, Volume 116, Issue 1, pp 157-202
Date: 29 Nov 2005

Smooth Projective Planes

Abstract

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to \(\mathbb{CP}^{2}\) . We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory of classical projective planes.

*Thanks to Robert Bryant and John Franks.