Moduli spaces of sheaves on Hirzebruch surfaces, Poisson geometry, and integrable systems

Abstract

This research project has two aims. The first is to investigate Poisson structures on Hilbert schemes of points on Hirzebruch surfaces. We believe that this goal can be achieved by using the description of these Hilbert schemes in terms of quiver varieties given by Bartocci, Bruzzo, Lanza, and Rava, and by adapting to our needs some techniques of noncommutative Poisson geometry developed by Van den Bergh (double brackets) and Crawley-Boevey (H0-Poisson structures). This would also clarify some resultsof Bottacin of late 1990's about Poisson structures on Hilbert schemes over Poisson surfaces. The second aim of this project is to complete the ADHM description of moduli spaces of framed sheaves on Hirzebruch surfaces, accomplished so far only in the rank 1 case by Bartocci, Bruzzo, Lanza, and Rava. It should be possible to achieve this result by suitably using the monadic description of these moduli spaces given earlier by Bartocci, Bruzzo, and Rava. Once obtained the ADHM data, we plan to employ them to answer two questions. First: are moduli spaces of framed sheaves on Hirzebruch surfaces (non-trivially) Poisson? Sala has recently solved the issue for the second Hirzebruch surface, by showing that the moduli spaces under consideration are even symplectic. A second natural question is: can our moduli spaces beinterpreted as integrable systems? As in Nakajima's work on framed sheaves on the projective plane, it is quite likely that ADHM data will allow us to find the first integrals of motion. (AU)