Are "ample" and "positive" line bundle the same concept? A line bundle is ample if some power of it is very ample. A line bundle is positive if the chern class in $H^2(X,\mathbb{Z})$ is represented by a Kahler metric in $H^{1,1}(X,\mathbb{Z})$.(Regarded as elements in $H^{1,1}(X,\mathbb{C})$ through the map $H^2(X,\mathbb{Z})\to H^2(X,\mathbb{C})$)
Are ample and positive the same concept for projective algebraic manifolds?
 A: A line bundle $L$ on a complex manifold is positive if there is an Hermitian metric $h$ on $L$ such that $\frac{1}{2\pi}\Theta(L,h)$ is a Kahler form. By Kodaira's embedding theorem if $L$ is a positive line bundle on a compact Kahler manifold the $L^{\otimes m}$ induces an embedding $f:X\rightarrow\mathbb{P}^n$ for some $m>0$ and $L^{\otimes m} = f^*\mathcal{O}_{\mathbb{P}^n}(1)$. Therefore $L$ is ample. Conversely if such an embedding exists the pullback of the Fubini-Study metric on $*\mathcal{O}_{\mathbb{P}^n}(1)$ determine a positive metric on $L^{\otimes m}$ and hence on $L$.
Similarly, if $L$ has an Hermitian metric such that $c_1(L) = \frac{i}{2\pi}\Theta(L,h)$ is non-negative then $L$ is nef. On the other hand it could happen that a nef bundle does not have such a metric, see for instance Example $1.7$ in this paper http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/dps1.pdf.
If $\omega$ is a Kahler form on $X$, then $L$ is nef if and only if for any $\epsilon >0%$ there is an Hermitian metric $h_\epsilon$ on $L$ such that $\frac{i}{2\pi}\Theta(L,h_\epsilon)>-\epsilon \omega$. Note that a compact Kahler manifold may not contain any curve. However, in this way one can define nefness on any compact Kahler manifold.
A: Yes. For ample implies positive, use the fact that $c_1(O(1))$ on projective space is the Kähler form of the Fubini-Study metric, and then restrict to $X$. For the converse, you need the Kodaira embedding theorem (in fact, this is more or less the content of that theorem). 
