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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Pure Maths Colloquium: Hodge type theorems for arithmetic manifolds associated to orthogonal and unitary groups.

Presented by John Millson, Maryland

17 August 2015 16:00 in PH30 (Physics)

I will speak about two papers with Nicolas Bergeron and Colette Moeglin (and one paper with Bergeron, Zhiyuan Li and Moeglin). For each n between 1 and p, the (standard) arithmetic manifolds M associated to the orthogonal groups SO(p,q) resp. the unitary groups SU(p,q) contain many totally geodesic submanifolds N of codimension nq associated to embedded subgroups SO(p-n,q) resp SU(p-n,q). In the 1980's Steve Kudla and I proved that the cohomology classes dual to such N could be represented by differential forms constructed using the Weil or oscillator representation. More precisely, for the case of SO(p,q), the theory of the oscillator representation provides an integral transform ( "geometric theta lifting") from the space of holomorphic Siegel modular forms for Sp(2n,R) and weight (p+q)/2 to harmonic differential nq-forms on the above manifolds M. Recently using Arthur's work on the Selberg trace formula we proved that the geometric theta lifting was onto. In my talk I will explain the basic differential-geometric principles behind the geometric theta lifting.

The theory of the previous paragraph has the following applications:

1.For the case of SO(p,1,) p >3,, the next-to-top homology group H_{p-1}(M) for the standard arithmetic real hyperbolic p-manifolds M is spanned by totally-geodesic hypersurfaces.

2. For the case of SU(p,1), the Hodge and Tate conjectures hold away from the middle third of cohomological degrees.

3. For the case of SO(2,19), the Noether-Lefschetz conjecture of Maulik and Pandharapande holds (Noether-Lefschetz divisors span the Picard group).

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