Characterizations of projective spaces and quadrics by strictly nef bundles

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 $g\ge 2$.

arXiv commentThis paper is superseded by arXiv:1801.09191