Let F̄ be a Hermitian vector bundle on an arithmetic variety X over ℤ. We prove an inequality between the L2-norm of an element in H1(X,FV) and arithmetic Chern classes of F under certain stability condition. This is a higher dimensional analogue of a result of C. Soulé for Hermitian line bundles on arithmetic surfaces. We observe that our result is related to a conjectural inequality of Miyaoka-Yau type.
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