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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 $\mathbb P^n(\mathbbC)\ (n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset\mathbbP^n(\mathbb C)$. 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 $\mathbbP^n(\mathbb C)$ in $N$-subgeneral position with respect to $V$. Our resultimplies 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 theleast of common multiple of ${\rm deg} Q_1,\ldots,{\rm deg}Q_q$.

Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersuraces of projective varieties in subgeneral position