Fano Fibrations and Twisted K\"ahler-Einstein Metrics I

TL;DR

This paper explores the geometric structure of Fano fibrations and their relation to twisted K"ahler-Einstein metrics, focusing on the construction of specific forms and metric decompositions relevant to K"ahler-Ricci flows.

Contribution

It introduces a new $(1,1)$-form related to Fano fibrations and demonstrates its role in the twisted K"ahler-Einstein equation and Chern class decompositions.

Findings

Constructed a $(1,1)$-form related to the Weil-Petersson metric.

Proved the twisted K"ahler-Einstein equation involving this form.

Showed Chern class decompositions for the fibration.

Abstract

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to K\"ahler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact K\"ahler 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 K\"ahler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted K\"ahler-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.