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I thought that balanced manifolds were hermitian manifolds such that $d\omega^{dimX−1}=0$, where $\omega$ is the fundamental $(1,1)$-form associated to the hermitian metric. Is this equivalent to your definition?
On the other hand, of course the "global approach" two doesn't work in higher dimension, since the difference of two exceptional divisors coming form the blow-up of two different points is never zero in cohomology. But it was so simple, that I wanted to post it anyway. :)
I mean, you have to arrange it a little bit, but nothing mysterious... Just use a tubular neighborhood of the exceptional divisor E on the blown-up manifold $\tilde X$ and extend the natural metric of $\mathcal O_E(−E)$ in an arbitrary way to a metric on $\mathcal_{\tilde X} O(-E)$.
Thanks very much Sándor, do you think that if I find a more elementary proof, which does not rely on deep theorems such as the Basepoint-free one, could be interesting?
