According to mirror symmetry, the geometry of a given Fano
manifold endowed with some extra data, including an arbitrary Kähler
class, should be reflected in a mirror Landau-Ginzburg model, i.e. a
noncompact complex manifold endowed with a nonconstant holomorphic
function. On the other hand, a fundamental notion for constructing
moduli of Fano manifolds is K-polystability, i.e. positivity of the
Donaldson-Futaki invariants for nonproduct test-configurations. In
this talk I will introduce the problem of characterising K-polystable
Kähler classes on a Fano in terms of their mirror Landau-Ginzburg
models. I will then discuss some first concrete results in the case of
slope stability for del Pezzo surfaces. The computations involve the
particular “large complex structure limit” of the Landau-Ginzburg
model corresponding to scaling the Kähler class on the Fano, which
acts trivially on K-polystability.