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 Kahler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map Theta from a domain in the complexified Kahler cone of X to a well-defined, separated moduli space 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 M parametrises K-stable manifolds and the domain of Theta is endowed with the pullback of a Weil-Petersson form.
Some applications of canonical metrics to Landau–Ginzburg models / Stoppa, Jacopo. - In: JOURNAL OF THE LONDON MATHEMATICAL SOCIETY. - ISSN 0024-6107. - 111:4(2025), pp. 1-49. [10.1112/jlms.70148]
Some applications of canonical metrics to Landau–Ginzburg models
Stoppa, Jacopo
2025-01-01
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 Kahler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map Theta from a domain in the complexified Kahler cone of X to a well-defined, separated moduli space 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 M parametrises K-stable manifolds and the domain of Theta is endowed with the pullback of a Weil-Petersson form.File | Dimensione | Formato | |
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