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Toric separable geometries and extremal Kähler metrics

Roland Púček

TL;DR

This work introduces toric separable geometries, a unifying framework that encodes all known explicit extremal toric Kähler metrics through factorization structures and a separation-of-variables philosophy. The authors derive explicit formulae for the Laplacian, Ricci potential, and scalar curvature within this framework, recasting the extremality problem as a PDE in momentum coordinates. They prove that extremal solutions are rational functions in one variable with denominators fixed by the chosen factorization structure and numerators of bounded degree, and they systematically solve the extremality equations for broad SV structures, including new extremal metrics arising from the product Segre-Veronese case. The results provide a versatile toolbox for constructing and classifying extremal toric metrics, with potential applications to stability, compactifications, and geometric flows. Overall, the paper advances a separation-of-variables approach to Calabi-type variational problems in Kähler geometry, expanding the landscape of explicit extremal examples and illuminating how combinatorial factorization data govern analytic solvability.

Abstract

This paper introduces the framework of (local) toric separable geometries, where toric separable Kähler geometries come in families, each uniquely determined by an underlying factorization structure. This unifying framework captures all known explicit Calabi-extremal toric Kähler metrics, previously constructed through diverse methods, as two distinct families corresponding to the simplest factorization structures: the product Segre and the Veronese factorization structure. Crucially, the moduli of typical factorization structures has a positive dimension, revealing an immensely rich and previously untapped landscape of toric separable geometries. The scalar curvature of toric separable geometries is computed explicitly, necessary conditions for the PDE governing extremality are derived, and new extremal metrics are obtained systematically. In particular, for a $2m$-dimensional toric separable geometry, solutions of the PDE are necessarily $m$-tuples of rational functions of one variable belonging to an at most $(m+2)$-dimensional real vector space and whose denominators are determined by the factorization structure. Toric separable geometries serve as a separation of variables technique and are well-suited for the analytic study of geometric PDEs.

Toric separable geometries and extremal Kähler metrics

TL;DR

This work introduces toric separable geometries, a unifying framework that encodes all known explicit extremal toric Kähler metrics through factorization structures and a separation-of-variables philosophy. The authors derive explicit formulae for the Laplacian, Ricci potential, and scalar curvature within this framework, recasting the extremality problem as a PDE in momentum coordinates. They prove that extremal solutions are rational functions in one variable with denominators fixed by the chosen factorization structure and numerators of bounded degree, and they systematically solve the extremality equations for broad SV structures, including new extremal metrics arising from the product Segre-Veronese case. The results provide a versatile toolbox for constructing and classifying extremal toric metrics, with potential applications to stability, compactifications, and geometric flows. Overall, the paper advances a separation-of-variables approach to Calabi-type variational problems in Kähler geometry, expanding the landscape of explicit extremal examples and illuminating how combinatorial factorization data govern analytic solvability.

Abstract

This paper introduces the framework of (local) toric separable geometries, where toric separable Kähler geometries come in families, each uniquely determined by an underlying factorization structure. This unifying framework captures all known explicit Calabi-extremal toric Kähler metrics, previously constructed through diverse methods, as two distinct families corresponding to the simplest factorization structures: the product Segre and the Veronese factorization structure. Crucially, the moduli of typical factorization structures has a positive dimension, revealing an immensely rich and previously untapped landscape of toric separable geometries. The scalar curvature of toric separable geometries is computed explicitly, necessary conditions for the PDE governing extremality are derived, and new extremal metrics are obtained systematically. In particular, for a -dimensional toric separable geometry, solutions of the PDE are necessarily -tuples of rational functions of one variable belonging to an at most -dimensional real vector space and whose denominators are determined by the factorization structure. Toric separable geometries serve as a separation of variables technique and are well-suited for the analytic study of geometric PDEs.

Paper Structure

This paper contains 25 sections, 27 theorems, 203 equations, 2 figures.

Table of Contents

  1. Background
  2. Factorization structures
  3. Segre-Veronese factorization structures
  4. Products and decomposable elements
  5. Contact geometry
  6. Sasaki geometry
  7. Toric contact geometry
  8. Separable geometries and their scalar curvatures
  9. Separable contact geometries
  10. Separable CR geometries
  11. Separable Kähler geometries
  12. Examples
  13. Laplace operator
  14. Scalar curvature
  15. Extremality equation
  16. ...and 10 more sections

Key Result

Lemma 1.3.2

[lemma]tensors split Let $\varphi(\mathfrak{h})\otimes T+ S\otimes\chi(\mathfrak{g}) \hookrightarrow V_1^*\otimes\cdots\otimes V_m^*\otimes W_1^*\otimes\cdots\otimes W_n^*$ be a product factorization structure. Then for some 1-dimensional subspaces $I \subset V_1^*\otimes\cdots\otimes V_m^*$ and $K \subset W_1^*\otimes\cdots\otimes W_n^*$ if and only if

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (75)

  • Definition 1.1.1
  • Remark 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Definition 1.2.1
  • Remark 1.2.2
  • Example 1.2.3
  • Example 1.2.4
  • Remark 1.2.5
  • Example 1.2.6
  • ...and 65 more