Balanced line bundles and equivariant compactifications of homogeneous spaces

TL;DR

This paper introduces the concept of balanced line bundles to explain geometric inequalities related to Manin's conjecture, providing a framework applicable to many equivariant compactifications of homogeneous spaces.

Contribution

It develops a general geometric framework using balanced line bundles and proves key inequalities for a broad class of equivariant compactifications.

Findings

Established inequalities among geometric invariants

Introduced the notion of balanced line bundles

Applied framework to various equivariant compactifications

Abstract

Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety in terms of its global geometric invariants. The strongest form of the conjecture implies certain inequalities among geometric invariants of the variety and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces.