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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:enter image description here

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.

Michael
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  • What exactly do you mean by saying "$\lambda$ is bounded"? – Ted Shifrin Jun 27 '19 at 19:04
  • @TedShifrin: Honestly, I dont really know. I am just trying to somehow make sense out of this. But its a term I encountered in multiple sources, e.g. Griffith & Harris.; – Michael Jun 27 '19 at 22:12
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    If $Y$ is non-compact, it will not have a compact neighborhood (though you can probably still use a patching argument in that case). – Max Jun 28 '19 at 04:25
  • @TedShifrin: I was thinking it means something like $|\lambda(V(\cdot),W(\cdot))|$ is bounded for any pair of vector fields $V,W$ or maybe $-\lambda(IV(\cdot),W(\cdot))$ – Michael Jun 28 '19 at 19:55
  • I think the idea should be like this: Let $U$ be such a compact neighbourhood. For every point $x\in U$, find an open neighbourhood $V_x$, such that for a $C_x$ $C_x\tau^*\omega_X+\lambda$ is positive around $V_x$. Since $U$ is compact, you just need finitely many $C_x$ and then just take the greatest one. – Notone Jun 30 '19 at 12:58

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