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

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.