Left and Right in small-x neutrino DIS

The color dipole analysis of nuclear effects in charge current DIS is presented. The emphasis is put on the pronounced effect of left-right asymmetry of shadowing in neutrino-nucleus DIS at small values of Bjorken $x$. Strikingly different scaling behavior of nuclear shadowing for the left-handed andright-handed $W^+$ is predicted. Large, about 25%, shadowing in the $Fe$ structure functions is predicted, which is important for a precise determination of the P-odd nucleon structure functions $xF^{\nu(\bar\nu)}_3$ and $\Delta x F_3=xF_3^{\nu}-xF_3^{\bar \nu}$.