Let be the affine plane regarded as a toric variety with an action of the 2-dimensional torus . We study the equivariant Chow ring of the punctual Hilbert scheme with equivariant coefficients inverted. We compute base change formulas in 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 in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme parametrizing nested punctual subschemes of degree and is irreducible.
Soit le plan affine muni de sa structure de variété torique via l’action du tore de dimension deux. Nous étudions l’anneau de Chow équivariant du schéma de Hilbert . 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 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 et est irréductible.
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
@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 = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 65 (2015) no. 3, pp. 1201-1250. doi : 10.5802/aif.2955. https://aif.centre-mersenne.org/articles/10.5802/aif.2955/
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