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Degenerations of Negative Kähler-Einstein Surfaces

Holly Mandel

Abstract

Every compact Kähler manifold with negative first Chern class admits a unique metric $g$ such that $\text{Ric}(g) = -g$. Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler-Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang (2019) in the Calabi-Yau case, I construct a Kähler-Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.

Degenerations of Negative Kähler-Einstein Surfaces

Abstract

Every compact Kähler manifold with negative first Chern class admits a unique metric such that . Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler-Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang (2019) in the Calabi-Yau case, I construct a Kähler-Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.
Paper Structure (42 sections, 17 theorems, 303 equations, 4 figures)

This paper contains 42 sections, 17 theorems, 303 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kähler-Einstein metrics
  3. Degenerations of negative Kähler-Einstein metrics
  4. Kähler Reduction
  5. Kähler metrics with Hamiltonian symmetry
  6. The reduced Kähler-Einstein equation
  7. From $X$ to $(\chi,h)$
  8. From $(\chi,h)$ to $X$
  9. Deriving a linearized equation for $\delta h$.
  10. Constructing an Approximate Solution
  11. Solving the linearized equation for $\delta h$
  12. Separation of variables
  13. ODE solution for each eigenvalue.
  14. Finding a decaying solution for $\lambda \not \in \Sigma$
  15. Extension to $\Sigma$
  16. ...and 27 more sections

Key Result

Theorem 1.1

Fix a complex curve $D$ with $c_1(D) < 0$ and integers $k_- \geq 0$, $k_+ \leq 0$. Let $(\mathcal{C}_{\pm},g_{\mathcal{C}_{\pm}})$ be the Calabi model space over $k_- \mathcal{N}_D$ and $-k_+ \mathcal{N}_D$, respectively. Then there exists $\alpha_0 \in (0,1)$, a manifold $\mathcal{M}$ with boundary

Figures (4)

  • Figure 1: The construction of $(\mathcal{M},\omega_{\text{KE},T})$. The base space is the manifold $D \times I_z$. The diameter of $D$ is bounded below independently of $T$ and blows up near the singular points. $\mathcal{M}$ is a singular $S^1$ fibration over $D \times I$. The size of the $S^1$ fiber, given by $h^{-1}$, decreases to $0$ near the singular points. The degree of the restriction of the $S^1$ fibration to $D$ is given by $k_-$ for $z < 0$ and $-k_+$ for $z > 0$ and changes at $z = 0$ because there are $k_--k_+$ singular points.
  • Figure 2: Kähler reduction. Locally, the total space is decomposed into an $S^1$ fibration over $D \times I_z$. The $S^1$-invariant metric on $X$ is specified by the fiber size, given by $h^{-1}$, and the $z$-family $\tilde{\omega}(z)$ of metrics on $D$.
  • Figure 3: The Calabi model space. Let $(D,\omega_D)$ be a compact negative Kähler-Einstein curve with canonical bundle $K_D$ and $\Vert \cdot \Vert_D$ a Hermitian metric on $K_D$ with curvature $\omega_D$. The Calabi model space is the tubular neighborhood $T_D = \lbrace -1/2n < z < 0 \rbrace \subset nK_D$ equipped with the metric $\omega = i \partial \bar{\partial} F(-\Vert \cdot \Vert_D^{2n})$.
  • Figure 4: The weight function $\rho^{(2)}_{\delta,\nu,\mu}$ as a function of $z$ along two cross sections $(p,t) \in D \times I$ for $t \in (-1,0)$. In the first case, the cross section goes through the singular point $(p_D,0)$, while in the second case, it takes the form $\lbrace (q,t), t \in (-1,0) \rbrace$ for some point $q \in D$ such that $r_w((q,0)) > C_3$. Parameters are chosen in accordance with Theorem \ref{['schauderGlobal']}, and in addition we are assuming that $\nu + 2 < \delta$, though the opposite may be true.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 28 more