2.1 Classification of compact homogeneous Kähler manifolds

According to [Reference Matsushima30], it was proved that the centralizer of S in G is R. So the maximal tori containing S must be contained in R. In the following, we fix a maximal torus T with $S\subset T\subset R$ .

Notice that $\mathfrak {g}^{\mathbb {C}}$ admits a root space decomposition,

$$ \begin{align*} \mathfrak{g}^{\mathbb{C}}= \mathfrak{t}^{\mathbb{C}} \oplus \sum_{\alpha\in \Delta^+} (\mathfrak{g}_\alpha \oplus \mathfrak{g}_{-\alpha}). \end{align*} $$

We introduce a normalized adapted basis $\{X_\alpha , Y_\alpha \}$ of each pair of root spaces $\mathfrak {g}_\alpha \oplus \mathfrak {g}_{-\alpha }$ , which satisfies the following:

1. (a) Let $\mathfrak {g}_\alpha $ and $\mathfrak {g}_{-\alpha }$ denote the complex eigen-spaces corresponding to the roots $\alpha $ and $-\alpha $ . Then

   (2.1) $$ \begin{align} X_\alpha-iY_\alpha= E_{\alpha}\in \mathfrak{g}_\alpha\quad\text{and} \quad X_\alpha+iY_\alpha=E_{-\alpha}\in \mathfrak{g}_{-\alpha}. \end{align} $$

2. (b) $X_\alpha $ and $Y_\alpha $ are normalized in the sense that,

   (2.2) $$ \begin{align} [X_\alpha,Y_\alpha]=- H_\alpha, \end{align} $$
   where $H_\alpha $ satisfies $\beta ( H_\alpha )= (\beta ,\alpha )/2i$ for each root $\beta $ . Consider the inner product induced by the negative Killing form. One can easily check that $|X_\alpha |^2=|Y_\alpha |^2=1/2$ . This normalization will be applied in Section 3.1 to calculate the differential of invariant 1-forms.

Then the compact real sub-algebra $\mathfrak {g}$ can be decomposed as follows:

(2.3) $$ \begin{align} \mathfrak{g} = \mathfrak{t}\oplus \sum_{\alpha \in \Delta^+} \mathbb{R} \langle X_\alpha, Y_\alpha \rangle. \end{align} $$

Next, we classify simply connected compact homogeneous Kähler manifolds by the structure of root system. Consider a parabolic subgroup P of $G^{\mathbb {C}}$ . Then P is determined by a subset of the simple root system. Let $(\Delta ^+)' \subset \Delta ^+$ be the subset of positive roots generated by $\Pi '$ . The corresponding Lie algebra $\mathfrak {p}$ can be decomposed as

(2.4) $$ \begin{align} \mathfrak{p}=\mathfrak{t}^{\mathbb{C}} \oplus \sum_{\alpha\in \Delta^+}\mathfrak{g}_{\alpha}\oplus \sum_{\alpha\in (\Delta^+)' } \mathfrak{g}_{-\alpha}=\mathfrak{b}\oplus \sum_{\alpha\in (\Delta^+)' } \mathfrak{g}_{-\alpha}, \end{align} $$

where $\mathfrak {b}$ is the Lie algebra of the Borel subgroup. The generalized flag variety with data $(G, \Pi , \Pi ')$ is defined to be the complex manifold $X = G^{\mathbb {C}} / P$ .

Assuming that $X = G^{\mathbb {C}} /P$ , we show that X is simply connected compact homogeneous Kähler manifold. Consider the maximal compact subgroup G of $G^{\mathbb {C}}$ . Then G acts transitively on X with stabilizer group $R=G \cap P$ . The Lie algebra of the stabilizer group R is

(2.5) $$ \begin{align} \mathfrak{r} = \mathfrak{t} \oplus \sum_{\alpha \in (\Delta^+)'} \mathbb{R} \langle X_\alpha, Y_\alpha \rangle. \end{align} $$

Then, topologically, $X\cong G/R$ , and its complex structure can also be described as follows. Let

(2.6) $$ \begin{align} D^+:=\Delta^+ \setminus (\Delta^+)'. \end{align} $$

Then $D^+$ is a closed subset of the root system in the sense that for any $\alpha , \ \beta \in D^+ $ , if $\alpha +\beta $ is a root, then $\alpha +\beta \in D^+$ . The tangent space of X at a distinguished point p can be identified with

(2.7) $$ \begin{align} T_p X = \sum_{\alpha \in D^+ } \mathbb{R} \langle X_\alpha , Y_\alpha \rangle. \end{align} $$

We also call $\{X_\alpha , Y_\alpha : \alpha \in D^+ \}$ a normalized adapted basis of X. There exists a natural R-invariant almost complex structure J on $T_p X$ given by

$$ \begin{align*}J(X_\alpha)=Y_\alpha, \quad J(Y_\alpha)=-X_\alpha.\end{align*} $$

Because J is R-invariant, it extends to a G-invariant almost complex structure on the whole tangent bundle $TX$ . The complexified tangent space at p splits into

(2.8) $$ \begin{align} \displaystyle T^{(1,0)}_p X = \sum_{\alpha \in D^+} \mathfrak{g}_\alpha, \quad T^{(0,1)}_p X = \sum_{\alpha \in D^+} \mathfrak{g}_{-\alpha}. \end{align} $$

One can check that J is an integrable almost complex structure because $D^+$ is closed (see [Reference Borel and Hirzebruch7, Section 12]), and the complex manifold $(X,J)$ is G-equivariantly biholomorphic to $G^{\mathbb {C}}/P$ .

