Kov\'acs' conjecture on characterization of projective space and hyperquadrics

TL;DR

This paper proves Kovács' conjecture that certain conditions on the exterior or symmetric powers of the tangent bundle characterize projective space and hyperquadrics, unifying several classical characterizations.

Contribution

It establishes a new characterization of projective space and hyperquadrics based on exterior and symmetric powers of the tangent bundle, generalizing previous results.

Findings

Proves Kovács' conjecture for exterior powers.

Proves a similar result for symmetric powers.

Unifies multiple classical characterizations of these varieties.

Abstract

We prove Kov\'acs' conjecture that claims that if the $p^{th}$ exterior power of the tangent bundle of a smooth complex projective variety contains the $p^{th}$ exterior power of an ample vector bundle then the variety is either projective space or the $p$-dimensional quadric hypersurface. We also prove a similar characterization involving symmetric powers instead of exterior powers. This provides a common generalization of Mori, Wahl, Cho-Sato, Andreatta-Wi\'sniewski, Kobayashi-Ochiai, and Araujo-Druel-Kov\'acs type characterizations of such varieties.