Log-Conformal Projective Manifolds

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.