Chow groups of zero- and higher zero-cycles

• Applicant: Dr. Morten Lüders
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 421164752

In algebraic geometry one studies sets of solutions of equations with given coefficients. Examples of such coefficients are the complex numbers, the rational numbers or the integers. Endowed with a certain structure, these sets of solutions are called varieties. In order to study a variety one can study its subspaces, i.e. subvarieties, and classify them up to an equivalence relation. In the theory of Chow groups, a classical theory of invariants, one classifies subspaces of a given variety up to rational equivalence. Two subspaces of a variety are rationally equivalent if one can be deformed into the other. The theory of Chow groups does not just give information on the geometry of the variety one studies but also on the coefficients which are used to define it. Using this theoretical approach to find out more about different coefficients such as the rational numbers, the p-adic numbers or the integers is an important goal of arithmetic geometry. Of particular interest is the Chow group of zero-cycles, i. e. points up to deformation, because of its computability. Many results about zero-cycles may be reduced to curves.Chow groups may be generalised to higher Chow groups. These higher Chow groups are a model for so called motivic cohomology which is a universal theory of invariants. Higher Chow groups are interesting as a refinement of the classical theory but also often make it possible to prove new results in the classical case. One part of higher Chow groups is given by higher zero-cycles. These are also called Chow groups of zero-cycles with coefficients in Milnor K-theory. Just like classical Chow groups of zero-cycles they may in many cases be computed but have not been studied in such depth. Furthermore they are related to other important theories like higher K-theory, class field theory and Kato conjectures. In our project we would like to study higher Chow groups of zero-cycles over different coefficients such as the p-adic, rational and complex numbers and explore the mentioned relations with other theories. We would also like to extend some results which are known for smooth varieties to cases in which singularities appear. In order to do so we develop deformation techniques for higher cycles, a decomposition of the diagonal for higher Chow groups over the complex numbers, a generalisation of the Levine-Weibel Chow group and new relations with Kato conjectures. In the last case we are particularly interested in the p-part. Furthermore we would like to study some mixed-characteristic phenomena in the local class field theory of schemes. In sum the project would substantially enlarge the picture of (motivic) invariants and we hope that this - in combination with new techniques we intend to learn during the stay - leads to new conjectures and results.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Dr. Matthew Morrow