New perspectives for canonical dimension

点击翻译按钮获取中文摘要

Finding rational points on an algebraic variety amounts to finding solutions of systems of polynomials equations with coefficients in a given field. Many questions concerning quadratic forms, central simple algebras, or more generally Galois cohomology, may be formulated as whether a certain projective homogeneous variety under a linear algebraic group has a rational point or not. Canonical dimension measures how far a variety is from having a rational point, and thus allows finer distinctions between varieties than the mere consideration of the existence of rational points. Accordingly, this notion provides valuable informations on quadratic forms and central simple algebras. Until now, the concept of canonical dimension has been mostly used in the context of projective homogeneous varieties. Its computation often involves tools inspired by algebraic topology, in particular cohomological operations and motivic decompositions.This project is organised along two directions: (a) Use canonical dimension and the associated methods in new contexts, outside the realm of linear algebraic groups. These techniques may be useful whenever one is trying to find rational points on an algebraic variety, a situation which is of course not limited to the study of projective homogeneous varieties, but in fact corresponds to a large part of arithmetic geometry.(b) Use methods coming from other parts of algebraic geometry to compute the canonical dimension of projective homogeneous varieties. This includes techniques from classical algebraic geometry, which are routinely used in arithmetic geometry, and Chow-Witt-theoretic methods, which are currently used for the study of projective modules.Explicit problems that will be investigated are: (1) Does the action of a finite p-group on the affine space over a field of characteristic different from p fix a rational point? (2) Compute the canonical dimension of the Severi-Brauer varieties. (3) Obtain new informations on the splitting patterns of quadratic forms. (4) Prove a quadratic refinement of Serre's vanishing conjecture.