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

Let X be a smooth complex projective variety. A recent conjecture of S. Kovacs states that if the p th-exterior power of the tangent bundle TX contains the pth-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 a common generalization of Mori's, Wahl's Andreatta-Wisniewski's, Kobayashi-Ochiai's, and Araujo-Druel-Kovacs's characterizations of these spaces. In support of this conjecture, I give an affirmative proof for varieties with Picard number 1. As an auxiliary result, I also prove a structure theorem for vector bundles that split in a certain uniform way on the rational curves covering a Fano variety. This result corrects, in fact generalizes, a previous well-known result in the literature.