On Projective Manifolds with Two P-bundle Structures

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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.


This article was originally published in Annals of Global Analysis and Geometry, volume 6, issue 1, in 1988. DOI: 10.1007/BF00054608

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Kluwer Academic Publishers