Equivariant cohomology of incidence Hilbert schemes and loop algebras

Details Equivariant cohomology of incidence Hilbert schemes and loop algebras Equivariant cohomology of incidence Hilbert schemes and loop algebras

2008-02-12

arxiv.org/abs/0802.1673v1

Mathematics

Algebraic Geometry

30 pages, 2 figures

Scientific paper

Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T = \C^*$ action. Let $\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct sum $$\Wft = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of $\hil{m,m+1}$. The algebra is related to the loop algebra of an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of $\Wft$. Our results are applied to the ring structure of the ordinary cohomology of $\hil{m,m+1}$ and to the ring of symmetric functions in infinitely many variables.

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