In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a positive $(1,1)$-form. A very ample is a strengthening of ample (no need to take powers). Is there a corresponding Kaehler notion, and in this case, is there a differential geometric proof that such a line bundle is generated by holomorphic sections? 



Kaehler analogue of very ample line bundle
J. P. Demailly, Multiplier ideal sheaves and analytic methods, p. 26, (3.15) A holomorphic line bundle $F$ over a compact complex manifold $X$ is 


a) *very ample* if the map $\Phi\_{|F|}\colon X \to \mathbb{P}^{N−1}$ associated to the complete linear system $|F| = \mathbb{P}(H^0(X,F))$ is a regular embedding (by this we mean in particular that the base locus is empty, i.e. $B\_{|F|} = \emptyset$).


b) *ample* if some multiple $mF$, $m > 0$, is very ample.


By Example 3.14, every ample line bundle $F$ has a smooth hermitian metric with positive definite curvature form; indeed, if the linear system $|mF|$ gives an embedding in projective space, then we get a smooth hermitian metric on $F^{\otimes m}$, and the $m$-th root yields a metric on $F$ such that $(i/2\pi)Θ(F )=(1/m)\Phi^\*\_{|mF}\omega\_{FS}$. Conversely, the Kodaira embedding theorem tells us that every positive line bundle $F$ is ample (see Exercise 5.14 for a straightforward analytic proof of the Kodaira embedding theorem).


