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

Abstract : 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 $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients inverted. We compute base change formulas in $A_{K}^*(Hilb^n(S))$ between the natural bases introduced by Nakajima, Ellingsrud and Str{\o}mme, and the classical basis associated with the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in $Hilb^n(S)$ in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme $Hilb^[n,n+1](S)$ parametrizing nested punctual subschemes of degree $n$ and $n+1$ is irreducible.
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Contributor : Laurent Evain <>
Submitted on : Friday, June 29, 2012 - 3:26:59 PM
Last modification on : Wednesday, April 1, 2020 - 12:20:06 AM

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  • HAL Id : hal-00713204, version 1
  • ARXIV : 1205.5470



Pierre-Emmanuel Chaput, Laurent Evain. On the equivariant cohomology of Hilbert schemes of points in the plane. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2015, 65 (3), pp.1201--1250. ⟨hal-00713204⟩



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