On the cohomology rings of Grassmann varieties and Hilbert schemes Mahir Bilen Can1 and Jeff Remmel2
[email protected] [email protected] July 20, 2017 Abstract By using vector field techniques, we compute the ordinary and equivariant coho- mology rings of Hilbert scheme of points in the projective plane in relation with that of a Grassmann variety. Keywords: Hilbert scheme of points, zeros of vector fields, equivariant cohomology MSC:14L30, 14F99, 14C05 1 Introduction The Hilbert scheme of points in the plane is a useful moduli that brings forth the power of algebraic geometry for solving combinatorial problems not only in commutative algebra but also in representation theory such as Garsia-Haiman modules and Macdonald polyno- mials [22]. Its geometry has deep connections with physics. For example, by considering the cohomology rings of all Hilbert schemes of points together, one obtains Fock representation of the infinite dimensional Heisenberg algebra, which has importance for string theorists (see arXiv:1706.02713v3 [math.AG] 24 Jul 2017 [30] and the references therein). We denote by Hilbk(X) the Hilbert scheme of k points in a complex algebraic variety 2 X. In this paper, we will be concerned with the cohomology ring of Hilbk(P ). Its additive structure was first described by Ellingsrud and Strømme in [16], where it was shown that 2 the Chern characters of tautological bundles on Hilbk(P ) are enough to generate it as a Z-algebra. In relation with the Virasoro algebra, a finer system of generators for the cohomology ring is described in [28] by Li, Qin, and Wang.