Generalized Kähler-Einstein metric along $\mathbb{Q}$-Fano fibration

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 Kähler-Einstein metric. We introduce the fiberwise Kähler-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 $$Ric({\omega }_X)={\pi }^{\ast }{\omega }_B+{\pi }^{\ast }({\omega }_{WP})+[\mathcal{N}]$$ on K-poly stable degeneration $\pi :X\to B$, where $Ric({\omega }_B)={\omega }_B$, we need to have Canonical bundle formula.

References in corpus (7)

• K-stability and Kähler-Einstein metrics
• The Kahler-Ricci flow through singularities
• Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds
• Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata's base point free theorem
• Compact moduli space of Kahler-Einstein Fano varieties
• K-stability implies CM-stability
• Orbifold regularity of weak Kahler-Einstein metrics