# A sufficient condition for a toric weak Fano 4-fold to be deformed to a Fano manifold

## TL;DR

This paper classifies smooth toric special weak Fano 4-folds and shows that most can be deformed into Fano manifolds, expanding understanding of their structure and deformation properties.

## Contribution

It introduces the concept of toric special weak Fano manifolds and provides a complete classification of smooth toric special weak Fano 4-folds.

## Key findings

- Almost all smooth toric special weak Fano 4-folds are deformable to Fano manifolds.
- The structure of toric special weak Fano manifolds is characterized by special primitive crepant contractions.
- The paper confirms the deformation connection between weak Fano and Fano manifolds in this class.

## Abstract

In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, we can confirm that almost every smooth toric special weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.