Toric mirrors and test configurations

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).