# Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

@article{Lehn1998ChernCO,
title={Chern classes of tautological sheaves on Hilbert schemes of points on surfaces},
author={Manfred Lehn},
journal={Inventiones mathematicae},
year={1998},
volume={136},
pages={157-207}
}
• M. Lehn
• Published 20 March 1998
• Mathematics
• Inventiones mathematicae
Abstract. We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbn(A2) with a ring of explicitly given differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilbn up to n=7, extending…
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## References

SHOWING 1-10 OF 31 REFERENCES

This paper is based on author's lectures at Kyoto University in 2010 Summer, and in the 6th MSJ-SI `Development of Moduli Theory' at RIMS in June 2013. The purpose of lectures was to review several
• Mathematics
• 1994
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• Mathematics
• 1996
We compute the intersection number between two cycles A and B of complementary dimensions in the Hilbert scheme H parameterizingr subschemes of given finite length n of a smooth projective surface S.
• Mathematics
• 1994
We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic
• Mathematics
• 1993
Let Xbe a smooth projective variety over the complex numbers C and X the Hubert scheme of subschemes of length 3 of JSf, which is known to be smooth. The additive structure of H* (P2, Z) has been
This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises
Several authors have been interested in the Hilbert scheme SPq:=Hilb"(S) parametrizing finite subschemes of length n on a smooth projective surface S. In EF 1] and I-F 2] Fogarty shows that S tnj is
The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various
• Mathematics
• 1987
Geir Ellingsrud 1 and Stein Arild Stromme 2 i Matematisk institutt, Universitetet i Oslo, Blindern, N-Oslo 3, Norway 2 Matematisk institutt, Universitetet i Bergen, N-5014 Bergen, Norway Although
The linear determinant.- Representation of n-fold sections by symmetric products.- Invertible sheaves and rational maps into C(g).- Construction of the Picard scheme of a family of curves.