nef Line bundles over Kähler manifolds
The answer to your question, at least with the definition that you suggest, is negative. The main trouble is that a compact Kähler manifold need not have any complete curve at all. This is the case for instance for a compact complex torus $\mathbb{C}^n/\Lambda$ when $\Lambda$ is a sufficiently general lattice, and $n > 1$.
However, a much better definition of nef in the Kähler case is precisely to ask that $c_1(L)$ is a limit of Kähler classes, because this has all the expected properties of nef classes on projective varieties.
Jean-Pierre Demailly has been considering such questions in his work since a long time. See especially on his webpage references [35], [40], [62], [65], [69].
J.P. Demailly. Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), 87–104, Lecture Notes in Math., 1507, Springer, Berlin, 1992.
J.P. Demailly, T. Peternell, and M. Schneider. Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3, 295–345 (1994).
J.P. Demailly, T. Peternell, and M. Schneider. Pseudo-effective line bundles on compact Kähler manifolds. Internat. J. Math. 12, 689–741 (2001).
J.P. Demailly, T. Eckl, and T. Peternell. Line bundles on complex tori and a conjecture of Kodaira. Comment. Math. Helv. 80, 229–242 (2005).
S. Boucksom, J.P. Demailly, M. Păun, and T. Peternell. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22, 201–248 (2013).
Even a definition of "nef" that works for general compact complex manifolds (or even spaces with singularities) is given. Since Hodge decomposition does not hold, one has to use Bott-Chern cohomology instead of the more usual de Rham cohomology. (As a matter of fact, all Chern classes are defined in these "richer" cohomology groups, which coincide with the Dolbeault groups in the Kähler case, but differ in general).
You will find most of the above stuff in his book "analmeth_book.pdf" (funniest file name ever, am I right) here.
- J.P. Demailly. Analytic methods in algebraic geometry. Surveys of Modern Mathematics, 1. International Press, Somerville, MA; Higher Education Press, Beijing, 2012. viii+231 pp.
The "agbook" introduces more preliminary material, see here.
- J.P. Demailly. Complex analytic and differential geometry. Monograph Grenoble, 1997.