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Some applications of canonical metrics to Landau-Ginzburg models

Jacopo Stoppa

TL;DR

This work constructs a holomorphic mirror-map Θ from a neighborhood of the complexified Kähler cone to a moduli space of polarized manifolds carrying canonical metrics, grounded in cscK theory and K-stability. For smooth del Pezzo surfaces with Gorenstein toric degenerations, the authors prove a complete realization of Θ and show the domain inherits a Weil–Petersson form, tying moduli geometry to the framed LG mirrors. The approach is demonstrated through explicit, case-by-case LG mirror constructions for a wide range of Fano surfaces and threefolds, using dual polytopes, base-locus analyses, Kempf–Ness stability, and Arezzo–Pacard gluing to produce families of cscK polarised manifolds; in many instances the resulting moduli are Hausdorff and Zariski separated, with discrete automorphism groups. The work also discusses HMS-type perspectives, deformation theory in the cscK setting, and cases where adiabatic or $K$-instability phenomena arise, highlighting both the reach and the limitations of current techniques in linking complex and Kähler moduli across mirror symmetry.

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 Kähler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map $Θ$ from a domain in the complexified Kähler 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 Kähler metrics. As a consequence $\mathfrak{M}$ parametrises $K$-stable manifolds and the domain of $Θ$ is endowed with the pullback of a Weil-Petersson form.

Some applications of canonical metrics to Landau-Ginzburg models

TL;DR

This work constructs a holomorphic mirror-map Θ from a neighborhood of the complexified Kähler cone to a moduli space of polarized manifolds carrying canonical metrics, grounded in cscK theory and K-stability. For smooth del Pezzo surfaces with Gorenstein toric degenerations, the authors prove a complete realization of Θ and show the domain inherits a Weil–Petersson form, tying moduli geometry to the framed LG mirrors. The approach is demonstrated through explicit, case-by-case LG mirror constructions for a wide range of Fano surfaces and threefolds, using dual polytopes, base-locus analyses, Kempf–Ness stability, and Arezzo–Pacard gluing to produce families of cscK polarised manifolds; in many instances the resulting moduli are Hausdorff and Zariski separated, with discrete automorphism groups. The work also discusses HMS-type perspectives, deformation theory in the cscK setting, and cases where adiabatic or -instability phenomena arise, highlighting both the reach and the limitations of current techniques in linking complex and Kähler moduli across mirror symmetry.

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.
Paper Structure (39 sections, 13 theorems, 241 equations)

This paper contains 39 sections, 13 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Compactified toric LG models
  3. Maps of moduli spaces
  4. del Pezzo surfaces
  5. HMS pairs
  6. Fano threefolds
  7. $X = T_{P_3} = \mathbb{P}^2$
  8. First step: dual polytope
  9. Second step: resolving the base locus
  10. Global quotient
  11. Kempf-Ness Theorem and Hilbert-Mumford Criterion
  12. Third step: crepant resolution
  13. $X = T_{P_{6a}} = S_6$
  14. Kempf-Ness Theorem
  15. $T_X = T_{P_9}$
  16. ...and 24 more sections

Key Result

Theorem 1.4

There exists a separated complex space $\mathfrak{M}$ which is a coarse moduli space for compact Kähler manifolds with discrete automorphisms, endowed with a cscK metric representing a fixed cohomology class (on the underlying smooth manifold). Moreover, $\mathfrak{M}$ carries a canonical Weil-Peter

Theorems & Definitions (28)

  • Definition 1.1: Przyjalkowski_CYLG, Definitions 6 and 7
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Fujiki-Schumacher FujikiSchumacher_moduli
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma : Section \ref{['AdiabaticSec']}
  • ...and 18 more