On possible Chern classes of stable bundles on Calabi-Yau threefolds

Supersymmetric heterotic string models, built from a Calabi-Yau threefold X endowed with a stable vector bundle V, usually lead to an anomaly mismatch between c2(V) and c2(X); this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In [M.R. Douglas, R. Reinbacher, S.-T. Yau, Branes, Bundles and Attractors: Bogomolov and Beyond, math.AG/0604597], a conjecture is stated which gives sufficient conditions on cohomology classes on X to be realized as the Chern classes of a stable reflexive sheaf V; a weak version of this conjecture predicts the existence of such a V if c2(V) is of a certain form. In this note, we prove that on elliptically fibered X infinitely many cohomology classes c∈H4(X,Z) exist which are of this form and for each of them a stable SU(n) vector bundle with c=c2(V) exists.