Let M be a five-dimensional manifold polarized by a very ample line bundle L. We show that a smooth A ∈ |L| cannot be a holomorphic P 1 -bundle over a smooth projective 3-fold Y , unless Y ∼ = P 3 and A ∼ = P 1 × P 3 . Introduction The aim of this note is to provide a proof of the following result. The proof is based on Lemma 1.1 and on a result due to Lanteri and Struppa [9] . To give a motivation for this result, recall the general conjecture that for a P dbundle p : A → B, over a projective

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... anifold B of dimension b, which is an ample divisor on a manifold M, the condition d ≥ b − 1 should be true (see [3, Section 5.5]). Our result gives some more evidence to the conjecture, proving it in the special case when d = 1, b = 3 and A is a very ample divisor. Note also that a much weaker version of Proposition 0.1 was claimed in [5, p. 216] and used in an essential way in [2] . We work on the complex field C and use the standard terminology in algebraic geometry. In particular, we use the additive notation for line bundles, and we denote the canonical bundle of a projective manifold M by K M . We refer to [3] and [2] for any background material we need.