On the equivariant cohomology of Hilbert schemes of points in the plane

Chaput, Pierre-Emmanuel; Evain, Laurent

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 $A_K^{*}\left(S^{\left[n\right]}\right)$ du schéma de Hilbert $S^{\left[n\right]}$. 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 $S^{\left[n\right]}$ 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 $A_K^{*}\left(S^{\left[n\right]}\right)$ of the punctual Hilbert scheme $S^{\left[n\right]}$ with equivariant coefficients inverted. We compute base change formulas in $A_K^{*}\left(S^{\left[n\right]}\right)$ 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 $S^{\left[n\right]}$ 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.

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