We investigate whether a projective manifold V which is a P-bundle over a projective manifold X with the same homology type of P^r can admit another P-bundle structure over some projective manifold Y; morover, in the affirmative case we find restrictions on the topology of Y. Among the corollaries we prove that if V is a P-bundle over P^4 and over a 4-fold Y not of general type, then either V=P^4 \times 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.

On projective nanifolds with two P-bundle structures / A. Lanteri, D.C. Struppa. - In: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY. - ISSN 0232-704X. - 6:1(1988), pp. 39-45. [10.1007/BF00054608]

### On projective nanifolds with two P-bundle structures

#### Abstract

We investigate whether a projective manifold V which is a P-bundle over a projective manifold X with the same homology type of P^r can admit another P-bundle structure over some projective manifold Y; morover, in the affirmative case we find restrictions on the topology of Y. Among the corollaries we prove that if V is a P-bundle over P^4 and over a 4-fold Y not of general type, then either V=P^4 \times 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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projective manifold; projective bundle
Settore MAT/03 - Geometria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/187122
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