Generalized flag varieties are simply connected and the proof of this fact will be given in Remark 2.2. There exist Kähler forms on $(X,J)$ , as discussed in Section 2.3. Hence, each generalized flag variety is a simply connected compact homogeneous Kähler manifold of G.

Conversely, given a simply connected compact homogeneous Kähler manifold X that admits a transitive holomorphic action by a simply connected compact semisimple Lie group G with data $(R, T, S)$ and a point $p \in X$ as above, let $\Delta (\mathfrak {t}, \mathfrak {r})$ denote the root system of $\mathfrak {r}^{\mathbb {C}}$ with respect to $\mathfrak {t}^{\mathbb {C}}$ . This can be viewed as a subset of $\Delta $ . Define

$$ \begin{align*}D := \Delta\setminus\Delta(\mathfrak{t}, \mathfrak{r}).\end{align*} $$

The invariant complex structure J on X determines the set of positive roots $D^+$ as follows. Since the complexified tangent space of X is identified with

$$ \begin{align*}T_p^{\mathbb{C}} X= \sum_{\gamma\in D} \mathfrak{g}_\gamma,\end{align*} $$

the R-invariance of J implies that J preserves each root space. Then $D^+$ is defined to be

$$ \begin{align*}D^+:= \{\gamma\in D : Jv=iv\;\,\text{for all}\;\,v\in \mathfrak{g}_\gamma\}.\end{align*} $$

The closedness of $D^+$ follows from the integrability of J. In addition, we can choose a simple root system $\Pi '$ in $\Delta (\mathfrak {t},\mathfrak {r})$ . Then $D^+$ and the positive roots, $\Delta ^+(\mathfrak {t}, \mathfrak {r})$ , generated by $\Pi '$ determine a positive root system $\Delta ^+(\mathfrak {t}, \mathfrak {r})$ in $\Delta (\mathfrak {t}, \mathfrak {g})$ , i.e.,

$$ \begin{align*}\Delta^+(\mathfrak{t},\mathfrak{g})= D^+ \cup \Delta^+ (\mathfrak{t}, \mathfrak{r}).\end{align*} $$

The set $\Pi '$ can be extended to a simple root system, $\Pi $ , of $\Delta (\mathfrak {t}, \mathfrak {g})$ such that $\Pi $ generates the positive roots $\Delta ^+(\mathfrak {t},\mathfrak {g})$ . More details on root systems can be found in [Reference Borel and Hirzebruch7, Sections 13.6 and 13.7]. Based on this discussion, the Lie algebra $\mathfrak {r}$ can be written in terms of the simple root set $\Pi '$ as in (2.5). Thus, X can be identified with the generalized flag variety associated with the data $(G, \Pi , \Pi ')$ .

The notion of a generalized flag variety is actually independent of the choice of a simple root system. By [Reference Adams2, Section 5.13], the different simple root systems are identified by the action of the Weyl group, and then the associated generalized flag varieties are isomorphic via conjugation by an element of G. Hence, without loss of generality, we can fix a simple root system $\Pi $ at the beginning, and then each generalized flag variety of G is classified in terms of a subset of $\Pi $ .

In conclusion, we have proved the following theorem, which should be well-known to experts.

Theorem 2.1 Let X be a compact Kähler manifold. Assume that X is homogeneous under a simply connected compact semisimple Lie group G. Fix a system of simple roots $\Pi $ of $\mathfrak {g}^{\mathbb {C}}$ . Then X is G-equivariantly biholomorphic to the generalized flag variety of type $\Pi '$ for some subset $\Pi '\subset \Pi $ , i.e.,

$$ \begin{align*}X \cong G^{\mathbb{C}}/P,\ \text{where}\ P\ \text{is the parabolic subgroup of}\ G^{\mathbb{C}}\ \text{determined by}\ \Pi'.\end{align*} $$

Remark 2.2 In fact, all homogeneous manifolds discussed in this paper are simply connected. More precisely, let X be a compact Kähler manifold homogeneous under a simply connected semisimple compact Lie group G. Then X is simply connected. This fact follows quickly from the fiber bundle $R \hookrightarrow G \to X$ . Let $X=G/R$ , and let S be the connected center of R. According to the statement $(*)$ , R is the union of all the maximal tori containing S, which implies that R is connected. The fiber bundle structure $R \hookrightarrow G \to X$ induces a long exact sequence of homotopy groups

$$ \begin{align*} \ldots \rightarrow \pi_1(G) \rightarrow \pi_1(X) \rightarrow \pi_0(R)\rightarrow \dots, \end{align*} $$

and this tells us that $\pi _1(X)=0$ .

According to [Reference Borel and Remmert8, Satz I], each compact homogeneous Kähler manifold is the product of a flat complex torus and a simply connected compact homogeneous Kähler manifold. Furthermore, the connected component of the identity of the automorphism group of X is a semisimple Lie group [Reference Borel and Remmert8, Satz 4]. In conclusion, Theorem 2.1 classifies all compact homogeneous Kähler manifolds without torus part.

2.2 Classification of holomorphic line bundles

Consider a holomorphic line bundle L over X, with $\pi : L \rightarrow X$ . The holomorphic line bundle is said to be $G^{\mathbb {C}}$ -homogeneous (or in many articles $G^{\mathbb {C}}$ -linearizable) if there exists a $G^{\mathbb {C}}$ -action on L such that the projection $\pi $ is $G^{\mathbb {C}}$ -equivariant and the action is linear on fibers. In particular, we can construct $G^{\mathbb {C}}$ -homogeneous line bundle as follows.

