I am still trying to figure out how to construct positive line bundles on the blow up of a Kähler manifold and thus a Kähler form.
In Voisin's book (Hodge Theory and Complex Algebraic Geometry), it is written:
I fail to see the logic of the argument. Assuming $X$ is compact, then $\lambda$ is bounded, so everywhere where $\tau^* \omega_X$ is positive, $C \tau^* \omega_X + \lambda$ is positive for some fixed constant $C>0$. This is the case everywhere but on the tangent space of the fibers of $\tau$. But there $\lambda$ is positive and $\tau^* \omega $ vanishes.
Did I understand it right, that the argument is this simple?
The additional complexity should then come from the assumption of $X$ not neccesarily being compact. But I could still apply the same argument as above replacing $X$ with $K$, a compact nbhd of $Y$ where $\lambda$ is zero.
The fact that I did not use compactness of $Y$ at all, indicates that I did miss some subtlety. But I cannot figure out which one.
Update: I can make the argument work if $\lambda$ is semi-positive on a neighbourhood of $\tau^{-1}(Y)$.
Does it work without this assumption? I don't think so.