I’ve posted a new paper on the arXiv. Most of the results premiered at Oberwolfach, but since then I’ve added an interesting new example inspired by related work of Takehiko Yasuda, whom I met there. (Oberwolfach at work!)
The paper shows that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
Image: Palmerston, Foreign Office cat, sneaks up on cheese; from his DiploMog twitter feed
WAGS returns to UCLA.
The Fall 2017 edition of the Western Algebraic Geometry Symposium (WAGS) will take place the weekend of 14–15 October 2017 at IPAM on the UCLA campus, hosted by the UCLA Mathematics Department. Details are now on the conference website.
If you plan to attend but haven’t yet registered, please register. It’s free, and knowing who’s coming will allow us to ensure that:
- We have enough space.
- We have enough coffee.
- We have enough food.
- We have a name tag ready for you, so that the conference is successful in helping you meet fellow mathematicians and helping other mathematicians meet you.
- We can help our funder to demonstrate — with evidence — that they’re supporting a thriving enterprise.
Send questions to firstname.lastname@example.org. [UPDATE: I previously gave the wrong email address (starting fall17).]
Photo of the Powell Cat from the Daily Bruin. More about the Powell Cat on twitter.
My paper about the Tate conjecture for Bull. AMS is now available to view.
In it I survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely intertwined with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. I conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.
After returning the proofs to the AMS, it occurred to me that it could be helpful to readers if I recommended some available related videos. I was too slow for the AMS’s speedy production, however, so I make the recommendations here.
F. Charles. K3 surfaces over finite fields: insights from complex geometry (2015).
K. Madapusi Pera. Regular integral models for orthogonal Shimura varieties and the Tate conjecture for K3 surfaces in finite characteristic (2012).
D. Maulik. Finiteness of K3 surfaces and the Tate conjecture (2012).
Photo is from foldedspace.org.
The March for Science will be on Earth Day, Saturday, April 22, 2017, in 605 cities across the U.S. and beyond. Come out and march for truth against lies! Check out the web site for details of the march in your city. I’ll be marching in Houston.
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 2001.
The spring 2017 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at the University of British Columbia in Vancouver on the weekend 8–9 April 2017. For detailed information and to register, please see the conference website.
The excellent line-up of speakers is:
Jarod Alper (Washington)
Melody Chan (Brown)
Daniel Halpern-Leistner (Columbia)
Sam Payne (Yale)
Giulia Sacca (Stony Brook)
Bernd Sturmfels (Berkeley)
Ravi Vakil (Stanford, to be confirmed)
Image is from the Tremaine Arkley Croquet Collection in the UBC Library Rare Books and Special Collections via the UBC Library’s digitization blog.
Thanks 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. 2002.