Projective manifolds swept out by large dimensional linear spaces (Q1384442)

Main theorem. Let \(X\) be an \(n(\geq 2)\)-dimensional smooth projective variety in a projective space \(\mathbb{P}^N\). Assume that for each point \(x\) in \(X\), there exists an \(m\)-dimensional linear subspace \(P_x\) in \(X\) containing the point \(x\) with \(2m\geq n\). Moreover, suppose that the characteristic of the base field is zero. Then for a general point \(x\) the normal bundle \(N_{P_x/X}\) of \(P_x\) in \(X\) is isomorphic to one of the following: (1) \({\mathcal O}_{\mathbb{P}^m}^{\oplus a}\bigoplus{\mathcal O}_{\mathbb{P}^m} (1)^{\oplus b}\) \((a\) and \(b\) are non-negative integers) (2) \(\Omega_{\mathbb{P}^m} (2)\) \((m\geq 2)\), (3) \(T_{\mathbb{P}^m}(-1)\) \((m\geq 2)\). Moreover, corresponding to the above cases, \(X\) is respectively as follows: (1) a \(\mathbb{P}^{n-a}\)-bundle over an \(a\)-dimensional smooth projective variety \(S\) where a general \(m\)-plane \(P_x\) is in the fiber of the canonical projection \((n\geq a\geq n/2)\); (2) an even-dimensional smooth hyperquadric; (3) the Grassmann variety \(\text{Gr}(m+1,1)\) parametrized by lines in \(\mathbb{P}^{m+1}\) with \(n=2m\), if \(m\) is even.