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.
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*Thanks to Robert Bryant and John Franks.
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Mckay, B. Smooth Projective Planes. Geom Dedicata 116, 157–202 (2005). https://doi.org/10.1007/s10711-005-9012-5
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DOI: https://doi.org/10.1007/s10711-005-9012-5