Classification of polarized manifolds admitting homogeneous varieties as ample divisors (Q957882)

The article under review presents a classification of smooth polarized varieties \((X,L)\) such that the linear system \(| L| \) contains a smooth member \(A\) that is homogeneous. In the first part of the paper, the author builds on earlier results of \textit{T. Fujita} [J. Math. Soc. Japan 34, 355--363 (1982; Zbl 0478.14002)] to state necessary conditions for a homogeneous variety \(A\) to be an ample divisor in a smooth variety \(X\). This yields a finite list of possibilities for \(A\). The following section examines explicitly each of these possibilities and determines for each A the possible varieties \(X\). The author also obtains the following corollary as a byproduct: If \(X\) is a projective bundle over a smooth curve \(C\) and if \(X\) is homogeneous, then \(X\) is isomorphic to \(\mathbb{P}^1\times \mathbb{P}^n\) or to \(E\times \mathbb{P}^n\) for \(E\) an elliptic curve.