| Résume | In the seventies, Hartshorne proposed several problems regarding the existence of low rank vector bundles on projective spaces. In rank 2 and characteristic 0, those bundles are notoriously difficult to produce, and Hartshorne conjectured that none should exist from dimension seven. In this talk, I will explain some motivations for this conjecture, and a new approach to the problem, by mean of toric sheaves.
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