An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane

We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{ G}_{ m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{ C}^{ 2}$ for a natural $\mathbb{ G}_{ m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.

Categories: math.AG math.RT