Kovács' conjecture on characterisation of projective space and hyperquadrics

TL;DR

The paper settles Kovács' conjecture by proving that if a smooth complex projective variety $X$ of dimension $n$ carries an ample vector bundle $\mathcal{E}$ of rank $r$ with $\bigwedge^p\mathcal{E}\subseteq\bigwedge^pT_X$ for some $p\le n$, then $X$ is either the projective space $\mathbb{P}^n$ or, when $p=n$, a smooth quadric $Q_p$, with $\mathcal{E}$ matching the natural twisted tangent or line-bundle structure on these varieties. The argument proceeds by deriving a nonzero section in $H^0(X, T_X^{\otimes pf}\otimes\det(\bigwedge^p\mathcal{E})^{-1})$ and applying the Druel–Paris framework to reduce to the two geometric possibilities, then a detailed analysis on $\mathbb{P}^n$ using line restrictions, splitting types, and the Elencwajg–Hirschowitz–Schneider classification of uniform bundles to pin down the exact form of $\mathcal{E}$. Snow's cohomology results on twisted differential forms on quadrics are used to exclude intermediate cases, cementing the dichotomy. The results unify and extend many classical characterizations of projective spaces and quadric hypersurfaces, with precise descriptions of $\mathcal{E}$ in each case.

Abstract

We prove Kovács' 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. This provides a common generalization of Mori, Wahl, Cho-Sato, Andreatta-Wiśniewski, Kobayashi-Ochiai, and Araujo-Druel-Kovács type characterizations of such varieties.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2: Mor79
• Theorem 1.3: Wah83, Dru04
• Theorem 1.4: AW01
• Theorem 1.5: KO73
• Theorem 1.6: ChoSato95
• Theorem 1.7: ADK08
• Theorem 1.8: DP[Theorem B]
• Lemma 2.1
• proof