# DAGA Seminar: Hyperpolygons and Moduli Spaces of Parabolic Higgs Bundle

## Thursday 1 December 2011 - Alessia Mandini

ARGOMENTI: Seminari

Thursday, December 1, 2011, h. 11:30, room 2BC60
Alessia Mandini (Instituto Superior Técnico, Lisbon University)
"Hyperpolygons and Moduli Spaces of Parabolic Higgs Bundle"

-Abstract
I will describe two families of manifolds: hyperpolygon spaces and moduli spaces of stable, rank-2, holomorphically trivial parabolic Higgs bundles over P1with fixed determinant and trace free Higgs field, proving the existence of an isomorphism between them. This relationship connecting two different fields of study allows us to benefit from techniques and ideas from each of these areas to obtain new results. In particular, using the study of variation of moduli spaces of parabolic Higgs bundles over a curve, we describe the dependence of hyperpolygon spaces X(?) and their cores on the choice of the parameter ?, and show that, when a wall is crossed, the hyperpolygon space suffers an elementary transformation in the sense of Mukai. If time permits, I will describe how one can take advantage of the geometric description of the core components of a hyperpolygon space to obtain explicit expressions for the computation of the intersection numbers of the core components of hyperpolygon spaces. Using our isomorphism we can obtain similar formulas for the nilpotent cone components of the moduli space of rank-2, holomorphically trivial parabolic Higgs bundles over P1with fixed determinant and trace-free Higgs field. This is joint work with Leonor Godinho.

Rif. int. A. Bertapelle