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Algebra Seminar
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Joost van Hamel
University of Sydney

Extended Albanese motives and the arithmetic of linear algebraic groups

Friday 23rd April, 12:05-12:55pm, Carslaw 373.

Let X be a non-singular (but not necessarily complete) algebraic variety. To X we associate a so-called generalised Albanese variety, which is a commutative group variety. I will explain its construction and I will explain how this generalised Albanese variety can be seen as a geometric representative of the maximal free abelian quotient of the fundamental group of X.

For a linear algebraic group G over a number field, the generalised Albanese variety captures important information about the arithmetic of G, provided the fundamental group of G is torsion-free. Otherwise we have to extend the generalised Albanese variety to something that represents the whole fundamental group of G (which happens to be abelian), rather than just the torsion-free quotient. This extension was defined group-theoretically by M. Borovoi, inspired by work of P. Deligne and R. Kottwitz. In this talk I will give a geometric construction of this `extended Albanese motive'.