Log-Conformal Projective Manifolds

Abstract

Let $(X,Δ)$ be a smooth complex projective simple normal crossing pair of dimension $n\ge 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+Δ$ is not nef, then exactly one of the following occurs: $Δ=\varnothing$ and $X\simeq Q^n$; $X\simeq\mathbb{P}^n$ and $Δ$ is a hyperplane; or $n=2m$ and $(X,Δ)$ admits a $(K_X+Δ)$-negative elementary contraction $ϕ:X\to Y$ generated by minimal rational curves of conformal degree one. In the third case, if $\dim Y>0$, then for some $1\le s\le m$ one has $\dim Y=m-s+1$ and the geometric generic fiber is $(\mathbb{P}^{m+s-1},H_1+\cdots+H_s)$, where the $H_i$ are hyperplanes in general position; over a dense open subset, $ϕ$ is a projective bundle and the horizontal boundary is the sum of $s$ relative hyperplanes. The case $s=1$ yields a rational maximal isotropic fibration with generic fiber $(\mathbb{P}^m,H)$; if $\dim Y=0$, then $ρ(X)=1$ and $-(K_X+Δ)$ is ample. If $K_X+Δ\equiv0$, then, under a Bochner extension principle, a restricted-holonomy condition, and trivial monodromy of the induced flat connection on $X\setminusΔ$, a logarithmic conformal tensor with trivial conformal line bundle forces $X\setminusΔ$ to be semi-abelian and $(X,Δ)$ to be its toroidal compactification.