Characterizations of projective spaces and quadrics by strictly nef bundles

TL;DR

This paper characterizes projective spaces and quadrics through the properties of strictly nef bundles, showing that certain nef conditions imply the variety is a projective space or quadric, with additional results on vector bundles over curves.

Contribution

It establishes new characterizations of projective spaces and quadrics based on strictly nef tangent and exterior power bundles, extending understanding of positivity conditions in algebraic geometry.

Findings

Strictly nef tangent bundle implies the variety is a projective space.

Strictly nef 2 TX implies the variety is 1 2 1 or a quadric.

On elliptic curves, strictly nef bundles are ample; on higher genus curves, Hermitian flat and strictly nef bundles coexist.

Abstract

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $\Lambda^2 TX$, then it is isomorphic to $\mathbb{P}^n$ or quadric $\mathbb{Q}^n$. We also prove that on elliptic curves, strictly nef vector bundles are ample, whereas there exist Hermitian flat and strictly nef vector bundles on any smooth curve with genus $g\geq 2$.