K-stability of relative flag varieties

TL;DR

This paper extends the understanding of K-stability for relative flag varieties, especially over Riemann surfaces, by developing new theoretical tools and computations involving Schur functors and filtered algebras.

Contribution

It introduces the concept of relative K-stability and applies combinatorial Chern class computations to analyze stability of flag varieties over different bases.

Findings

Strong results over Riemann surfaces

Partial results for higher-dimensional bases

Development of relative K-stability framework

Abstract

We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The strongest results are obtained when working over a Riemann surface. Weaker partial results are obtained for adiabatic polarisations over a base of arbitrary dimension. We develop the notion of relative K-stability which embeds the idea of working over a base variety into the theory of K-stability. Natural constructions in filtered algebras equip the collection of test configuration with extra structures. We illustrate these constructions with several examples.