Special Year Research Seminar
Talk #1: On the Non-abelian Hodge Correspondence for Higher-dimensional Quasiprojective Varieties | Talk #2: Monodromy of Lagrangian Fibrations
Speaker #1 (Tran): On a projective variety, Simpson showed that there is a homeomorphism between the moduli space of semisimple flat bundles and that of polystable Higgs bundles with vanishing Chern classes. Recently, Bakker, Brunebarbe and Tsimerman proved a version of this homeomorphism on a log smooth curve; for a log smooth variety of higher dimension, they got a continuous bijection. In this talk, I will give a sketch of their approach and give an argument that we do get a homeomorphism for arbitrary dimension.
Speaker #2 (Varvak): A compact hyperkahler manifold is a higher-dimensional analog of a K3 surface; Lagrangian fibrations of hyperkahler manifolds are higher-dimensional versions of elliptic fibrations of K3 surfaces. A result of Voisin shows that these fibrations yield irreducible real variations of weight one Hodge structures. We show that when the variation is not isotrivial, its underlying local system is irreducible over the complex numbers.