ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

4. Algebraic Geometry

1991 MS Classification: 14H10, 14D22 Cornalba, Maurizio, Università di Pavia, Italia Cohomology of moduli spaces of stable curves

I will illustrate some recent progress in the calculation of the cohomology of \overline{M}_{g,n}, the Deligne-Mumford-Knudsen compactification of the moduli space of genus g algebraic curves with n numbered punctures. In one direction, an elementary algebro-geometric calculation of the first, second, third, and fifth rational cohomology groups of \overline{M}_{g,n}, for all g and n, has been given by Arbarello and myself. The proof uses a Lefschetz-type argument to do induction on g and n, starting from a few cases that can be disposed of directly. In a different direction, considerable progress has been made, mostly by Getzler, in understanding the cohomology ring of \overline{M}_{g,n} for g = 1 and for g = 2 and low n. In particular, remarkable new relations satisfied by the tautological classes in the fourth cohomology of \overline{M}_{g,n} have been found by Getzler for g = 1, n = 4, and by Belorusski and Pandharipande for g = 2, n = 3.

ICM 1998 Berlin · Plenary and Invited Lectures