Generalized K\"ahler-Einstein metric along $\mathbb Q$-Fano fibration

TL;DR

This paper establishes the existence of relative Kähler-Einstein metrics along $Q$-Fano fibrations under certain stability and singularity conditions, introducing a fiberwise foliation and addressing complex Monge-Ampère equations.

Contribution

It introduces a framework for constructing canonical metrics on $Q$-Fano fibrations with Kawamata log terminal singularities, utilizing fiberwise Kähler-Einstein foliations and the canonical bundle formula.

Findings

Existence of relative Kähler-Einstein metrics under K-polystability.

Introduction of fiberwise Kähler-Einstein foliation.

Addressing the relative complex Monge-Ampère equation.

Abstract

In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative K\"ahler-Einstein metric. We introduce the fiberwise K\"ahler-Einstein foliation and we mention that the main difficulty to obtain higher estimates is to solve relative CMA equation along such foliation. We propose a program such that for finding a pair of canonical metric $(\omega_X,\omega_B)$, which satisfies in $$Ric(\omega_X)=\pi^*\omega_B+\pi^*(\omega_{WP})+[\mathcal N]$$ on K-poly stable degeneration $\pi:X\to B$, where $Ric(\omega_B)=\omega_B$, we need to have Canonical bundle formula.