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.

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