Given a one-dimensional holomorphic representation $\chi :P\rightarrow \mathbb {C}^*$ , a $G^{\mathbb {C}}$ -equivariant holomorphic line bundle $L_\chi $ over $X = G^{\mathbb {C}}/P$ can be constructed as follows:

(2.9) $$ \begin{align} L_\chi= G^{\mathbb{C}}\times_\chi \mathbb{C}= (G^{\mathbb{C}} \times \mathbb{C})/{\sim}, \end{align} $$

where $(gh, v)\sim (g,\chi (h)v)$ for all $g\in G^{\mathbb {C}}$ , $h\in P$ , $v\in \mathbb {C}$ . This admits a natural $G^{\mathbb {C}}$ -action given by

(2.10) $$ \begin{align} g \cdot [(l,v)]=[(gl,v)] \end{align} $$

for all $g\in G^{\mathbb {C}}$ , $l\in G^{\mathbb {C}}$ , $v \in \mathbb {C}$ . Conversely, given a $G^{\mathbb {C}}$ -equivariant holomorphic line bundle L over X, the stabilizer group P at the distinguished point $p\in X$ acts on the fiber $L_p$ , inducing a holomorphic character $\chi : P \rightarrow \mathbb {C}^*$ such that $L \cong L_\chi $ . In conclusion, there is a one-to-one correspondence between the $G^{\mathbb {C}}$ -homogeneous line bundles over X and the characters of P.

Let L be an arbitrary holomorphic line bundle over X. We claim that L is a $G^{\mathbb {C}}$ -homogeneous line bundle. To prove this claim, we need to borrow some results from algebraic geometry. Referring to [Reference Humphreys21, Section 21.3], X is a projective variety. According to a well-known result from GAGA [Reference Serre33], holomorphic line bundles over a projective variety are algebraic; in other words, the line bundle L is algebraic over X. Moreover, since $G^{\mathbb {C}}$ is a simply connected semisimple Lie group, by [Reference Popov31, Proposition 1], the Picard group of $G^{\mathbb {C}}$ is trivial. According to the key fact that $\operatorname {Pic} G^{\mathbb {C}} =0 $ , we can construct a character $\chi : P \rightarrow \mathbb {C}^*$ such that $L \cong L_\chi $ (see [Reference Popov31, Theorem 4 or Section 5] for details). Thus, we have the following proposition:

Fixing a simple root system $\Pi $ of G, let $X\cong G^{\mathbb {C}}/ P$ , where the parabolic subgroup P is determined by a subset of simple roots $\Pi '\subset \Pi $ as in (2.4). Let $\Pi =\{\alpha _1,\ldots , \alpha _n\}$ be ordered in such a way that $\alpha _i\in \Pi '$ if and only if $i=k+1,\ldots , n$ . Let $\{\omega _1,\ldots , \omega _n\}$ be the set of fundamental weights corresponding to $\Pi $ , which are defined by

$$ \begin{align*} \frac{2(\omega_i,\alpha_j)}{(\alpha_j,\alpha_j)}=\delta_{ij}\;\,(1\leq i,j\leq n). \end{align*} $$

Here, the inner product is the bilinear form induced by the Killing form of $\mathfrak {g}$ . The lattice generated by $\{\omega _1,\ldots ,\omega _n\}$ , i.e., all vectors of the form $\sum _{n_i\in \mathbb {Z}} n_i \omega _i$ , is called the lattice of algebraically integral weights. Since G is a simply connected semisimple Lie group, the analytically integral weights coincide with the algebraically integral weights (see [Reference Knapp24, Chapter IV.7]). Thus, each algebraically integral weight induces a character of the maximal torus T. Recall that $\mathfrak {s},\mathfrak {s}^*$ denote the Lie algebra of S and its dual space, respectively. More precisely,

$$ \begin{align*} \mathfrak{s}=\{H\in \mathfrak{t} : \alpha (H)=0, \ \forall \alpha \in \Delta(\mathfrak{t},\mathfrak{r}) \},\quad \mathfrak{s}^*=\{\beta\in \mathfrak{t}^* : (\alpha,\beta )=0, \ \forall \alpha \in \Delta(\mathfrak{t},\mathfrak{r}) \}. \end{align*} $$

Then $(\mathfrak {s}^{\mathbb {C}})^*=\mathbb {C} \langle \omega _1,\ldots ,\omega _k\rangle $ and the intersection of the weight lattice with $(\mathfrak {s}^{\mathbb {C}})^*$ is $\mathbb {Z}\langle \omega _1,\ldots , \omega _k\rangle $ . We call the elements of this sublattice integral weights on X. Given such an element $\lambda $ , there exists an associated character $\chi ^\lambda : S^{\mathbb {C}}\rightarrow \mathbb {C}^*$ defined by

(2.11) $$ \begin{align} \chi^\lambda(\exp{v})=\exp{ \lambda(v)} \end{align} $$

for all $v \in \mathfrak {s}^{\mathbb {C}} \subset \mathfrak {t}^{\mathbb {C}}$ . The character $\chi ^\lambda $ can be extended to P in the following way. The Lie subalgebra $\mathfrak {p}$ can be decomposed as follows:

$$ \begin{align*} \mathfrak{p} & =\underbrace{\mathfrak{s}^{\mathbb{C}}\oplus \sum_{\beta\in D^+} \mathfrak{g}_{\beta}}_{\mathfrak{p}_1} \oplus \underbrace{\sum_{\alpha\in \Pi'} \mathbb{C}\langle H_\alpha \rangle \oplus \sum_{\alpha\in \Pi' } (\mathfrak{g}_{\alpha}\oplus \mathfrak{g}_{-\alpha})}_{\mathfrak{p}_2}\\ &= \mathfrak{s}^{\mathbb{C}} \oplus \mathfrak{n}\ \oplus \mathfrak{p}_2, \end{align*} $$

where $ \mathfrak {p}_1$ is the solvable part of $\mathfrak {p}$ , $\mathfrak {n} $ is the nilpotent part of $\mathfrak {p}_1$ , and $\mathfrak {p}_2$ is the semisimple part of $\mathfrak {p}$ . Each integral weight $\lambda $ on X can be extended to a complex Lie algebra homomorphism $\sigma :\mathfrak {p} \rightarrow \mathbb {C}$ by defining the extension to be $0$ on both $\mathfrak {n}$ and $\mathfrak {p}_2$ . Then the corresponding holomorphic character $\chi ^\sigma $ extends $\chi ^\lambda $ from $S^{\mathbb {C}}$ to P. By abuse of notation, we will denote this extension by $\chi ^\lambda $ . The following proposition shows that all characters of P arise by extension in this way.

