Log-Conformal Projective Manifolds

Abstract

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

30 pages