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On K-stability of Calabi-Yau fibrations

Masafumi Hattori

TL;DR

The paper establishes a precise equivalence between uniform adiabatic K-stability of Calabi–Yau fibrations over curves and log-twisted K-stability of the base pair with discriminant and moduli divisors. It develops a framework linking base stability to total-space stability via delta-invariants and a J-stability criterion extended to singular settings, enabling existence results for cscK metrics on smooth total spaces. A key contribution is showing that delta-invariant convergence translates base K-stability into adiabatic stability of the fibration, with concrete applications to rational elliptic surfaces. The results provide a robust method to detect adiabatic stability from base data and to construct moduli in the K-stable regime, impacting the study of canonical metrics and moduli of polarized fibrations.

Abstract

We show that Calabi-Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are K-stable in the log-twisted sense. Moreover, we prove that there are cscK metrics for such fibrations when the total spaces are smooth.

On K-stability of Calabi-Yau fibrations

TL;DR

The paper establishes a precise equivalence between uniform adiabatic K-stability of Calabi–Yau fibrations over curves and log-twisted K-stability of the base pair with discriminant and moduli divisors. It develops a framework linking base stability to total-space stability via delta-invariants and a J-stability criterion extended to singular settings, enabling existence results for cscK metrics on smooth total spaces. A key contribution is showing that delta-invariant convergence translates base K-stability into adiabatic stability of the fibration, with concrete applications to rational elliptic surfaces. The results provide a robust method to detect adiabatic stability from base data and to construct moduli in the K-stable regime, impacting the study of canonical metrics and moduli of polarized fibrations.

Abstract

We show that Calabi-Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are K-stable in the log-twisted sense. Moreover, we prove that there are cscK metrics for such fibrations when the total spaces are smooth.
Paper Structure (17 sections, 45 theorems, 206 equations)

This paper contains 17 sections, 45 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. K-stability and its criteria
  3. K-stability and the cscK problem on fibrations
  4. Main results
  5. Preliminaries
  6. Birational geometry and MMP
  7. K-stability
  8. Log-twisted K-stability
  9. Lc-trivial fibrations
  10. Criterion for J-stability without test configurations
  11. Adiabatic K-stability versus log-twisted K-stability
  12. Adiabatic K-stability implies log-twisted K-stability
  13. Convergence of the $\delta$-invariant
  14. Applications to Rational Elliptic Surfaces
  15. Auxiliary results on log-twisted Fano pairs
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $f:(X,\Delta,H)\to (C,L)$ be a polarized klt-trivial fibration such that $C$ is a smooth curve. Let $M$ be the moduli divisor and $B$ the discriminant divisor on $C$. Then, $(X,\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$ if and only if $(C,B,M,L)$ is log-twisted K-stable. Moreove

Theorems & Definitions (120)

  • Definition 1.1
  • Conjecture 1.2: Miranda, Mi3
  • Remark 1.3
  • Theorem 1.1
  • Theorem 1.2: Theorems \ref{['can']}, \ref{['stp1']}
  • Theorem 1.3
  • Theorem 1.4: Theorems \ref{['ctn']}, \ref{['ctnd']}
  • Corollary 1.5: Corollary \ref{['K-fib']}
  • Definition 2.1: Log pair
  • Example 2.2
  • ...and 110 more