Algebraic Geometry Seminar
Tuesday, November 14, 2000, 4:00pm, 120 Physics
Matthew Cushman (Kromos Technologies/University of Chicago)
The Motivic Fundamental Group
Abstract:- The fundamental group of a topological space is usually defined in
terms of homotopy classes of based loops. The group structure is
given by composition of loops. If X is a complex algebraic variety,
one has an underlying topological space, and hence a fundamental group.
Hain showed that the nilpotent completion of the group ring of this
topological fundamental group carries a mixed Hodge structure.
Grothendieck defined a fundamental group for schemes defined over any
field. Applying this to a complex algebraic variety, one obtains the
profinite completion of the topological fundamental group. This group
comes with a natural action of the absolute Galois group of the field
of definition.
The above indicates that varieties over fields of characteristic zero
have two notions of fundamental group, armed with either a Galois
action or a mixed Hodge structure. This is similar to the situation
with homology and cohomology groups, where one has both an etale and
Betti version carrying Galois actions and Hodge structures. An important
guiding principle is that both of these versions of homology and
cohomology should come from an underlying ``motivic'' theory. This is
a homology and cohomology theory for algebraic varieties over a field
k taking values the abelian tensor category of mixed motives over
k, denoted M_k. There should be functors from M_k
to both the category of Galois modules and mixed Hodge structures.
When applied to the motivic homology of a variety X, these functors
should yield the etale homology or Betti homology of X. In this
way, motives glue these two different theories together more strongly
than just the comparison isomorphisms. Nori has recently given a definition of the category of
mixed motives. In this talk, we will show how this category relates to
the fundamental group.
In fact, more generally there is a motivic version of paths between
two different points x and y of X which is important for
applications.
We also show that the multiplication and comultiplication maps are
motivic, and compare with Hain's theory.
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