  Let $A$ be an annular end of a complete minimal surface $S$ in $\mathbb R^m$
and let $V$ be a $k$-dimension projective subvariety of $\mathbb P^n(\mathbb
C)\ (n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset\mathbb
P^n(\mathbb C)$. In this paper, we establish a modified defect relation of $g$
on the annular end $A$ for $q$ hypersurfaces $\{Q_i\}_{i=1}^q$ of $\mathbb
P^n(\mathbb C)$ in $N$-subgeneral position with respect to $V$. Our result
implies that the image $g(A)$ cannot omit all $q$ hypersurfaces
$Q_1,\ldots,Q_q$ if $g$ is nondegenerate over $I_d(V)$ and
$q>\frac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)}$, where $M=H_V(d)-1$ and $d$ is the
least of common multiple of ${\rm deg} Q_1,\ldots,{\rm deg}Q_q$.
