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

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

2010-12-09

arxiv.org/abs/1012.2043v2

Mathematics

Algebraic Geometry

Version 2 updates: (1) Added reference to abstract, (2) Updated remark 4.2

Scientific paper

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.

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