Project description
On the trail of novel theorems related to the Riemann–Roch theorem
The Riemann–Roch theorem has played a definitive role in complex analysis and algebraic geometry for more than 150 years. It was first proved by Riemann as Riemann's theorem in 1857 and modified shortly thereafter by Riemann's student Gustav Roch, with specific application to Riemann surfaces, one of the most important concepts in higher level mathematics. Since then, it has been generalised, and its applicability has increased even further. The EU-funded RRMAP project is exploring mathematical techniques based on discrete and integral versions of Riemann–Roch as well as other related theorems and their application to the solution of arithmetic problems.
Objective
Our project “Riemann-Roch and Motives for Arithmetic Problems” aims to develop techniques in the area of Motives and the Riemann-Roch to attack arithmetic problems. To be more concrete we aim to attack:
- The integral Riemann-Roch: At SGA VI Grothendieck developed his landmark Riemann-Roch result stating an integral version of it as an open question. Later on, research of Fulton, MacPherson and Pappas raised Grothendieck original conjecture to a more complete statement related to traces, which is known today only in the complex geometric setting. We aim to prove this conjecture in its full generality.
-The discrete Riemann-Roch: At SGA5 Grothendieck proved his wellknown Ogg-Shafarevich formula computing the Euler characteristic of a constructible sheaf over curve in terms of the genus, the Swan conductor and therank. This formula plays a central role in the original strategy to prove the Weyl conjectures. Grothendieck also conjectured that this formula would fit into a Riemann-Roch type theorem for the K-group of étale constructible sheaves and general schemes, which he called the “discrete Riemann-Roch”. We aim to attack this theorem from the motivic point of view.
-Intersection theory in the arithmetic setting: A major objective of Algebraic Geometry is to define a product algebraic cycles for
in the arithmetic setting. So far, this product has being defined with rational coefficients. The first definition, due to Gillet-Soulé, was achieved throughout the Adam’s operations, the Adams Riemann-Roch and the
Grothendieck-Riemann-Roch. We aim to explore some of Gillet-Soulé’s ideas and the arithmetic bivariant integral version of the Riemann-Roch to explore a definition of the intersection product of cycles after killing certain torsion on the Chow groups related to the codimension of the cycle
Fields of science
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Programme(s)
Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
28006 Madrid
Spain