K-stability and large complex structure limits

Abstract

We discuss how, under suitable assumptions, a K\"ahler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the K\"ahler form. We exploit the observation that this scaling trivially preserves $K$-stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces.