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\(\mathbb{P}^1\)-bundles admitting another smooth morphism of relative dimension one. (English) Zbl 1304.14054

Let \(X\) be a complex manifold of dimension \(\geq 2\), with Picard number one and \(\pi:Z \to X\) a \(\mathbb{P}^1\)-bundle over \(X\) (i.e., \(X\) is the projectivization of a rank two vector bundle on \(X\)). Suppose moreover that \(Z\) admits another smooth morphism, say \(\phi: Z \to Y\), of relative dimension one. Then, in the main result of this paper (see Theorem 1.1), the author proves that \(\phi: Z \to Y\) is also a \(\mathbb{P}^1\)-bundle over \(Y\) and, moreover, is completely classified: \(Z\) is the complete flag manifold of type \(A_2\), \(B_2\) or \(G_2\). This result generalizes Theorem 6.5 in [R. Muñoz et al., Kyoto J. Math. 54, No. 1, 167–197 (2014; Zbl 1295.14038)] where some extra assumption on the topology of \(X\) is needed (fourth Betti number \(b_4=1\)) and where \(\phi\) is imposed to be a \(\mathbb{P}^1\)-bundle.
An application to simplify the proof of a partial result on the Campana-Peternell conjecture is also provided (see Theorem 1.3).

MSC:

14J45 Fano varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M17 Homogeneous spaces and generalizations

Citations:

Zbl 1295.14038
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References:

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