Log-Conformal Projective Manifolds

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