Smooth projective varieties with the ample vector bundle \(\overset {2}\bigwedge T_ X\) in any characteristic
type: Item
instance of: scholarly article
title: Smooth projective varieties with the ample vector bundle \(\overset {2}\bigwedge T_ X\) in any characteristic
zbMATH Open document ID: 0832.14031
author: Koji Cho
author: Ei-ichi Sato
published in: Journal of Mathematics of Kyoto University
publication date: 1996-03-05T00:00:00Z
review text: Let \(X\) be an \(n\)-dimensional smooth projective variety such that \(\bigwedge^2 T_X\) is ample. Here the authors prove that if either \(n \geq 5\) or \(n \geq 3\) and the algebraically closed base field has characteristic 0, then \(X\) is isomorphic either to \(\mathbb{P}^n\) or to a hyperquadric. The proof (as usual in similar type of results) uses Mori theory to produce families of rational curves contained in \(X\). However, due to positive characteristic technical problems, the authors must be very skilful; the trouble comes from cuspidal rational curves. The authors use also heavily the Fulton-Hansen connectedness theorem. They start an attempt to classify Fano manifolds in terms of their length: \(\text{length} (X) : = \min \{(-K_X \cdot C) |C\) is a rational curve in \(X\}\).
reviewed by: E. Ballico
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 784731
zbMATH Keywords: characterization of projective variety
zbMATH Keywords: Fano manifold
zbMATH Keywords: hyperquadric
zbMATH Keywords: Mori theory
zbMATH Keywords: Fulton-Hansen connectedness theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518837
OpenAlex ID: W1523485522
DOI: 10.1215/KJM/1250518837
description: scientific article; zbMATH DE number 784731