Quantum function algebras as quantum enveloping algebras

Inspired by a result in [Ga], we locate two $k[q,q^{-1}]$-integer forms of $F_q[SL(n+1)]$, along with a presentation by generators and relations, and prove that for $q=1$ they specialize to $U({\mathfrak{h}})$, where ${\mathfrak{h}}$ is the Lie bialgebra of the Poisson Lie group $H$ dual of $SL(n+1)$; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $F_q[SL(n+1)]$, both related to the classical PBW theorem for $U({\mathfrak{h}})$.Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final version, as it appears in the journal printed version, BUT for a formula in Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this very pre(post)print have been correcte

Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold $X$. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on $X$, which are shown to be canonical in $K$-theory

Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.Comment: 20 pages, 0 figure

Del Pezzo surfaces as Springer fibres for exceptional groups

We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups