Characterizations of Projective Spaces and Hyperquadrics via Positivity Properties of the Tangent Bundle

TL;DR

This paper proves a conjecture that characterizes projective spaces and hyperquadrics based on positivity properties of the tangent bundle, specifically for varieties with Picard number one.

Contribution

It confirms Kovács's conjecture for smooth complex projective varieties with Picard number one, linking tangent bundle positivity to classical geometric characterizations.

Findings

Proves Kovács's conjecture for Picard number one varieties.

Shows that certain positivity conditions imply the variety is a projective space or hyperquadric.

Extends classical characterizations of these spaces through tangent bundle properties.

Abstract

Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kov\'acs states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then $X$ is either a projective space or a smooth quadric hypersurface. This conjecture is appealing since it is a common generalization of Mori's, Wahl's, Andreatt\ a-W\'isniewski's, and Araujo-Druel-Kov\'acs's characterizations of these spaces. In this paper I give a proof affirming this conjecture for varieties with Picard number 1.