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.

We give an explicit description of an extremal nbd $X \supset C \simeq \mathbb{P}^1$ of type $k2A$. We also give a criterion for $X$ to be a flipping contraction and an explicit description of the contraction and the flip.

We classify all the projective elliptic fibrations defined over a unit polydisc whose discriminant loci are contained in a union of coordinate hyperplanes, up to the bimeromorphic equivalence relation. If the monodromies are unipotent and if general singular fibers are not of multiple type, then we can construct relative minimal models.