A counterexample to a conjecture due to Douglas, Reinbacher and Yau

Abstract

In “Branes, Bundles and Attractors: Bogomolov and Beyond”, by Douglas, Reinbacher and Yau, the authors state the following conjecture: Consider a simply connected surface X with ample or trivial canonical line bundle. Then, the Chern classes of any stable vector bundle with rank r ≥ 2 satisfy 2 r c 2 − ( r − 1 ) c 1 2 − r 2 12 c 2 ( X ) ≥ 0 . The goal of this short note is to provide two sources of counterexamples to this strong version of the Bogomolov inequality.