The Betti numbers of the Hilbert scheme of points on a smooth projective surface

Fields, Strings, Matrices and Symmetric Products

“…We will only be able to fully justify these definitions (because that's what it is at this point) from the string theory considerations that we present in the next section. For the moment let us just state that in particular cases where the symmetric product allows for a natural smooth resolution (as for the algebraic surfaces studied in [16] where the Hilbert scheme provides such a resolution), we expect the orbifold definition to be compatible with the usual definition in terms of the smooth resolution.…”

BPS invariants of $\CN=4$ gauge theory on Hirzebruch surfaces

“…S-duality suggests that the generating function of the BPS invariants (3.2) should exhibit modular properties if the gauge group is SU (r) or U (r). They tested this in various cases, for example for sheaves with rank r = 1 [11], and r = 2 on P 2 [18,35,36]. The generating functions for rank 1 were found to be genuine (weakly) holomorphic modular forms.…”

On the Hilbert scheme of points of an almost complex fourfold

“…Many beautiful works ( [3], [7], [8], [1]) have been done towards the understanding of their cohomology and Hodge structures-it appears to be built by universal constructions on several copies of the tensor product of H*{X) with shifts of degree-and a very interesting structure in the infinite dimensional space (Dfe.H^Hilb^X)), endowed with the actions of H*(X) on it induced by the natural incidence correspondences Zk,n C Hilb^X) x Hilb^^X) x X, has been found in [12].…”