Monthly Archives: February 2015

New paper: Adjoint functors on the derived category of motives

robert-leighton-cat-on-a-couch-dreaming-about-being-on-the-other-side-of-the-couch-new-yorker-cartoonI’ve posted a new paper here (and on the arXiv). Voevodsky’s derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper I study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. I find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups in various situations. For some base fields, I can determine exactly which adjoint functors exist.

Drawing by Robert Leighton, from The New Yorker

Leave a comment

Filed under math

New paper: Hypersurfaces that are not stably rational

gorey-catsI’ve posted a new paper here (and on the arXiv). It uses the Chow group of algebraic cycles to study a fundamental question in algebraic geometry: which hypersurfaces are stably rational varieties. The result is that for all d at least about 2n/3, a very general complex hypersurface of degree d and dimension n is not stably rational. This is a wide generalization of Colliot-Thélène and Pirutka’s theorem that very general quartic 3-folds are not stably rational. In a vague sense it uses the same machine as last week’s paper.

Drawing by Edward Gorey via Goreyana

Leave a comment

Filed under math

New paper: Complex varieties with infinite Chow groups modulo 2

UPDATEVideo of a talk in which I focus on the idea of the ‘machine’ mentioned below (from the IAS workshop held 9-13 March 2015).

ourcatsite-com_ray-bradburyI’ve posted a new paper here (and on the arXiv). It contains an outline of a general machine for studying Chow groups mod p of a complex variety. This turns out to be an effective way of attacking finiteness problems about algebraic cycles.

Leave a comment

Filed under math