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