### 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'.