Simple normal crossing Fano varieties and log Fano manifolds

TL;DR

This paper classifies certain smooth projective varieties called log Fano manifolds with simple normal crossing divisors, focusing on cases with specific log Fano index and Picard number, extending previous classifications of logarithmic Fano threefolds.

Contribution

It provides a classification of n-dimensional log Fano manifolds with nonzero divisors under specific index and Picard number conditions, generalizing Maeda's work on threefolds.

Findings

Classified log Fano manifolds with r ≥ n/2 and ρ(X) ≥ 2

Classified log Fano manifolds with r ≥ n-2

Extended the classification of logarithmic Fano threefolds

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\geq n/2 with \rho(X)\geq 2 or r\geq n-2. This result is a partial generalization of the classification of logarithmic Fano threefolds by Maeda.