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 Riemann-Roch theorem. Here I want to look at another method, and convince you of its utility. The idea is to use the Atiyah-Bott 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).

Copyright © 2003 UBC Mathematics Department