Log-Conformal Projective Manifolds

TL;DR

This paper classifies smooth complex projective pairs with a nondegenerate logarithmic conformal tensor, revealing their structure based on the nefness of the canonical bundle and the existence of special fibrations.

Contribution

It provides a classification of such pairs, identifying conditions under which they are projectively equivalent to known varieties or admit specific fibrations.

Findings

If $K_X+ abla$ is not nef, then $X$ is either a quadric, projective space with a hyperplane, or admits a rational maximal isotropic fibration.

When $K_X+ abla$ is numerically trivial, the pair is semi-abelian and its compactification is toroidal.

The existence of a trivial conformal line bundle under these conditions implies the pair's structure is highly constrained.

Abstract

Let $(X,\Delta)$ be a smooth complex projective simple normal crossing pair of dimension $n\geq 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+\Delta$ is not nef, then precisely one of the following mutually exclusive alternatives occurs: either $\Delta=\varnothing$ and $X\simeq Q^n$; or $X\simeq \mathbb{P}^n$ and $\Delta$ is a hyperplane; or $n=2m$ is even and $(X,\Delta)$ admits a rational maximal isotropic fibration whose geometric generic fibre is the log pair $(\mathbb{P}^m,H)$. If $K_X+\Delta\equiv 0$, then, under a Bochner extension principle and an irreducibility assumption on the restricted holonomy of a complete Ricci-flat K\"ahler metric on $M:=X\setminus \Delta$, the existence of a logarithmic conformal tensor with trivial conformal line bundle forces $M$ to be semi-abelian and $(X,\Delta)$ to be its toroidal compactification.