane regarded as a toric variety with an action of the 2-dimensional torus T. We study the equivariant Chow ring A K * (S [n] ) of the punctual Hilbert scheme S [n] with equivariant coefficients inverted. We compute base change formulas in A K * (S [n] ) between the natural bases introduced by Nakajima, Ellingsrud and Str mme, and the classical basis associated to the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in S [n] in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme S 0 [n,n+1] parametrizing nested punctual subschemes of degree n and n+1 is irreducible.

DOI : 10.5802/aif.2955
Classification : 14C05, 14C15
Keywords: equivariant cohomology, Hilbert schemes, Chow ring
Mot clés : cohomologie quivariante, Schma de Hilbert, Anneau de Chow
Chaput, Pierre-Emmanuel 1, 2 ; Evain, Laurent 3

1 Domaine Scientifique Victor Grignard 239, boulevard des Aiguillettes Universit Henri Poincar Nancy 1 B.P. 70239 54506 Vandoeuvre-ls-Nancy Cedex (France)
2 Institut Elie Cartan de Nancy (France)
3 Universit d’Angers Facult des Sciences Dpartement de maths. 2, Boulevard Lavoisier 49045 Angers Cedex 01 (France)
@article{AIF_2015__65_3_1201_0,
     author = {Chaput, Pierre-Emmanuel and Evain, Laurent},
     title = {On the equivariant cohomology of {Hilbert} schemes of points in the plane},
     journal = {Annales de l'Institut Fourier},
     pages = {1201--1250},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     doi = {10.5802/aif.2955},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2955/}
}
TY  - JOUR
AU  - Chaput, Pierre-Emmanuel
AU  - Evain, Laurent
TI  - On the equivariant cohomology of Hilbert schemes of points in the plane
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 1201
EP  - 1250
VL  - 65
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2955/
DO  - 10.5802/aif.2955
LA  - en
ID  - AIF_2015__65_3_1201_0
ER  - 
%0 Journal Article
%A Chaput, Pierre-Emmanuel
%A Evain, Laurent
%T On the equivariant cohomology of Hilbert schemes of points in the plane
%J Annales de l'Institut Fourier
%D 2015
%P 1201-1250
%V 65
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2955/
%R 10.5802/aif.2955
%G en
%F AIF_2015__65_3_1201_0
Chaput, Pierre-Emmanuel; Evain, Laurent. On the equivariant cohomology of Hilbert schemes of points in the plane. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1201-1250. doi : 10.5802/aif.2955. http://archive.numdam.org/articles/10.5802/aif.2955/

[1] Białynicki-Birula, A. Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 24 (1976) no. 9, pp. 667-674 | MR | Zbl

[2] Briançon, Joël Description de H ilb n C{x,y}, Invent. Math., Volume 41 (1977) no. 1, pp. 45-89 | DOI | MR | Zbl

[3] Brion, M. Equivariant Chow groups for torus actions, Transform. Groups, Volume 2 (1997) no. 3, pp. 225-267 | DOI | MR | Zbl

[4] Bulois, Michael; Evain, Laurent Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras (http://arxiv.org/abs/1306.4838)

[5] Chaput, Pierre-Emmanuel; Evain, Laurent On the equivariant cohomology of Hilbert schemes of points in the plane (http://arxiv.org/abs/1205.5470)

[6] Cheah, Jan Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math., Volume 183 (1998) no. 1, pp. 39-90 | DOI | MR | Zbl

[7] Edidin, Dan; Graham, William Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math., Volume 120 (1998) no. 3, pp. 619-636 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.3edidin.pdf | DOI | MR | Zbl

[8] Ellingsrud, Geir; Strømme, Stein Arild On the homology of the Hilbert scheme of points in the plane, Invent. Math., Volume 87 (1987) no. 2, pp. 343-352 | DOI | MR | Zbl

[9] Evain, L. The Chow ring of punctual Hilbert schemes on toric surfaces, Transform. Groups, Volume 12 (2007) no. 2, pp. 227-249 | DOI | MR | Zbl

[10] Evain, Laurent Irreducible components of the equivariant punctual Hilbert schemes, Adv. Math., Volume 185 (2004) no. 2, pp. 328-346 | DOI | MR | Zbl

[11] Fulton, William Intersection theory, 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], 2, Springer-Verlag, Berlin, 1998, pp. xiv+470 | DOI | MR | Zbl

[12] Gaffney, Terence; Lazarsfeld, Robert On the ramification of branched coverings of P n , Invent. Math., Volume 59 (1980) no. 1, pp. 53-58 | DOI | MR | Zbl

[13] Grothendieck, Alexander Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249-276 | Numdam | Zbl

[14] Lazarsfeld, Robert Kendall BRANCHED COVERINGS OF PROJECTIVE SPACE, ProQuest LLC, Ann Arbor, MI, 1980, pp. 45 Thesis (Ph.D.)–Brown University | MR

[15] Lehn, Manfred Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math., Volume 136 (1999) no. 1, pp. 157-207 | DOI | MR | Zbl

[16] Nakajima, Hiraku Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999, pp. xii+132 | MR | Zbl

[17] Schiffmann, Olivier; Vasserot, Eric The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of 𝔸 2 (http://arxiv.org/abs/0905.2555) | MR

[18] Vasserot, Eric Sur l’anneau de cohomologie du schéma de Hilbert de C 2 , C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 1, pp. 7-12 | DOI | MR | Zbl

Cité par Sources :