Seminar by Gergely Berczi (University of Oxford)

Title: GIT for positively graded groups and applications

2016.04.01 | Jane Jamshidi

Date Thu 07 Apr
Time 15:30 16:30
Location 1531-219 (Aud. D4)

Abstract:

The orbit space for unipotent group actions on projective varieties is often too complicated to define a well-behaved quotient. In most applications, however, the acting group H is not unipotent but it contains a C^* which normalises the maximal unipotent subgroup U of H and acts with positive weights on the Lie algebra of U.
We call these groups positively graded. After a short review of Mumford's GIT for reductive groups I will explain how the key geometric and computational features of GIT can be extended to positively graded groups. I will briefly mention two applications: construction of moduli of unstable vector bundles over a curve and the moduli of representations of quivers with multiplicities. I will then demonstrate through an explicit example how the topology of non-reductive moduli spaces can be recovered and I will explain new iterated residue formulas for classical problems such as Thom polynomials of singularities, the Green-Griffiths-Lang hyperbolicity conjecture and enumerative geometry problems of counting hypersurfaces with prescribed singularities in an ample linear system over a projective variety. As a special case I will show a new iterated residue formula for the number of nodal curves on a surface which is different from the Gottsche formula.
This review is based on joint works from the last few years with A. Szenes, F. Kirwan, T. Hawes and B. Doran.