Characterizations of Projective Space and Applications to Complex Symplectic Manifolds

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.