Probl\`{e}me de Lehmer relatif dans un tore : cas des hypersurfaces

We tackle the “relative” Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its obstruction index over ${\text{Q}}^{ab}$, the maximal abelian extension of $\text{Q}$. We prove up to $\text{eps}$ the sharpest conjecture that can be formulated.