Applications of the Ohsawa-Takegoshi Extension Theorem to Direct Image Problems

TL;DR

This paper applies the Ohsawa-Takegoshi extension theorem to address problems in algebraic geometry, providing bounds on direct image sheaves and extending results on Fano manifolds and rational connectedness.

Contribution

It offers new bounds on the global generation of direct image sheaves and generalizes key theorems to Kawamata log terminal cases using extension techniques.

Findings

Effective bounds on the global generation of direct image sheaves

Affirmative answer to a question by Demailly-Peternell-Schneider

Generalization of results on weak Fano manifolds and rational connectedness

Abstract

In the first part of the paper, we study a Fujita-type conjecture by Popa and Schnell, and give an effective bound on the generic global generation of the direct image of the twisted pluricanonical bundle. We also point out the relation between the Seshadri constant and the optimal bound. In the second part, we give an affirmative answer to a question by Demailly-Peternell-Schneider in a more general setting. As an application, we generalize the theorems by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal cases, and refine a result by Broustet and Pacienza on the rational connectedness of the image.