Log-Conformal Projective Manifolds

Abstract

Let $(X,\Delta )$ be a smooth complex projective simple normal crossing pair of dimension $n\ge 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+\Delta$ is not nef, then exactly one of the following occurs: $\Delta =\varnothing$ and $X\simeq Q^n$; $X\simeq {\mathbb{P}}^n$ and $\Delta$ is a hyperplane; or $n=2m$ and $(X,\Delta )$ admits a $(K_X+\Delta )$-negative elementary contraction $\varphi :X\to Y$ 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, $\varphi$ 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 $\rho (X)=1$ and $-(K_X+\Delta )$ is ample. If $K_X+\Delta \equiv 0$, then, under a Bochner extension principle, a restricted-holonomy condition, and trivial monodromy of the induced flat connection on $X\setminus \Delta$, a logarithmic conformal tensor with trivial conformal line bundle forces $X\setminus \Delta$ to be semi-abelian and $(X,\Delta )$ to be its toroidal compactification.

arXiv comment19 pages