Category Archives: Susie

Now it’s about stacks (new paper)

I’ve posted an updated version of a paper on the arXiv, Hodge theory of classifying stacks (initial version posted 10 March 2017). We compute the Hodge and de Rham cohomology of the classifying space BG (defined as sheaf cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology. This is part of the grand theme of bringing topological ideas into algebraic geometry to build analogous organizing structures and machines.

In the first version of the paper, I defined Hodge and de Rham cohomology of classifying spaces by thinking of BG as a simplicial scheme. In the current version, I followed Bhargav Bhatt‘s suggestion to make the definitions in terms of the stack BG, with simplicial schemes as just a computational tool. Although I sometimes resist the language of algebraic stacks as being too abstract, it is powerful, especially now that so much cohomological machinery has been developed for stacks (notably in the Stacks Project). (In short, stacks are a flexible language for talking about quotients in algebraic geometry. There is a “quotient stack” [X/G] for any action of an algebraic group G on a scheme X.)

Using the language of stacks led to several improvements in the paper. For one thing, my earlier definition gave what should be considered wrong answers for non-smooth groups, such as the group scheme μp of pth roots of unity in characteristic p. Also, the paper now includes a description of equivariant Hodge cohomology for group actions on affine schemes, generalizing work of Simpson and Teleman in characteristic zero. There is a lot of room here for further generalizations (and calculations).

Photo: Susie the cat in Cambridge, November 2000.

Leave a comment

Filed under math, Susie

Very old paper: The cohomology ring of the space of rational functions

susie-papersThanks to Claudio Gonzales, who requested it, and to MSRI Librarian Linda Riewe, who found and scanned it, my 1990 MSRI preprint “The cohomology ring of the space of rational functions” is available on my webpage.

Abstract: We consider three spaces which can be viewed as finite-dimensional approximations to the 2-fold loop space of the 2-sphere, Ω2S2. These are Ratk(CP1), the space of based holomorphic maps S2→S2; Bβ2k, the classifying space of the braid group on 2k strings; and Ck(R2, S1), a space of configurations of k points in R2 with labels in S1. Cohen, Cohen, Mann, and Milgram showed that these three spaces are all stably homotopy equivalent. We show that these spaces are in general not homotopy equivalent. In particular, for all positive integers k with k+1 not a power of 2, the mod 2 cohomology ring of Ratk is not isomorphic to that of Bβ2k or Ck. There remain intriguing questions about the relation among these three spaces.

Since 1990, a few papers have built on this preprint, including:

J. Havlicek. The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218 (1995), 179–190.

D. Deshpande. The cohomology ring of the space of rational functions (2009). arXiv:0907.4412

Photo: Susie the cat in Cambridge c. 2001.

Leave a comment

Filed under math, Susie

New paper: Rationality does not specialize among terminal varieties

IMG-20150805-00382I’ve posted a new paper on the arXiv. A limit of rational varieties need not be rational, even if all varieties in the family are projective and have at most terminal singularities. This shows that a result of de Fernex and Fusi’s does not extend to higher dimensions.

Photo: Susie the cat in Westwood.

Leave a comment

Filed under math, Susie

New paper: The integral cohomology of the Hilbert scheme of two points

IMG-20140924-00178I’ve posted a new paper on the arXiv. (I haven’t found an apt cat picture, so have just used a photo of Susie the cat, who supervised the writing of the paper closely.)

The Hilbert scheme X^{[a]} of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X. The integral cohomology of X^{[a]} is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology.

The results of this paper are used in Voisin’s work on the universal CH_0 group of cubic hypersurfaces, because the crucial point there is to study the 2-torsion in the Chow group.

Photo: Susie the cat in Princeton.

Leave a comment

Filed under math, Susie

IAS: getting settled

Using the study in Institute housing

As we start the second week of the fall term at the Institute, the backbone for the Topology of Algebraic Varieties program has been settled. (This forms a small subset of all the relevant mathematical activities at the Institute and the university.)

The regular weekly round of program activities is:
Tuesday
11:00 am Preprint seminar, organized by János Kollár
2:00 pm Talk 1 of the double seminar joint with Princeton
3:30 pm Talk 2 of the double seminar joint with Princeton

Wednesday
11:15 am Program seminar

Some specific events of interest are:
Workshop on Fundamental Groups and Periods, IAS, October 13–17
Barry Mazur: Minerva Lectures, Princeton University, October 14, 15, 17
AGNES, University of Pennsylvania, October 31–November 2

Details and more information on program activities are available on the Institute website and on the informal program website.

Leave a comment

Filed under math, Susie

Compositio meeting

IMG-20121219-00062

Update on other commitments: supervising the grading of exams
IMG-20121219-00062

Reviewing the course
IMG-20130608-00166

3 Comments

December 19, 2012 · 8:10 pm