Proof The character $\chi : P\rightarrow \mathbb {C}^*$ induces a Lie algebra homomorphism $\sigma : \mathfrak {p}\rightarrow \mathbb {C}$ . It suffices to prove that $\sigma $ is trivial if restricted to $\mathfrak {n}$ and $\mathfrak {p}_2$ . Notice that $\mathfrak {p}_1$ is a solvable Lie algebra and $\mathfrak {n}=[\mathfrak {p}_1,\mathfrak {p}_1]$ . The restriction of $\sigma $ to $\mathfrak {n}$ must be trivial as $\mathbb {C}$ is an abelian Lie algebra. Since $\mathfrak {p}_2$ is a semisimple Lie algebra, for each root $\alpha \in D^+$ , there exist $X_{\alpha }\in \mathfrak {g}_\alpha $ , $X_{-\alpha }\in \mathfrak {g}_{-\alpha }$ and $H_{\alpha }=[X_\alpha ,X_{-\alpha }]$ such that

$$ \begin{align*} \mathfrak{l}_{\alpha}:=\mathbb{C}\langle H_\alpha,X_{\alpha},X_{-\alpha}\rangle\cong \mathfrak{s}\mathfrak{l}(2,\mathbb{C}). \end{align*} $$

By the representation theory of $\mathfrak {s}\mathfrak {l}(2,\mathbb {C})$ , the only possible one-dimensional representation of $\mathfrak {l}_\alpha $ is trivial. Since the restriction of $\sigma $ to each $\mathfrak {l}_\alpha $ is trivial, we conclude that $\sigma |_{\mathfrak {p}_2}=0$ .

Summarizing, we have now proved that every integral weight $\lambda \in \mathbb {Z}\langle \omega _1,\ldots , \omega _k \rangle $ induces a character, $\chi ^\lambda : P\rightarrow \mathbb {C}^*$ , hence a homogeneous line bundle $L_\lambda $ . Conversely, every holomorphic line bundle L is of the form $L\cong L_\lambda $ for some $\lambda \in \mathbb {Z}\langle \omega _1, \ldots , \omega _k \rangle $ . Recall that an integral weight $\lambda $ on X is called dominant if $\lambda =\sum _{i=1}^kn_i\omega _i$ with $n_i>0$ . By the Highest Weight Theorem, for each dominant integral weight $\lambda $ , there exists a unique finite-dimensional irreducible complex representation $V(\lambda )$ of G with highest weight $\lambda $ . The Bott–Borel–Weil Theorem [Reference Bott9, Section 7] states that the space of global sections of $L_\lambda $ is isomorphic to $V(\lambda )$ as a G-module. Also by the Bott–Borel–Weil Theorem, the global sections of $L_\lambda $ induce an embedding $X\hookrightarrow \mathbb {P}(V(\lambda )^*)$ , so $L_\lambda $ is very ample. In conclusion, we have the following.

2.3 Invariant closed (1,1)-forms on X

In this section, we classify the G-invariant closed $(1,1)$ -forms on X, which is needed for the proof of Theorem B.

Throughout this section, let X be a compact homogeneous Kähler manifold with respect to $(G,R)$ . Fixing a distinguished point $p\in X$ , let $\{X_\alpha , Y_\alpha : \alpha \in D^+\}$ be a normal adapted basis of $T_p X$ as defined in (2.1) and (2.2). Then, we write $\{\eta _\alpha , \xi _\alpha : \alpha \in D^+ \}$ for the dual basis of $T_p^* X$ .

The results that we will discuss here mainly come from [Reference Besse6, Chapter 8]. A slight difference is that we rewrite the results in more explicit way. Recall that tangent vectors at p can be identified with infinitesimal transformation at p by elements in $\mathfrak {g}$ . If $A\in \mathfrak {g}$ , we define $V_A$ as a vector field on X by

$$ \begin{align*} V_A(x)=\frac{d}{dt}\Big|_{t=0}\exp(tA)(x), \qquad \text{ for all } x\in X. \end{align*} $$

We call $V_A$ the fundamental vector field related to A. Then, by quick calculation, the Lie bracket of two fundamental vector fields is also fundamental as

(2.12) $$ \begin{align} [V_A, V_B]=-V_{[A,B]}. \end{align} $$

The reason we introduce fundamental vector fields is to test invariant 2-forms on X. Recall that the negative Killing form defines a G-invariant metric $(\cdot ,\cdot )$ on $\mathfrak {g}$ . Fixing an element $S\in \mathfrak {s}$ , we can define a 2-form, $\omega _S(V_A, V_B) = (S, [A,B])$ . According to the standard calculation (see [Reference Besse6, Proposition 8.66]), $\omega _S$ is a G-invariant, closed, real $(1,1)$ -form. It is observed that the only nonvanishing terms are

(2.13) $$ \begin{align} \omega_S (X_\alpha, Y_\alpha) = (S, [X_\alpha, Y_\alpha]) = (S, -H_\alpha) = \frac{i}{2} \alpha(S). \end{align} $$

The observation (2.13) indicates that we can write down $\omega _S$ explicitly in terms of the covector basis $\{\eta _\alpha ,\xi _\alpha : \alpha \in D^+\}$ at $p\in X$ . Precisely,

(2.14) $$ \begin{align} \omega_S=\sum_{\alpha\in D^+} C_S(\alpha) \eta_\alpha\wedge \xi_\alpha,\qquad C_S(\alpha)=\frac{i}{2}{\alpha(S),} \end{align} $$

where $C_S$ can be viewed as a linear function of $\mathfrak {s}^*$ (generated by elements in $D^+$ ).

