  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 $\Q^{ab}$, the maximal abelian extension of
$\Q$. We prove up to $\eps$ the sharpest conjecture that can be formulated.
