Residue currents of monomial ideals

We compute residue currents of Bochner-Martinelli type associated with a monomial ideal I, by methods involving certain toric varieties. In case the variety of I is the origin, we give a complete description of the annihilator of the currents in terms of the associated Newton diagram. In particular, we show that the annihilator is strictly included in I, unless I is defined by a complete intersection. We also provide partial results for general monomial ideals.