Given a G-invariant closed real $(1,1)$ -form, $\omega $ , in the following, we prove that $\omega $ can be written as in (2.14) related to some S. Since $\omega $ is G-invariant, for each $A,B,N\in \mathfrak {g}$ ,

$$ \begin{align*} 0 = (L_{V_N}\omega)(V_A, V_B) = {V_N}(\omega(V_A, V_B)) - \omega([V_N,V_A],V_B) - \omega(V_A, [V_N,V_B]). \end{align*} $$

In particular, by taking N to be an element in $\mathfrak {t}$ and $A,B$ to be elements of $\{X_\alpha , Y_\alpha , \alpha \in D^+\}$ , noting that $V_N(\omega (V_A, V_B)) = 0$ , we have:

• • For two different roots in $D^+$ , $\alpha , \beta $ , $\omega (X_\alpha , X_\beta )=\omega (X_\alpha , Y_\beta )=\omega (Y_\alpha ,Y_\beta )=0.$

• • The only nonvanishing case are $\displaystyle \omega (X_\alpha ,Y_\alpha )=C_\alpha , \alpha \in D^+$ .

It suffices to show that $C_\alpha $ , a function defined in $D^+$ , can extend linearly to $\mathfrak {s}^*$ . Since $2$ -form $\omega $ and inner product in $\mathfrak {g}$ can be extended linearly to complex field, consider the following complex vectors:

(2.15) $$ \begin{align} U=X_\alpha-iY_\alpha, \quad V=X_\beta-iY_\beta,\quad W=X_{\alpha+\beta}+iY_{\alpha+\beta}, \end{align} $$

where $\alpha , \beta , \alpha +\beta \in D^+$ and $X_{\alpha ,\beta ,\alpha +\beta }$ , $Y_{\alpha ,\beta ,\alpha +\beta }$ are taking from the normal adapted basis as in (2.1) and (2.2). Viewing U, V, W as elements in $\mathfrak {g}\otimes \mathbb {C}$ , the G-invariant inner product satisfies

$$ \begin{align*} (W,[U,V])=([W,U],V). \end{align*} $$

Therefore, it is easy to check $U,V,W$ satisfies the following:

$$ \begin{align*} [U,V]=\lambda \overline{W}, \quad [V,W]=\lambda \overline{U}, \quad [W,U]=\lambda \overline{V}. \end{align*} $$

By closedness and G-invariance of $\omega $ , we have

(2.16) $$ \begin{align} d\omega (U, V, W) = &\ \omega([U, V], W)+\omega([V, W], U)+\omega([W, U], V) \nonumber \\ = &- \lambda\big(\omega(\overline{W}, W)+\omega(\overline{U}, U)+\omega(\overline{V}, V)\big)=0. \end{align} $$

Inserting (2.15) into (2.16), we have

$$ \begin{align*} \omega(X_{\alpha+\beta}, Y_{\alpha+\beta}) = \omega(X_\alpha,Y_\alpha) +\omega(X_\beta, Y_\beta), \end{align*} $$

hence, $C_{\alpha +\beta }=C_{\alpha }+C_\beta \ (*)$ . Noticing that $C_{\bullet }$ defines a function on $D^+$ , the condition $(*)$ implies that $C_{\bullet }$ can be extended to a linear function on $\mathfrak {s}^*$ as $D^+$ generates $\mathfrak {s}^*$ . In other words, there exists an element $S\in \mathfrak {s}$ such that

$$ \begin{align*} C_\alpha=\frac{i}{2}\alpha(S). \end{align*} $$

In conclusion, we have the following proposition.

Proposition 2.6 implies a bijection from $\mathfrak {s}$ to the space of G-invariant closed real $(1,1)$ -forms on X, $S\rightarrow \omega _S$ . There is a special element of the G-invariant closed real $(1,1)$ -forms, Kähler–Einstein form, and the related element in $\mathfrak {s}$ is given by ${S_{KE}=2\sum _{\alpha \in D^+} H_{\alpha }.}$ So we can write $\omega _{KE}$ explicitly at the distinguished point $p\in X$ with respect to the normal adapted basis. Indeed, set $\delta $ be the sum of all positive roots in $D^+$ , then

(2.17) $$ \begin{align} \omega_{KE}=\frac{1}{2}\sum_{\alpha \in D^+} (\alpha,\delta)\eta_\alpha \wedge \xi_\alpha. \end{align} $$

3.1 Invariant $1$ -forms on M

Let X be a homogeneous compact Kähler manifold as before, and $L=L_\lambda $ , a homogeneous line bundle over X corresponding an integral weight $\lambda $ . Given the data $(X,L)$ , in this subsection, we will determine all left-invariant 1-forms on the unit circle bundle M of X.

