# Ample subvarieties and rationally connected fibrations

## TL;DR

This paper investigates how rationally connected fibrations extend from submanifolds with ample normal bundles to their ambient varieties, providing new extension and classification results under positivity assumptions.

## Contribution

It establishes conditions under which rational connected fiber structures on submanifolds extend to the ambient variety, including applications to Mori contractions and classifications.

## Key findings

- Extension theorem for Mori contractions of fiber type
- Classification of varieties with submanifolds as projective bundles or quadrics
- Relations between rational connected structures on submanifolds and ambient varieties

## Abstract

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.