Characterizations of projective spaces and hyperquadrics

TL;DR

This paper characterizes projective spaces and hyperquadrics by conditions on tensor powers of tangent bundles and ample vector bundles, providing a new criterion for identifying these classical varieties.

Contribution

It introduces a novel characterization of projective spaces and hyperquadrics based on tensor powers of tangent bundles and ample vector bundles.

Findings

If the r-th tensor power of the tangent bundle contains the determinant of an ample vector bundle of rank ≥ r, then X is a projective space or a smooth quadric.

Provides a new geometric criterion for classifying certain algebraic varieties.

Enhances understanding of the structure of varieties with specific tangent bundle properties.

Abstract

In this paper we prove that if the r-th tensor power of the tangent bundle of a smooth projective variety X contains the determinant of an ample vector bundle of rank at least r, then X is isomorphic either to a projective space or to a smooth quadric hypersurface.