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As far as I understand, Calabi ansatz is (in particular) a way to produce Kahler metrics on total spaces of line bundles (or their disk subbudles) over Kahler manifolds of the following form:

Calabi Ansatz. Let $p:(L,h)\to (M,\omega_M)$ be a Hermitian line bundle over a Kahler manifold $M$. Consider on the total space $L$ the following two-form:

$$\omega_L=p^*(\omega_M)+dd^cf(t).$$

Here $t=t(v)=\log|v|_h$ is the log of the norm function on $L$ defined by $h$ and $f\in C^{\infty}(\mathbb R^1)$.

Question. As far as I understand, $\omega_L$ is Kahler on some disk sub-bundle of the total space $L$ provided $f$ satisfies certain (convexity?) conditions. Are you aware of a good reference on this that would give these conditions on $f$? (I am aware of a few articles, like Hwang-Singer (Transaction of AMS 2002) but would like something addressing my question more directly)

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One of application of Calabi-ansatz is choosing inital metric to run the Kahler Ricci flow, So if you want to using minimal model program and apply Kahler Ricci flow to find canonical metric study of such Calabi-ansatz would be very important. In fact study of Calabi-ansatz give an effective way to find inital metric and its connection with semi-flat metric. The question is still open

See the paper of

Jian Song, Yuan Yuan , Metric Flips with Calabi Ansatz, Geometric and Functional Analysis February 2012, Volume 22, Issue 1, pp 240–265

See proposition of 3.2 for study of Sasaki-Ricci flow and finding suitable inital metric via Calabi-Ansatz

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  • $\begingroup$ Thanks Hassan for the reference. I would be really glad to have something like proposition 2.1 in the paper you cite for the case of line bundles over any variety. Maybe after all I'll need to look in Calabi's paper... $\endgroup$ – aglearner Apr 25 '17 at 9:22
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I think that the best reference is Calabi's paper: Métriques Kahleriennes et fibrés holomorphes, Annales Scientifiques de l’Ecole ´ Normale Supérieure, 12(1979), 268-294. (NUMDAM, doi:10.24033/asens.1367)

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