Some applications of canonical metrics to Landau-Ginzburg models

TL;DR

This paper explores the construction of a map from the K"ahler cone of Fano varieties to a moduli space of polarized manifolds with canonical metrics, linking mirror symmetry and K-stability.

Contribution

It provides a complete construction for del Pezzo surfaces and partial results for certain Fano threefolds, connecting mirror symmetry with canonical metrics and moduli spaces.

Findings

Constructed a map from the K"ahler cone to the moduli space of polarized manifolds.

Proved the moduli space parametrizes K-stable manifolds.

Endowed the domain with the pullback of the Weil-Petersson form.

Abstract

It is known that a given smooth del Pezzo surface or Fano threefold $X$ admits a choice of log Calabi-Yau compactified mirror toric Landau-Ginzburg model (with respect to certain fixed K\"ahler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map $\Theta$ from a domain in the complexified K\"ahler cone of $X$ to a well-defined, separated moduli space $\mathfrak{M}$ 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 K\"ahler metrics. As a consequence $\mathfrak{M}$ parametrises $K$-stable manifolds and the domain of $\Theta$ is endowed with the pullback of a Weil-Petersson form.