Some applications of canonical metrics to Landau-Ginzburg models

Jacopo Stoppa

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arXiv · 2406.08613

Some applications of canonical metrics to Landau-Ginzburg models

Jacopo Stoppa

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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.

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Jacopo Stoppa. 2024-06-12. Some applications of canonical metrics to Landau-Ginzburg models. https://arxiv.org/abs/2406.08613

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