# Strictness of the inequality relating the Iitaka dimension and algebraic dimension

For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension $\kappa(X,L)$ of $L$ over $X$, the number of algebraic independent meromorphic functions $\alpha(X)$ on $X$ and the dimension of $X$. If $\alpha(X)$ equals $\dim(X)$, i.e. if $X$ is Moishezon, there exists a big line bundle $L$ on $X$, i.e. such that all the inequalities are actually equalities.

I don't expect it to be true, but I wonder if there always exists a line bundle that "sees all the meromorphic functions". I failed in searching a manifold with strictly bigger algebraic dimension than Iitaka dimension for all line bundles (simply because I'm not able to compute $\sup_{L\to X}\{\kappa(X,L)\}$, unless there exists a big line bundle).

Consequently, I'm asking for a manifold $X$ such that for all its line bundles $L$, $\kappa(X,L)<\alpha(X)$. Do you know some? There is no word for line bundles with $\kappa(X,L)=\alpha(X)$, is it?

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Given a compact complex manifold $X$, there exists a meromorphic map $F: X\dashrightarrow Z$ to a projective manifold such that every meromorphic function on $X$ is the pull back under $F$ of a rational function on $Z$. The pull-back of an ample line-bundle on $Z$ realizes the equality between algebraic and Iitaka's dimensions.