Singular Todd classes of tautological sheaves on Hilbert schemes
of points on a smooth surface
Let X be a quasi-projective, smooth complex algebraic surface, with
the Hilbert scheme of n points on X, so that the (rational)
cohomology of all these Hilbert schemes together can be generated by the
cohomology of X in terms of Nakajima creation operators. Given an algebraic
vector bundle V on X, there exist universal formulae for the
characteristic classes of the associated tautological vector bundles
in terms of the Nakajima creation operators and the
corresponding characteristic classes of V. But in general the corresponding
coefficients are not known.
Based on the derived equivalence of Bridgeland-King-Reid and work of Haiman
and Scala, we give an explicit formula in case of the singular Todd classes,
but in terms of Nakajima creation operators of the
delocalized equivariant cohomology of all Xn with its natural Sn-action.
- Lisa Traynor:
Lagrangian Cobordisms between Legendrian Submanifolds
Smooth cobordisms play an important role in topology. Lagrangian cobordisms between Legendrian submanifolds are smooth
cobordisms that satisfy additional geometric conditions imposed by symplectic and contact structures. From a qualitative perspective,
Lagrangian cobordisms have more topological rigidity than smooth cobordisms: for example, any Lagrangian cobordism between a Legendrian
unknot and a Legendrian trefoil must have genus equal to 1. From a quantitative perspective, Lagrangian cobordisms are at times very
flexible while at other times have some rigidity: between some pairs of Legendrian submanifolds it is possible to find arbitrarily short
Lagrangian cobordisms, while between other pairs of Legendrians there is a positive lower bound to the length of a Lagrangian cobordism.
This is joint work with Joshua Sabloff.