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Collapsing of $ALH^*$-Gravitational Instantons

Yu-Shen Lin, Ryosuke Takahashi

Abstract

We showed that a sequence of ALH*-gravitational instantons from pairs consisting of a weak del Pezzo surface and a smooth anti-canonical divisor towards a large complex structure limit introduced by Collins, Jacobs and the first author collapsing to a punctured plane with a special Kahler metric, which can be viewed as a non-compact version of the collapsing result of Gross-Wilson. We provide a partial compactification of the moduli space of pointed ALH*-gravitational instantons with respect to the pointed Gromov-Hausdorff topology and locally is a polyhedron complex.

Collapsing of $ALH^*$-Gravitational Instantons

Abstract

We showed that a sequence of ALH*-gravitational instantons from pairs consisting of a weak del Pezzo surface and a smooth anti-canonical divisor towards a large complex structure limit introduced by Collins, Jacobs and the first author collapsing to a punctured plane with a special Kahler metric, which can be viewed as a non-compact version of the collapsing result of Gross-Wilson. We provide a partial compactification of the moduli space of pointed ALH*-gravitational instantons with respect to the pointed Gromov-Hausdorff topology and locally is a polyhedron complex.
Paper Structure (12 sections, 23 theorems, 99 equations)

This paper contains 12 sections, 23 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. $ALH^*$-Gravitational Instantons
  3. Local Models
  4. The Calabi ansatz
  5. Generalized Semi-Flat Metrics
  6. $ALH^*$-Gravitational Instantons
  7. Construction of Ansatz
  8. Untwisted Ansatz Metrics
  9. Ansatz Twisted by B-Fields
  10. Synergy
  11. A Priori Estimates
  12. Collapsing Limits of $ALH^*$-Gravitational Instantons and Partial Compactification of Pointed Moduli Space of $ALH^*$-Gravitational Instantons

Key Result

Theorem 1.1

CJL Let $\check{X}$ be the complement of a smooth anti-canonical divisor $\check{D}$ in a weak del Pezzo surface $\check{Y}$ of degree $d$ with the Tian-Yau metric $\check{\omega}$. Denote by $\check{\Omega}$ a meromorphic $2$-form on $\check{Y}$ with a simple pole along $\check{D}$ such that $\chec admits an elliptic fibration $\pi:X\rightarrow \mathbb{C}$ and it can be compactified to a rational

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4: see Theorem \ref{['thm: compactify 1']}(4)
  • Theorem 1.5: =Theorem \ref{['thm: compactify 1']}
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 37 more