There is an elementary but very striking result which asserts that the quotient of the complex projective plane by complex conjugation is the 4-dimensional sphere. A short while ago Arnold and independently Atiyah and Witten proved that the quotient of the quaternionic projective plane by a certain circle action is a 7-dimensional sphere. In the first part of the talk we extend the above two results to the Cayley projective plane and provide a unifying proof for all three projective planes. Every projective plane over a normed real division algebra has a natural
complexification. These complexified projective planes are precisely the four Severi varieties in complex projective spaces. In the second part of the talk we extend the above results to the other Severi varieties. The resulting fibrations exhibit some interesting interplay between complex algebraic geometry and differential geometry.