# Hilbert schemes and W algebras

@article{Li2001HilbertSA, title={Hilbert schemes and W algebras}, author={Wei-Ping Li and Zhenbo Qin and Weiqiang Wang}, journal={arXiv: Algebraic Geometry}, year={2001} }

We construct geometrically the generating fields of a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes of n points on a projective surface for all n. We compute explicitly the commutators among a set of linear basis elements of the W algebra, and identify this algebra with a $W_{1+\infty}$-type algebra. A precise formula of certain Chern character operators, which is essential for the construction of the W algebra, is established in terms of the…

## 21 Citations

### Ideals of the cohomology rings of Hilbert schemes and their applications

- Mathematics
- 2002

We study the ideals of the rational cohomology ring of the Hilbert scheme X (n) of n points on a smooth projective surface X. As an application, for a large class of smooth quasi-projective surfaces…

### A ug 2 00 2 IDEALS OF THE COHOMOLOGY RINGS OF HILBERT SCHEMES AND THEIR APPLICATIONS

- Mathematics
- 2009

We study the ideals of the rational cohomology ring of the Hilbert scheme X [n] of n points on a smooth projective surface X . As an application, for a large class of smooth quasi-projective surfaces…

### GENERATING SERIES IN THE COHOMOLOGY OF HILBERT SCHEMES OF POINTS ON SURFACES

- Mathematics
- 2006

In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the…

### Hilbert schemes and symmetric products: a dictionary

- Mathematics
- 2001

Given a closed complex manifold $X$ of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings $\orbsym$ of the symmetric products. We…

### Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

- Mathematics
- 2006

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When X is C 2 , this was done in [13, 21] by realising this ring as a degeneration of the…

### On Okounkov's conjecture connecting Hilbert schemes of points and multiple q-zeta values

- Mathematics
- 2015

We compute the generating series for the intersection pairings between the total Chern classes of the tangent bundles of the Hilbert schemes of points on a smooth projective surface and the Chern…

### K-Theory of Hilbert Schemes as a Formal Quantum Field Theory

- Mathematics
- 2018

We define a notion of formal quantum field theory and associate a formal quantum field theory to K-theoretical intersection theories on Hilbert schemes of points on algebraic surfaces. This enables…

### Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

- MathematicsPure and Applied Mathematics Quarterly
- 2022

We prove that the complex cobordism class of any hyper-Kähler manifold of dimension 2n is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of K3…

### LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2020

Abstract The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle…

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Given a closed complex manifold $X$ of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings $\orbsym$ of the symmetric products. We…

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