Central extensions and Riemann-Roch theorem on algebraic surfaces

TL;DR

This paper develops a new approach to the Riemann-Roch theorem on algebraic surfaces using central extensions of the general linear group over adeles, providing a local adelic decomposition and alternative calculations.

Contribution

It introduces a novel method connecting central extensions of adelic groups with the Riemann-Roch theorem on surfaces, bypassing the Noether formula.

Findings

Derived a local adelic decomposition for Euler characteristic differences

Provided two calculations leading to the Riemann-Roch theorem on surfaces

Established a link between central extensions and algebraic surface invariants

Abstract

We study canonical central extensions of the general linear group of the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. By these central extensions and adelic transition matrices of a rank $n$ locally free sheaf of ${\mathcal O}_X$-modules we obtain the local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf ${\mathcal O}_X^n$. Two various calculations of this difference lead to the Riemann-Roch theorem on $X$ (without the Noether formula).