On Sun-Zhang's theory of Fano fibrations -- weighted volumes, moduli and bubbling Fano fibrations

TL;DR

This paper reviews and extends Sun-Zhang's theory of Fano fibrations, introducing new methods for computing weighted volumes and proposing conjectural constructions related to degenerating Fano fibrations.

Contribution

It provides new computational techniques for weighted volumes and conjectures a novel algebro-geometric bubbling construction for Fano fibrations.

Findings

Methods for computing weighted volumes using Laplace transforms and Gamma-functions

Introduction of a conjectural bubbling construction for degenerating Fano fibrations

Review and extension of Sun-Zhang's framework on Fano fibrations

Abstract

We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction (``bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.