  We study abelian dominance for confinement in terms of the local gluon
properties in the maximally abelian (MA) gauge in a semi-analytical manner with
the help of the lattice QCD. The global Weyl symmetry persistently remains as
the relic of SU($N_c$) in the MA gauge, and provides the ambiguity on the
electric and magnetic charges. We derive the criterion on the SU($N_c$)-gauge
invariance in terms of the residual symmetry in the abelian gauge. In the
lattice QCD, we find microscopic abelian dominance on the link variable for the
whole region of $\beta$ in the MA gauge. The off-diagonal angle variable, which
is not constrained by the MA-gauge fixing condition, tends to be random besides
the residual gauge degrees of freedom. Within the random-variable approximation
for the off-diagonal angle variable, we prove that off-diagonal gluon
contribution to the Wilson loop obeys the perimeter law in the MA gauge, and
show exact abelian dominance for the string tension, although small deviation
is brought by the finite size effect of the Wilson loop in the actual lattice
QCD simulation.