Recall the G-action on L defined by restricting the $G^{\mathbb {C}}$ -action of (2.10) to ${G \subset G^{\mathbb {C}}}$ . For each homogeneous line bundle L, there is a natural G-invariant Hermitian metric h induced by the standard Hermitian metric in $\mathbb {C}$ . In particular, according to construction of homogeneous line bundle in (2.9), let $q_0= \overline {(g,z)} \in L_\lambda $ for $(g,z)\in G^{\mathbb {C}} \times \mathbb {C}$ , then $h(q_0,q_0)=|z|^2$ . The Hermitian metric h induces a radian function r on L. Then, G acts transitively on each level set of r, an $S^1$ bundle of X, denoted as $M(r)$ . Away from zero level set, there is a canonical invariant vector field, $\partial /\partial r$ on L pointing in radius direction. We shall find the set of all invariant vector fields on each level set $M(r)$ . Notice that, in general, left-invariant vector fields on G are not well-defined over $M(r)$ , as the left action by the stabilizer group on the tangent space at one point can be nontrivial. Let $M=M(1)$ and let $\mathcal {T}_M$ be the space of all global G-invariant vector fields over M. $\mathcal {T}_M$ contains at least one element, $X_0=J(\partial /\partial r)$ , generating a circle action on each fiber. The other elements of $\mathcal {T}_M$ strongly depend on the base manifold X and the integral weight $\lambda $ . According to (2.7), the tangent space at a distinguished point $p\in X$ can be identified with a subspace of $\mathfrak {g}$ . Then, we can choose a normal adapted basis $\{X_\alpha ,Y_\alpha \}_{\alpha \in D^+}$ as in (2.1) and (2.2).

Based on the choice of $\{H_\alpha , X_\alpha , Y_\alpha \}$ as in (2.1) and (2.2), one can easily check the following Lie algebra structure:

$$ \begin{align*} [H_\alpha, X_\alpha]=-\frac{|\alpha|^2}{2} Y_\alpha, \quad [H_\alpha, Y_\alpha]= \frac{|\alpha|^2}{2} X_\alpha, \quad [X_\alpha, Y_\alpha]= - H_\alpha. \end{align*} $$

Let q be a distinguished point in M with $\pi (q) =p$ , then the tangent space at q can be identified with $\mathbb {R}\langle X_\alpha , Y_\alpha , \alpha \in D^+ \rangle \oplus \mathbb {R} X_0$ in the following sense. Consider the G-equivariant bundle projections,

with $\tilde {\pi }(e)=q\in M$ . At the distinguished point $q\in M$ with $\pi (q)=p\in X$ , assume that the stabilizer group at $p\in X$ is R and the stabilizer group at $q\in M$ is $R_0$ . By G-action on L, $g\in R_0$ if and only if,

$$ \begin{align*}g(q)= g(e, \theta)= (e, \chi^\lambda(g) \theta),\end{align*} $$

which implies that $R_0= \ker \chi ^\lambda : R\rightarrow S^1 \subset \mathbb {C}^*$ These projections induce the mapping on tangent spaces $\tilde {\pi }_*:\mathfrak {g}\rightarrow T_qM$ by

(3.1) $$ \begin{align} N\in \mathfrak{g} \mapsto \frac{d}{dt}\Big|_{t=0}\tilde{\pi}\circ\exp(tN). \end{align} $$

Similarly, we can define the mapping $\pi _*:\mathfrak {g}\rightarrow T_p X$ . By abusing notation, we write $X_\alpha , Y_\alpha $ instead of $\tilde {\pi }_*(X_\alpha )$ , $\tilde {\pi }_*(Y_\alpha )$ and $\pi _*(X_\alpha ),\ \pi _*(Y_\alpha )$ as $X_\alpha $ , $Y_\alpha $ . Notice that the left-invariant vector fields on G can be identified with $\mathfrak {g}$ . To determine the space of invariant vector fields, $\mathcal {T}_M$ , on $M_1$ , we observe that for any $R_0$ -invariant vector $v\in T_q M$ and $g_1(q)=g_2(q)$ , then $g_1=g_2 r$ with $r\in R_0$ and

$$ \begin{align*} (g_1)_*(v)=(g_2 r)_*(v)=(g_2)_* r_*(v)=(g_2)_*(v). \end{align*} $$

So each $R_0$ -invariant vector of $T_q M$ determines a left-invariant vector fields on M. There is an one-to-one correspondence between $\mathcal {T}_M$ and the $R_0$ -invariant space of $T_q M$ . In particular, in the case that $X=G/T$ with T a maximal torus of G, we have the following proposition.

Proof of Proposition 3.1

Let $R_0$ be the stabilizer group at $q\in M$ . Notice that $R_0$ is the kernel of character $\chi ^\lambda : R\rightarrow S^1\subset \mathbb {C}$ related to the weight $\lambda $ . Let $ V\in T_q M$ be an invariant vector. According to decomposition (2.7) of $T_q M$ , the vector V can be written as

$$ \begin{align*} V=\sum_{\alpha\in D^+}V_\alpha, \qquad V_\alpha\in \mathbb{R}\langle X_\alpha, Y_\alpha \rangle \text{ and } V_\alpha\ne 0. \end{align*} $$

Notice that $R_0$ preserves each $\mathbb {R}\langle X_\alpha , Y_\alpha \rangle $ . If a vector $V_\alpha = aX_\alpha +bY_\alpha \in E_{\pm \alpha }$ is invariant under $R_0$ action, then, since J is $R_0$ invariant, which means that $J(aX_\alpha +bY_\alpha )=aY_\alpha -bX_\alpha $ is also $R_0$ invariant. Therefore, all vectors in the space generated by $\langle aX_\alpha +bY_\alpha ,-b X_{\alpha }+a Y_\alpha \rangle =\langle X_\alpha , Y_\alpha \rangle = \mathbb {R}\langle X_\alpha , Y_\alpha \rangle $ are $R_0$ invariant. So we only need to determine the set of all $\alpha \in \Delta ^+$ such that $X_\alpha $ , $Y_\alpha $ are invariant under $ R_0 = \ker \chi ^\lambda $ .

