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GIT for positively graded groups and applications

Gergely Berczi
(University of Oxford)
Torsdag, 7 april, 2016, at 15:30-16:30, in Aud. D4 (1531-219)
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.
Organiseret af: QGM
Kontaktperson: Jørgen Ellegaard Andersen