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Content archived on 2024-04-30

Equivariant riemann-roch type theorems in arakelov geometry


Research objectives and content
Let f be a molphism between algebraic schemes. The theorerms of Grothendieck- and Lefschetz-Riemaml-Roch allow to compute the push-forward morphism induced by f on algelbraic K-groups. The theorem of Grothendieck-Riemann-Roch has been genelalized in two directions. On the one hand, one can consider an equivariant situation where a group scheme G acts on all objects involved. On the other hand, there is an arithmetic Riemann-Roch theorem in the context of Arakelov geometry. The objective of the project is to develop the basic setup of Arakelov geometry and arithmetic K-theory in an equivariant context and to give generalizations of the Lefschetz-Riemann-Roch and the equivariant Grothendieck-Riemann-Roch the- Orems. More concretely, the aim is to consider the following points: ? Define equivariant arithmetic K-groups of arithmetic G-schemes and establish their basic properties (products. A-structure. etc.). ? Define an equivariant determinant of cohomology and equip it with a.Quillen metric. More generally define a push forward morphism on equivariant arith- metic K-groups and establish its basic functorial properties. For example, is there an excess intersection formula? ? Find and prove arithmetic Lefschetz-Riemann-Roch formulas and an equivari- ant arithmetic Grothendieck-Riemann-Roch theorem. Is there a localization theorem for euivariant arithmetic K-groups? It is planned to consider in a first step simple situations like relative curves and projective spaces and to put restrictive conditions on the group schemes.
Training content (objective, benefit and expected impact)
In the last years. there has been increasing interest in equivariant and arithmetic Riemann-Roch theorems. The project will allow me to enter this very active area of research. It will benefit from my experience with the methods of Arakelov geometry from my previous work. It is intendend to improve my knowledge of representation theory and equivaliant geometry The special program on 'Arithmetic Geometry' at the Newton institute provides the right atmosphere to work on the proposed project.

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