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From Calabi's extremal metrics to scalar-flat Kähler cones

Vestislav Apostolov, Abdellah Lahdili, Chung-Ming Pan

Abstract

We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal Kähler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827.

From Calabi's extremal metrics to scalar-flat Kähler cones

Abstract

We prove that for any smooth polarized complex -dimensional manifold which admits an extremal Kähler metric in , and for any integer large enough (in terms of a bound depending on ), the -dimensional complex cone with section admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere of sufficiently large dimension admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827.

Paper Structure

This paper contains 13 sections, 16 theorems, 97 equations.

Table of Contents

  1. Kähler cones and Sasaki manifolds
  2. Sasaki structures
  3. Kähler cones vs Sasaki structures
  4. Sasaki join
  5. Sasaki structures with constant transversal scalar curvature versus scalar-flat Kähler cones
  6. Weighted cscK metrics
  7. $(v,w)$-cscK metrics
  8. Sasaki structures with constant transversal scalar curvature as $(v,w)$-weighted cscK metrics
  9. A LeBrun--Simanca stability theorem
  10. A family of Einstein--Hilbert functionals
  11. Locally uniform convergence of the Einstein--Hilbert functional
  12. Variation of $\mathop{\mathrm{\mathbf{EH}}}\nolimits_{\lambda,a}$
  13. Construction of scalar-flat cones

Key Result

Theorem A

In the setup of Problemmain-problem, there exists a positive integer $k_0=k_0(X, L)$, such that for any integer $k \geq k_0$, the Kähler cone $\mathcal{Y} = (\mathcal{L}^{-1})^{\times}$ associated to $\mathcal{X}= X \times {\mathbb P}^k$ polarized by the line bundle $\mathcal{L}= \pi_X^* L \otimes \

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Example 1.1: (Quasi-)regular Sasaki structures
  • Proposition 1.2
  • Remark 1.3
  • Example 1.4: (Quasi-)regular Kähler cones
  • Definition 1.5: The Sasaki join construction
  • Lemma 1.6
  • Definition 2.1: Lahdili_2019
  • Definition 2.2: Extremal Kähler metrics
  • ...and 27 more