Toric mirrors and test configurations

TL;DR

This paper explores the relationship between K-stability invariants of toric manifolds and their mirror Landau-Ginzburg models, revealing how mirror symmetry influences stability notions.

Contribution

It establishes a connection between Donaldson-Futaki invariants and mirror Landau-Ginzburg models, introducing conditions for subleading terms and linking mirror symmetry to K-stability concepts.

Findings

Invariants relate via expansions involving base loci of linear systems.

Conditions identified for when terms are subleading.

Mirror symmetry considerations naturally lead to Z-stability notions.

Abstract

We obtain results that relate Donaldson-Futaki type invariants (that is, the numerical invariants used to define K-stability for general polarised manifolds) for a toric polarised manifold and for a compactification of its mirror Landau-Ginzburg model, nearby the large volume limit. In general, these have the form of expansions containing terms which involve the base loci of certain linear systems determined by the Landau-Ginzburg potential (as expected from known constructions of compactified mirrors), and we give a condition under which these terms are subleading. As an application we show that recently proposed notions of K-stability involving elements of the extended K\"ahler moduli space, i.e. Z-stability for polarised varieties, appear naturally from considerations of mirror symmetry (as a mirror to classical K-stability).