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Using a canonical linear embedding of the algebra 
${{G}^{\infty }}\left( \Omega\right)$ of Colombeau generalized functions in the space of 
$\overline{\mathbb{C}}$ -valued 
$\mathbb{C}$ -linear maps on the space 
$D\left( \Omega\right)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class 
${{G}^{\infty }}$ . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.
