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The Hodge conjecture for complex ball quotients


Speaker: John Millson, University of Maryland
Location: 2112 MSB
Start time: Thu, May 8 2014, 3:10PM

I will explain the geometry behind my paper "The Hodge Conjecture and Arithmetic Quotients of Complex Balls" with Nicolas Bergeron and Colette Moeglin of the University of Paris. This paper may be found on the archive arXiv:1306.1515. We prove the Hodge and Tate conjectures for compact arithmetic quotients of the complex n-ball in cohomological degrees in cohomological degrees away from a band of width 2n/3 centered around the middle dimension n. We also prove the generalized Hodge conjecture (characterizing push-forwards of classes on algebraic subvarieties) in low degrees in the form first conjectured by Hodge but later shown to be incorrect (in general) by Grothendieck. There are two parts to this paper - a geometry part and a representation theoretic part dealing with the Weil representation and the trace formula . If time permits I will explain what Weil's motivation in constructing the Weil representation was - namely in explaining Siegel's construction of a theta function associated to an indefinite integral valued quadratic form in n variables and what he did with it. This construction is hard to do since there are infinitely many integer vectors where the form is negative and infinitely many integer vectors where the form is positive.