Let $\mathfrak {r}_0=\ker \lambda $ , it’s easy to see that $\mathfrak {r}_0$ is the Lie algebra of the stabilizer group $R_0$ . Let $T_0= R_0 \cap T$ associated with Lie algebra $\mathfrak {t}_{0}=\mathfrak {r}_0\cap \mathfrak {t}$ . Since $X_\alpha $ , $Y_\alpha $ is $T_0$ invariant,

(3.2) $$ \begin{align} [H_0,X_\alpha]=0=[H_0,Y_\alpha],\qquad \text{for all } H_0\in \mathfrak{t}_0. \end{align} $$

Noting that

$$ \begin{align*}[H_0, X_\alpha]= -i\alpha(H_0) Y_\alpha, \qquad [H_0, Y_\alpha] = i \alpha(H_0) X_\alpha.\end{align*} $$

Hence, we have $\ker \alpha = \mathfrak {t}^0 = \ker \lambda |_{\mathfrak {t}}$ , hence $\lambda $ is proportional to $\alpha $ .

Let $Z \in \mathfrak {r}_0$ , then $X_\alpha ,Y_\alpha $ are $R_0$ invariant if and only if

$$ \begin{align*} \tilde{\pi}_*[Z, X_\alpha]=\tilde{\pi}_*[Z,Y_\alpha]=0. \end{align*} $$

Notice that $\mathfrak {r}_0 = \mathfrak {t}_0 \oplus \sum _{\beta \in \Delta ^+(\mathfrak {t},\mathfrak {r})}\mathbb {R}\langle X_\beta , Y_\beta \rangle $ , where $\mathfrak {t}_0$ is kernel of $\lambda $ restricted in $\mathfrak {t}$ . Then, $X_\alpha $ and $Y_\alpha $ is $R_0$ -invariant is equivalent to (3.2) and

$$ \begin{align*} \tilde{\pi}_*[Z_\beta,X_\alpha]=\tilde{\pi}_*[Z_\beta,Y_\alpha]=0,&\qquad \text{for all } Z_\beta\in \mathbb{R}\langle X_\beta, Y_\beta \rangle,\ \beta\in \Delta^+(\mathfrak{t},\mathfrak{r}). \qquad (**) \end{align*} $$

According to [Reference Knapp24, Theorem 6.6] and the definition of $\tilde {\pi }_*$ , $(**)$ is equivalent to $\alpha + \beta ,\ \alpha -\beta \notin D^+ \cup (-D^+)$ . Indeed, $\alpha +\beta $ and $\alpha -\beta $ are not roots. Assume that

$$ \begin{align*}\alpha +\beta= \gamma \in \Delta(\mathfrak{t},\mathfrak{r}),\end{align*} $$

then, $\alpha =\gamma - \beta \in \Delta (\mathfrak {t},\mathfrak {r})$ . But $\alpha \in D^+$ , which leads to a contradiction.

Example 3.2 Let $X=SU(3)/T^{2} $ . Recall the basic notions of semisimple Lie group $SU(3)$ . The Lie algebra $\mathfrak {su}(3)$ is the set of trace zero skew-Hermitian matrices of order $3$ . A Cartan sub-algebra $\mathfrak {t}$ is the Lie algebra of diagonal matrices in $\mathfrak {su}(3)$ . The set of positive roots with respect to $\mathfrak {t}$ consists of three elements $\{\alpha ,\beta ,\gamma \}$ , and the normal adapted basis is given by $X_{\alpha ,\beta ,\gamma },\ Y_{\alpha ,\beta ,\gamma }$ .

Consider a distinguished point $q\in M$ with tangent space generated by $\{X_0,\ X_{\alpha ,\beta ,\gamma },\ Y_{\alpha ,\beta ,\gamma }\}$ . Now, we only focus on the subspace $V\subset T_q M$ generated by $\{X_{\alpha ,\beta , \gamma }, Y_{\alpha , \beta , \gamma }\}$ . To simplify the notation in calculation, we introduce a complex coordinate system in V; precisely, $z_{\alpha }=X_\alpha +i Y_{\alpha }$ . Let $\sigma _i$ denote the ith element of diagonal matrices. Then, $\alpha ,\beta ,\gamma $ can be expressed as, $ \alpha =\sigma _1- \sigma _2 ,\quad \beta =\sigma _1- \sigma _3,\quad \gamma =\sigma _2-\sigma _3. $ Hence, let $\{\alpha ,\gamma \}$ be the simple root system of $SU(3)$ and $\beta =\alpha +\gamma $ . The fundamental weights are given as follows:

$$ \begin{align*} \omega_1=\frac{2}{3}\sigma_1-\frac{1}{3} \sigma_2- \frac{1}{3}\sigma_3,\qquad \omega_2=\frac{1}{3}\sigma_1+\frac{1}{3} \sigma_2 -\frac{2}{3}\sigma_3. \end{align*} $$

Let $\mathcal {O}(p,q)$ denote the line bundle corresponding to the integral weight $\lambda =p\omega _1+q\omega _2$ . In the case of $(p,q)\ne 0$ . Noting that kernel of $\lambda = p\omega _1+ q\omega _2$ is $\mathbb {R}\langle q t_1 - (p+q) t_2\rangle $ . Then, the $S^1$ action is given by $T_\theta ^{p,q}=\operatorname {diag}(e^{iq \theta }, e^{-i(p+q)\theta },e^{ip\theta })$ and if we represent the action on complex coordinate system $(z_\alpha , z_\beta , z_\gamma )$ , we have

$$ \begin{align*} T^{p,q}_\theta(z_\alpha,z_\beta,z_\gamma)=(e^{-i(p+2q)\theta}z_\alpha, e^{i(p-q)\theta} z_\beta, e^{i(2p+q)\theta}z_\gamma ). \end{align*} $$

