Fano threefolds with affine canonical extensions

TL;DR

This paper proves that smooth Fano threefolds with an affine canonical extension of their tangent bundle are necessarily rational homogeneous, revealing a deep connection between affine structures and geometric classification.

Contribution

It establishes a classification result linking affine canonical extensions to rational homogeneity in Fano threefolds, a novel insight in algebraic geometry.

Findings

Fano threefolds with affine canonical extensions are rational homogeneous.

The structure of the tangent bundle extension determines the geometry of the threefold.

Affine canonical extensions impose strong symmetry conditions on the threefold.

Abstract

Let $M$ be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that $M$ is rational homogeneous.