Colloquium
3:00 p.m., Friday (September 12, 2003)
Math Annex 1100
Aaron Bertram
University of Utah
Localizing to get Relations in Cohomology
Continuous maps of compact manifolds generate cohomology
classes by pushing forward and pulling back. For example,
an embedding:
f:X \rightarrow Y
generates the class f_*1 (``the class of X'') in
the cohomology of Y, and more mysteriously, f^*f_*1
(``the Euler class of X in Y'') in the cohomology of X.
If the manifolds are also algebraic varieties, and the maps
are regular, one can also push forward coherent sheaves and
take Chern classes to generate even more (and more mysterious)
cohomology classes.
Moduli spaces in algebraic geometry tend to have lots of maps
among them, giving rise to lots of cohomology classes
(the ``tautological classes''). So here's a relevant question.
When are two classes generated by maps cohomologous? That is,
what are the relations among tautological classes?
One standard method for finding relations involves hard calculations
with the Grothendieck RiemannRoch theorem. Here I want to look
at another method, and convince you of its utility. The idea is
to use the AtiyahBott localization theorem. In its most crude form,
this involves taking a compact manifold Z with a group action and
a ``forgetful'' map, that is an equivariant map from Z to the
manifold X (equipped with the trivial group action):
f:Z \rightarrow X.
Localization then gives relations among certain classes generated
by the maps F \rightarrow X from the fixed loci of the group action.
Surprisingly, this method (or rather a refinement of it) gives a
wealth of relations among tautological classes on moduli spaces of
pointed curves. I'll illustrate this with the case of the
(compactified) moduli space of elliptic curves, and the famous
relation:
\lambda ={\Delta \over 12}
on that space.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
