Holomorphic quasi-positive line bundles on a complex manifold $M$ are line bundles whose chern class can be represented by a closed $(1,1)$-form which is quasi-positive, that is, non-negative at all points on M and strictly positive at least at one point on $M$. Positive line bundles are defined similarly.
Now my question is: Are holomorphic quasi-line bundles on a compact complex manifold $M$ positive? The answer is negative, because there are non-projective Moishezon manifolds. A Moishezon manifold has a quasi-positive line bundle. So, quasi-positive line bundles on a non-projective Moishezon manifold cannot be positive.
So now I ask a related question: Are holomorphic quasi-line bundles on a compact Kähler manifold $M$ positive? The answer seems to be negative. But can one give an example, that is, a quasi-positive line bundle that is not positive? Or prove it, if it is true. Thank you.