Manfred Lehn

Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

Abstract. We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface within the framework of Nakajimas oscillator algebra. This leads to an identification of the cohomology ring of Hilbn(A2) with a ring of explicitly given differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilbn up to n=7, extending computations of Severi, LeBarz, Tikhomirov and Troshina and give a conjecture for the generating series.

Singular symplectic moduli spaces

Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O’Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension ≥4 they do no admit projective symplectic resolutions.

Symmetric groups and the cup product on the cohomology of Hilbert schemes

Let C(Sn) be the Z-module of integer valued class functions on the symmetric group Sn. We introduce a graded version of the con- volution product on C(Sn) and show that there is a degree preserving ring isomorphism C(Sn) ! H ∗ (Hilb n (A 2); Z) to the cohomology of the Hilbert scheme of points in the complex affine plane.

On the symplectic eightfold associated to a Pfaffian cubic fourfold

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.

Irreducibility of the punctual quotient scheme of a surface

AbstractIt is shown that the punctual quotient schemeQlr parametrizing all zero-dimensional quotients

ON THE COTANGENT SHEAF OF QUOT-SCHEMES

\mathcal{O}_{A^2 }^{ \oplus ^r } \to T

The geometry of moduli spaces of sheaves

of lengthl and supported at some fixed point O∈A2 in the plane is irreducible.

The geometry of moduli spaces of sheaves Aspects of Mathematics

We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension 4 and 6.