Exemples de faisceaux cohérents sans résolution localement libre en dimension 3

Abstract

In 1982, Schuster proved that for any compact complex surface X, every coherent sheaf F on X has global resolutions⋯→En→En−1→⋯→E1→E0→F→0 such that the Eiʼs are locally free (vector bundles). In 2002, Voisin proved that this is false for some Kähler compact complex manifolds of dimension ⩾3. In this Note, we give some new examples of non-Kähler compact complex manifolds of dimension 3 and coherent sheaves F on X having no global resolution by vector bundles. The proof that these sheaves do not admit a locally free resolution is very different from Voisinʼs argument.