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Foldable fans, cscK surfaces and local K-moduli

Carl Tipler

Abstract

We study the moduli space of constant scalar curvature Kähler surfaces around the toric ones. To this aim, we introduce the class of foldable surfaces : smooth toric surfaces whose lattice automorphism group contain a non trivial cyclic subgroup. We classify such surfaces and show that they all admit a constant scalar curvature Kähler metric (cscK metric). We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modeled on a finite quotient of a toric affine variety with terminal singularities.

Foldable fans, cscK surfaces and local K-moduli

Abstract

We study the moduli space of constant scalar curvature Kähler surfaces around the toric ones. To this aim, we introduce the class of foldable surfaces : smooth toric surfaces whose lattice automorphism group contain a non trivial cyclic subgroup. We classify such surfaces and show that they all admit a constant scalar curvature Kähler metric (cscK metric). We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modeled on a finite quotient of a toric affine variety with terminal singularities.
Paper Structure (18 sections, 17 theorems, 131 equations, 13 figures)

This paper contains 18 sections, 17 theorems, 131 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background on cscK metrics and their moduli
  3. Extremal Kähler metrics
  4. Local moduli of polarised cscK manifolds
  5. Foldable toric surfaces
  6. Lattice automorphisms and fans
  7. Classification of foldable surfaces
  8. The local cscK moduli around a foldable surface
  9. Deformation theory of toric surfaces
  10. The toric GIT quotient
  11. Singularities
  12. Examples
  13. A smooth example : $Y_4$
  14. A singular example : $Y_3$
  15. A non--Gorenstein example
  16. ...and 3 more sections

Key Result

Proposition 1.2

Let $N$ be a rank two lattice and $\Sigma$ a complete smooth fan in $N_\mathbb{R}$. Then $\mathrm{Aut}(N,\Sigma)$ is isomorphic to one of the groups in the following set Moreover, any group in the above list is isomorphic to the lattice automorphism group of some complete two dimensional smooth fan.

Figures (13)

  • Figure 1: Fan $\Sigma_1'$ of $\mathbb{F}_2$
  • Figure 2: Fan $\Sigma_3'$ of $\mathbb{P}_2$
  • Figure 3: Fan $\Sigma_4'$ of $\mathbb{C}\mathbb{P}^1\times\mathbb{C}\mathbb{P}^1$
  • Figure 4: Fan $\Sigma_2'$ : iterated blow-up of $\mathbb{P}^1\times\mathbb{P}^1$
  • Figure 5: Fan $\Sigma_6'$ : blow-up of $\mathbb{P}^2$ along its three fixed points.
  • ...and 8 more figures

Theorems & Definitions (45)

  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1: matsushima57Lichn
  • ...and 35 more