Some applications of canonical metrics to Landau–Ginzburg models

Abstract It is known that a given smooth del Pezzo surface or Fano threefold admits a choice of log Calabi–Yau compactified mirror toric Landau–Ginzburg model (with respect to certain fixed Kähler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map from a domain in the complexified Kähler cone of to a well‐defined, separated moduli space of polarised manifolds endowed with a canonical metric. We prove a complete result for del Pezzos and a partial result for some special Fano threefolds. The construction uses some fundamental results in the theory of constant scalar curvature Kähler metrics. As a consequence parametrises ‐stable manifolds and the domain of is endowed with the pullback of a Weil–Petersson form.