On Projective Manifolds with Two P-bundle Structures
We investigate whether a projective manifold V which is a P 1-bundle over a projective manifold X with the same homology type of P r can have another P-bundle structure over some projective manifold Y; moreover, in the affirmative case we find restrictions on the topology of Y. Among the corollaries we prove that if V is a P l-bundle over P 4 and over a 4-fold Y not of general type, then either V = P 4 x P 4, V = P(T P 4), or, possibly, Y is a rational Fano 4-fold many properties of which are known. Further generalizations naturally arising from the geometry of flag manifolds are discussed.
Lanteri, A., & Struppa, D. C. (1988). On projective manifolds with two P-bundle structures. Annals of Global Analysis and Geometry, 6(1), 39-45. doi: 10.1007/BF00054608
Kluwer Academic Publishers