On projective manifolds with two \({\mathbb{P}}\)-bundle structures (Q1119703)

\textit{E. Sato} proved in J. Math. Kyoto Univ. 25, 445-457 (1985; Zbl 0587.14003) that a projective manifold having two different structures as a projective bundle over a projective space is either a product \({\mathbb{P}}^ r\times {\mathbb{P}}^ s\) or the projectivization \({\mathbb{P}}(T_{{\mathbb{P}}^ r})\) of the tangent bundle to \({\mathbb{P}}^ r.\) Here a slightly bigger class of manifolds is studied. In fact, it is investigated whether a projective manifold V which is a \({\mathbb{P}}^{\ell}\)-bundle over a projective manifold X of the same homology type as \({\mathbb{P}}^ r\) can have another projective bundle structure over a manifold Y and if so how to find restrictions on Y. The main result is, that if V is a \({\mathbb{P}}^{\ell}\)-bundle over X (X as above), which is also a \({\mathbb{P}}^{\ell -k-1}\)-bundle over a projective manifold Y, for \(r\geq \ell \leq 2k+1\), \(k\geq -1\) and \(\ell -k\geq 2\), then Y satisfies \(H_ i(Y)=0\) for i\ odd, \(H_ i(Y)={\mathbb{Z}}\) for i\ even \(\leq 2(\ell -k)-2\) and \(H_ i(Y)={\mathbb{Z}}^ 2\) for i\ even 2(\(\ell - k)\leq i\leq r.\) Some examples and corollaries are given.