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Fano Fibrations and Twisted Kähler-Einstein Metrics I

Alexander Bednarek

TL;DR

The paper develops a base-geometry framework for Fano fibrations arising from finite-time singularities of the Kähler-Ricci flow. It constructs a Weil–Petersson–type $(1,1)$-form $ ext{ω}_{WP, ext{λ}}$ on the base from fibrewise volume data with prescribed fibre Ricci curvature, and proves the base carries twisted Kähler–Einstein metrics $ ext{ω}_B$ (and $ ext{ω}_B'$) governed by $ ext{Ric} ext{ω}_B=- ext{ω}_B- ext{λ} ext{η}+ ext{ω}_{WP, ext{λ}}$ (and analogous variants). The work yields cohomological decompositions of $c_1(X)$ and $c_1(B)$ in terms of $ ext{ω}_{WP, ext{λ}}$ and fibre data, and extends the construction to singular fibres via a semi Kähler–Einstein framework, enabling a base-centric description of the collapsing KRF in this setting. Together with the sequel B25, these results illuminate how fibre deformation data control the base geometry during the KRF collapse and provide a toolkit for analyzing the limiting behavior.

Abstract

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold $X$ endowed with an ample, rational line bundle $L$ and non-nef canonical line bundle $K_X$, we construct a $(1,1)$-form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this $(1,1)$-form and, for a submersion, that the Chern classes of $X$ and the base manifold decompose in terms of this $(1,1)$-form.

Fano Fibrations and Twisted Kähler-Einstein Metrics I

TL;DR

The paper develops a base-geometry framework for Fano fibrations arising from finite-time singularities of the Kähler-Ricci flow. It constructs a Weil–Petersson–type -form on the base from fibrewise volume data with prescribed fibre Ricci curvature, and proves the base carries twisted Kähler–Einstein metrics (and ) governed by (and analogous variants). The work yields cohomological decompositions of and in terms of and fibre data, and extends the construction to singular fibres via a semi Kähler–Einstein framework, enabling a base-centric description of the collapsing KRF in this setting. Together with the sequel B25, these results illuminate how fibre deformation data control the base geometry during the KRF collapse and provide a toolkit for analyzing the limiting behavior.

Abstract

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold endowed with an ample, rational line bundle and non-nef canonical line bundle , we construct a -form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this -form and, for a submersion, that the Chern classes of and the base manifold decompose in terms of this -form.
Paper Structure (6 sections, 13 theorems, 104 equations)

This paper contains 6 sections, 13 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The Semi-Prescribed Ricci Curvature Form
  3. The $(1,1)$-Form $\omega_{WP,\lambda}$
  4. The Twisted Kähler-Einstein Metrics
  5. Cohomological Consequences for a Submersion
  6. Constructions for a Kähler-Einstein Fibration

Key Result

Theorem 1.1

There exists a $(1,1)$-form $\omega_{WP,\lambda}$ on $B \backslash S'$ defined locally in the open set $U \subseteq B \backslash S'$ by where $\Omega_{\Psi_b}$ is a local family of volume forms on $X_b$ over $U$ satisfying $\mathop{\mathrm{Ric}}\nolimits \Omega_{\Psi_b} = \lambda \omega_{0,b}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 13 more