Simple normal crossing Fano varieties and log Fano manifolds

Abstract

A projective log variety $(X,D)$ is called a log Fano manifold if $X$ is smooth and if $D$ is a reduced simple normal crossing divisor on $X$ with $-(K_X+D)$ ample. The $n$-dimensional log Fano manifolds $(X,D)$ with nonzero $D$ are classified in this article when the log Fano index $r$ of $(X,D)$ satisfies either $r\ge n/2$ with $\rho \left(X\right)\ge 2$ or $r\ge n-2$. This result is a partial generalization of the classification of logarithmic Fano $3$-folds by Maeda.