Good exposition of "Calabi ansatz"

Question

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

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

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

Here $t=t(v)=\log\lvert v\rvert_h$ 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 Kähler 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 (A momentum construction for circle-invariant Kähler metrics, Transactions of the AMS 2002), but would like something addressing my question more directly.)

Answer 1

One application of Calabi-Ansatz is choosing inital metric to run the Kahler–Ricci flow. So if you want to use 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 gives 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, Geometric and Functional Analysis, February 2012, Volume 22, Issue 1, pp 240–265.

See proposition 3.2 of Futaki - Momentum construction on Ricci-flat Kähler cones for study of Sasaki–Ricci flow and finding suitable inital metric via Calabi-Ansatz.

Answer 2

Very belated, but in the hope it's useful to posterity: The Calabi ansatz expresses a special type of circle-invariant Kähler metric in terms of its own moment map as sketched briefly below. The main limitations are the stringent curvature conditions required on the base metric $(M, \omega_{M})$ and the line bundle $(L, h)$. A slightly newer and more encompassing prospective resource might be the three-part series of papers on Hamiltonian $2$-forms Hamiltonian 2-forms in Kähler geometry, I General Theory, Hamiltonian 2-forms in Kähler geometry, II Global Classification, and Hamiltonian 2-forms in Kähler geometry, III Extremal metrics and stability by Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman.

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In the notation of the question, let $t$ denote the Hermitian norm function on the total space of $L$; let $(0, b)$ be an interval of real numbers and $\tau_{0}$ a number in $(0, b)$; and let $\varphi$ be a $C^{2}$ (or smoother) positive function on $(0, b)$. Equation (2.2) of the Transactions paper Hwang and Singer - A momentum construction for circle-invariant Kähler metrics directs us to define real-valued functions $\mu$ and $f$ by $$ t = \int_{\tau_{0}}^{\mu(t)} \frac{dx}{\varphi(x)},\qquad f(t) = \int_{\tau_{0}}^{\mu(t)} \frac{x\, dx}{\varphi(x)}. $$ Setting $\tau = \mu(t)$ and letting $\gamma$ denote the curvature form of $(L, h)$, we have an equality of closed $(1, 1)$-forms \begin{align*} \omega_{L} &= p^{*}\omega_{M} + dd^{c} f(t) \\ &= p^{*}\omega_{M} - \tau\, p^{*}\gamma + \frac{1}{\varphi(\tau)}\, d\tau \wedge d^{c}\tau. \end{align*} From the "momentum side," $\omega_{L}$ is positive as a closed $(1, 1)$-form on some annulus sub-bundle of $L$ if and only if $\varphi$ is positive on $(0, b)$ and the closed $(1, 1)$-form $\omega_{M} - b\gamma$ is positive on $M$. Conditions on $f$ may be read off from the formulas above.

Answer 3

I think that the best reference is Calabi's paper: Métriques kählériennes et fibrés holomorphes, Annales Scientifiques de l’École Normale Supérieure, 12(1979), 268–294. (NUMDAM, doi:10.24033/asens.1367)