Comparing natural volume forms on GL_n

TL;DR

This paper compares two natural volume forms on the algebraic group GL_n over Q, providing an explicit formula for their ratio, which enhances understanding of their relationship in algebraic geometry.

Contribution

It explicitly computes the comparison factor between the integral form and the product of primitive classes on GL_n, clarifying their relationship.

Findings

Derived an explicit formula for the ratio of the two volume forms.

Clarified the relationship between algebraic de Rham cohomology and integral forms.

Enhanced understanding of natural volume forms on algebraic groups.

Abstract

There are two natural choices for a volume form on the algebraic group Gl_n over Q: the first is the integral form (unique up to sign), the other is the product of the primitive classes in algebraic de Rham cohomology. We work out the explicit comparision factor between the two.