Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.

中文翻译：

---

从充足的子变体中扩展合理连接的纤维

使用有理曲线的变形理论，我们证明了当态射是具有有理连接的纤维的平滑（或轻度奇异）纤维化时，来自充足子变体的态射可扩展性的 Sommese 猜想。我们将此结果应用于 Fano fibrations 的上下文中，并证明了大量子变体上的投影丛和二次纤维化结构的分类定理。