The seminar normally meets from 4:10 to about 6:00 pm, with a break around 5 pm. We encourage speakers to make the first hour of each talk expository and introductory in nature, so that this part of the seminar will be accessible to graduate students and postdocs in other fields. Details and technicalities are welcome (but not required!) after the break.

For more information, contact  Rob Lazarsfeld ( rlaz@umich.edu ) .

Past Seminars: W09  F08  W08  F07 W07  F06  W06  F05 W05  F04  W04  F03 W03  F02  W02  F01 W01  F00  W00  F99  W99

Abstracts

Topology of the Hitchin fibration and Hodge theory of character varieties

Given a compact Riemann surface X of genus at least two, there are two algebraic varieties attached to it: the character variety Ch, and the Hitchin moduli space M. The non Abelian Hodge theorem asserts that they are diffeomorphic (but have different complex structures). While the rational cohomology rings H^*(Ch) and H^*(M) are isomorphic, the mixed Hodge structures are different and so are the weight filtrations, which therefore cannot possibly correspond via the non Abelian Hodge theorem.

In recent joint work with T. Hausel (Oxford) and L. Migliorini (Bologna) it is shown that the non Abelian Hodge theorem exchanges the weight filtration on H^*(Ch) with the (perverse) Leray filtration on H^*(M) for the Hitchin map h: M→Cⁿ. Moreover, curious symmetries observed on H^*(Ch) by number-theoretic means, turn out to be the more familiar Lefschetz and Poincaré symmetries for the map h. (The perverse Leray filtration is formally analogous to the Leray filtration associated with the Leray spectral sequence. However, as my on-line thesaurus states: "perverse = resistant to guidance or discipline.")

Rigidity properties of Fano varieties

I will discuss some deformation properties of Fano varieties. The general methods rely on the investigation of the variation of the cone of effective curves and, more generally, of the Mori chamber decomposition, which, according to Mori theory, encode information on the geometry of the variety. The talk is based on joint work with C. Hacon.

Deformations of zero-dimensional schemes

A natural question in the study of zero-dimensional schemes is to determine when such a scheme deforms to a disjoint union of points. I will discuss a syzygetic invariant which yields sharp information about this question. I will also give applications to the study of Hilbert schemes of points. This is joint work with M. Velasco.