Fano compactifications of contractible affine 3-folds with trivial log canonical divisors

TL;DR

This paper classifies certain smooth Fano 3-folds that compactify contractible affine 3-folds with trivial log canonical divisors, identifying 14 deformation classes and constructing explicit examples for each.

Contribution

It provides a complete classification of smooth Fano 3-folds admitting trivial log canonical divisors as compactifications of contractible affine 3-folds, including explicit constructions.

Findings

14 deformation equivalence classes identified

Explicit examples constructed for each class

Complete classification of such compactifications

Abstract

T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.