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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2015, Issue 702

# Bogomolov–Sommese vanishing on log canonical pairs

Patrick Graf
• Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany
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Published Online: 2013-05-28 | DOI: https://doi.org/10.1515/crelle-2013-0031

## Abstract

Let (X,D) be a projective log canonical pair. We show that for any natural number p, the sheaf ${\left({\Omega }_{X}^{p}\left(log⌊D⌋\right)\right)}^{**}$ 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 géométriques introduced by Campana. The main ingredients to the proof are the Extension Theorem of Greb–Kebekus–Kovács–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 Kähler differentials.

## About the article

Published Online: 2013-05-28

Published in Print: 2015-05-01

Funding Source: DFG

Award identifier / Grant number: Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 702, Pages 109–142, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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© 2015 by De Gruyter.