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Let $X$ be a compact generalized Sasakian CR manifold of dimension $2n-1$,$n\geqslant2$, and let $L$ be a generalized Sasakian CR line bundle over $X$equipped with a rigid semi-positive Hermitian fiber metric $h^L$. In this paperwe prove that if $h^L$ is positive at some point of $X$ and conditions Y(0) andY(1) hold at each point of $X$, then $L$ is big.Comment: 46 page

Existence of CR sections for high power of semi-positive generalized Sasakian CR line bundles over generalized Sasakian CR manifolds