Bogomolov-Sommese vanishing on log canonical pairs

TL;DR

This paper extends the Bogomolov-Sommese vanishing theorem to log canonical pairs, including orbifold structures, using advanced techniques like the MMP and residue maps, and provides applications to vanishing theorems.

Contribution

It generalizes the Bogomolov-Sommese vanishing theorem to log canonical pairs and orbifold structures, introducing new methods and applications in the field.

Findings

Reflexive logarithmic forms do not contain high Kodaira-Iitaka dimension subsheaves.

The theorem cannot be extended to spaces with Du Bois singularities.

A new vanishing theorem for log canonical pairs is established.

Abstract

Let (X, D) be a projective log canonical pair. We show that for any natural number p, the sheaf (Omega_X^p(log D))^** of reflexive logarithmic p-forms does not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds p. This generalizes a classical theorem of Bogomolov and Sommese. In fact, we prove a more general version of this result which also deals with the orbifoldes g\'eom\'etriques introduced by Campana. The main ingredients to the proof are the extension theorem of Greb-Kebekus-Kov\'acs-Peternell, a new version of the Negativity lemma, the Minimal Model Program, and a residue map for symmetric differentials on dlt pairs. We also give an example showing that the statement cannot be generalized to spaces with Du Bois singularities. As an application, we give a Kodaira-Akizuki-Nakano-type vanishing result for log canonical pairs which holds for reflexive as well as for K\"ahler differentials.