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?



Are "ample" and "positive" line bundle the same concept?
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). 


