AbstractIn this paper, we establish some modified defect relations for the Gauss map of a complete minimal surface into a ‐dimension projective subvariety with hypersurfaces of in ‐subgeneral position with respect to . In particular, we give the upper bound for the number if the image intersects each hypersurface a finite number of times and is nondegenerate over , where , that is, the image of is not contained in any hypersurface of degree with . Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.