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.