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Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

Andreas Höring (2014)

Annales de l’institut Fourier

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$ . Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf $Ω_{X} \otimes A$ is generically nef with respect to the polarisation $A$ . As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ ⊊ T_{X}$ is a foliation such that $det ℱ \equiv i_{ℱ} A$ , then $i_{ℱ}$ is at most the rank of $ℱ$ .

Two theorems on elementary contractions.

M. Andreatta, E. Ballico (1993)

Mathematische Annalen

Über die Grassmannmannigfaltigkeit G(1,4) und verwandte Mannigfaltigkeiten.

Norbert Kuhlmann (1985)

Mathematische Annalen

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(- K_{X})$ -degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$ ) of all rational curves through $x$ are at least $dim X + 1$ , then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(- K_{X})$ -degrees of all rational curves through $x$ are at least $dim X$ . Our study uses the projective variety $𝒞_{x} \subset ℙ T_{x} (X)$ , called the VMRT at $x$ , defined as the union of tangent directions to the rational curves...