Quasi Poisson structures and singular Poisson geometry of moduli spaces

02.05.2023 15:15 – 17:00

Moduli spaces typically exhibit singularities. For example, ordinary complex projective 3-space arises as some moduli space having a Kummer surface as its singular part. The theory of quasi Poisson structures provides some insight into such singularities, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of a symmetry group and a 2-tensor measure how that bracket fails to satisfy the Jacobi identity. The talk will be a guided tour through the workshop of quasi structures and will indicate how we can exploit them to gain insight into singularities in terms of the resulting Poisson geometry.

The theory yields, via reduction with respect to an appropriate momentum mapping, not necessarily non-degenerate ordinary Poisson structures on moduli spaces of representations, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and on moduli spaces of semistable holomorphic vector bundles as well as on Higgs bundles moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the Kähler form of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Also, this framework provides means to make sense of the idea of a Kähler space with singularities and of that of a hyperkähler space with singularities.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Tuesday 02.05.2023, Séminaire "Groupes de Lie et espaces de modules"

entrée libre