complex geometry - Tensor product of a line bundle with a large multiple of another positive line bundle also positive?

Let us prove that for an affine variety $X$ every line bundle $E$ is "positive" according to the chosen defintion. All we need to prove is that for any hermitian metric $g$ on $E$ with curvature $w$ there is a Kahler form $w_1$ on $X$ such that $w_1>-w$. Since $X$ is affine, for any $w_1$ we have $w_1=frac{i}{2pi}partialbarpartial (f_1)$ and changing the metric $g$ on $E$ by $ge^{f_1}$ we corresponing curvature will change from $w$ to $w+w_1$, which we assume to be positive.

So we need to show the existence of arbitrary large $w_1$. Since $X$ is affine and hence admits an embedding in $mathbb C^n$, it is enough to show this for $mathbb C^n$. Moreover, since $mathbb C^n=mathbb C^1times ...times mathbb C^1$ it is enought to prove the statement for $mathbb C^1$. Now, on $mathbb C^1$ every form of the shape $w_1=h_1dzwedge dbar z$ is Kahler for $h_1>0$ and we can chose $h_1$ as large as we wish.

The conclusion is that if one choses this definition, then each line bundle on an affine $X$ is positive, which sounds strange. So I am not sure what should be a reasonable definition of positivincess in non-compact case, if it exists at all.