Geometric class field theory with bounded ramification

TL;DR

This paper extends geometric class field theory to higher-dimensional varieties with bounded ramification, establishing a correspondence between divisors and abelian coverings, and proving a Roitman theorem with modulus.

Contribution

It generalizes class field theory to arbitrary dimension, including singular varieties and characteristic zero, with new reciprocity laws and existence theorems for ramified abelian coverings.

Findings

Establishes a bijection between relative Cartier divisors and compatible systems on curves.

Proves a Roitman theorem with modulus for higher-dimensional varieties.

Provides a reciprocity law and existence theorem for abelian coverings with bounded ramification.

Abstract

Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor on X with support in X\U. We consider a relative Chow group of modulus D, the Albanese variety of X of modulus D and the Abel-Jacobi map with modulus. We show that there is a 1-1 correspondence between relative Cartier divisors on X and compatible systems of relative Cartier divisors on curves in X. This allows us to prove a Roitman theorem with modulus, and we obtain a reciprocity law and an existence theorem for abelian coverings of X with ramification bounded by D. Changes to the previous version: X is of arbitrary dimension and not necessarily smooth, char(k) = 0 is included for the so called Skeleton Theorem and the Roitman Theorems, log as well as non-log versions are treated. The definition of the Chow group with modulus was inconsistent for singular curves, this is clarified now.