Let X be a complex, rationally connected, projective manifold. We show that X admits a modification [Formula: see text] that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of [Formula: see text]. For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification [Formula: see text] which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion [Formula: see text]. As applications we show various instances in which X is determined by [Formula: see text]. We also formulate a basic question about the birational invariance of ẽ(X, Y).
Characterization of Orlicz Sobolev Spaces
