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VOL. 35 | 2002 Characterizations of Projective Space and Applications to Complex Symplectic Manifolds
Koji Cho, Yoichi Miyaoka, N. I. Shepherd-Barron

Editor(s) Shigefumi Mori, Yoichi Miyaoka

Abstract

We obtain new criteria for a normal projective variety to be projective $n$-space. Our main result asserts that a normal projective variety which carries a closed, doubly-dominant, unsplitting family of rational curves is isomorphic to projective space. An immediate consequence of this is the solution of a long standing conjecture of Mori and Mukai that a smooth projective $n$-fold $X$ is isomorphic to $\mathbb{P}^n$ if and only if $(C, -K_X) \ge n + 1$ for every curve $C$ on $X$. As applications of the criteria, we study fibre space structures and birational contractions of compact complex symplectic manifolds.

Information

Published: 1 January 2002
First available in Project Euclid: 31 December 2018

zbMATH: 1063.14065
MathSciNet: MR1929792

Digital Object Identifier: 10.2969/aspm/03510001