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edited May 28, 2022 at 18:37

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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 threeHamiltonian 2-part seriesforms in Kähler geometry, I General Theory of, papersHamiltonian 2-forms in Kähler geometry, II Global Classification on, and Hamiltonian $2$2-forms in Kähler geometry, III Extremal metrics and stability by Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman.

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 paperTransactions 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.

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 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.

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.

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.

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created May 28, 2022 at 18:30

Andrew D. Hwang

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 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.