Let $M$ be a $5$-dimensional manifold polarized by a very ample line bundle $L$. We show that a smooth $A\in |L|$ cannot be a holomorphic $\pn 1$-bundle over a smooth projective threefold $Y$, unless $Y\cong\pn 3$ and $A\cong\pn 1 \times\pn 3$.
A note on ${\mathbb P}^1$-bundles as hyperplane sections
BELTRAMETTI, MAURO CARLO;
2005-01-01
Abstract
Let $M$ be a $5$-dimensional manifold polarized by a very ample line bundle $L$. We show that a smooth $A\in |L|$ cannot be a holomorphic $\pn 1$-bundle over a smooth projective threefold $Y$, unless $Y\cong\pn 3$ and $A\cong\pn 1 \times\pn 3$.
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