Abstract
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.
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Work partially supported by M.P.I. of the Italian Government.
Both authors are members of G.N.S.A.G.A. of the Italian C.N.R.
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Lanteri, A., Struppa, D.C. On projective manifolds with two P-bundle structures. Ann Glob Anal Geom 6, 39–45 (1988). https://doi.org/10.1007/BF00054608
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DOI: https://doi.org/10.1007/BF00054608