In conclusion, we have the following cases:

Notice that

$$ \begin{align*} \alpha= 2\omega_1-\omega_2, \quad \beta= \omega_1+\omega_2,\quad \gamma= -\omega_1+2\omega_2. \end{align*} $$

The integral weights in the above table, $-2n\omega _1+ n\omega _2$ , $n\omega _1 + n\omega _2$ , $-n\omega _1 +2n \omega _2$ are proportional to $\alpha ,\beta , \gamma $ , respectively, in agreement with Proposition 3.1.

The left-invariant $1$ -forms on M can be viewed as the dual space of left-invariant vector fields. More precisely, if we apply the previous notion of adapted basis at $T_q M$ , given by $\{X_0, X_\alpha , Y_\alpha; \alpha \in D^+\}$ , we write the dual basis of $T^*_qM$ as $\{\eta _0,\eta _\alpha , \xi _\alpha ;\alpha \in D^+\}.$ According to Proposition 3.1, the space of left-invariant vector fields is in one-to-one correspondence with the space generated by $\{X_0\}$ or $\{X_0, X_\alpha ,Y_\alpha \}$ for some $\alpha \in D^+$ . Consider the subset, $\{\eta _0\}$ or $\{\eta _0,\eta _\alpha ,\xi _\alpha \}$ of the dual basis, whose elements are $R_0$ invariant. Therefore, $\{\eta _0\}$ or $\{\eta _0,\eta _\alpha ,\xi _\alpha \}$ generates the space of G-invariant $1$ -forms over M.

3.2 The differentials of left-invariant 1-forms on M

According to Proposition 3.1, the left-invariant 1-forms are generated by $\{\eta _0 \}$ or $\{\eta _0, \xi _\alpha , \eta _\alpha \}$ . We only consider the second case in this subsection.

Let M be an $S^1$ bundle associated with line bundle L. Notice that there is a natural projection $\tilde {\pi }: G\rightarrow M$ . If we write $\Omega _1$ , $\Omega _2$ as the space of smooth $1$ -forms and $2$ -forms, respectively, then we have the following commutative graph:

(3.3)

Since the left-invariant vector fields are globally generated in G, with a natural basis corresponding to $\{H_\alpha ,X_\beta ,Y_\beta : \alpha \in \Pi ,\beta \in \Delta \}$ and its dual basis $\{h_\alpha ,\eta _\beta ,\xi _\beta :\alpha \in \Pi ,\beta \in \Delta \}$ , then the pull back of $\eta _\alpha $ , $\xi _\alpha $ in $\Omega _1(M)$ under $\tilde {\pi }$ are exactly $\eta _\alpha $ , $\xi _\alpha $ in $\Omega _1(G)$ . And the pull-back of $\eta _0$ is a certain combination of $h_\alpha $ determined by weight $\lambda $ . To calculate the differentials of left-invariant 1-forms, the technique is to apply the Maurer–Cartan equations on G with respect to the natural basis, Since Maurer–Cartan equations gives us the derivative of left-invariant $1$ -form on G, combining the commutative graph (3.3), we have the derivative of left-invariant $1$ -form on M.

Proposition 3.3 Let X be the homogeneous space and M, the $S^1$ bundle of X associated with $\lambda $ . Assuming that, at the distinguished point $q\in M$ , the space of left-invariant $1$ -forms can be identified with the spaces generated by $\{\eta _0,\eta _\alpha ,\xi _\alpha \}$ , $\alpha \in D^+$ . In this case, $\lambda $ is proportional to $\alpha $ , assuming that $\alpha =-l\lambda $ , then the derivative are given by

(3.4) $$ \begin{align} \ \ d\eta_0 &= - \frac{1}{2}\sum_{\alpha\in D^+} (\lambda,\alpha) \eta_\alpha\wedge \xi_\alpha,\qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

(3.5) $$ \begin{align} \qquad\qquad d\eta_\alpha &= -l \eta_0 \wedge \xi_\alpha-\frac{C^\alpha_{\beta,-\gamma}}{2}\sum_{\substack{\beta,\gamma\in D^+\\ \beta-\gamma=\alpha}} ( \eta_\beta\wedge \eta_\gamma+\xi_\beta\wedge \xi_\gamma )\nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad -\frac{C^\alpha_{\beta,\gamma}}{2}\sum_{\substack{\beta,\gamma\in D^+\\ \beta+\gamma=\alpha}}(\eta_\beta\wedge\eta_\gamma-\xi_\beta\wedge\xi_{\gamma}), \end{align} $$

(3.6) $$ \begin{align} \qquad\qquad d\xi_\alpha &= l \eta_0 \wedge \eta_\alpha + \frac{C^\alpha_{\beta,-\gamma}}{2}\sum_{\substack{\beta,\gamma\in D^+ \\ \beta-\gamma=\alpha}}(\eta_\beta\wedge \xi_\gamma-\xi_\beta \wedge \eta_\gamma )\nonumber \\ & \qquad\qquad\qquad\qquad\qquad\qquad -\frac{C^\alpha_{\beta,\gamma}}{2}\sum_{\substack{\beta,\gamma\in D^+\\ \beta+\gamma=\alpha}}(\eta_\beta\wedge\xi_\gamma+\xi_\beta\wedge\eta_{\gamma}), \end{align} $$

where the coefficients $C^\alpha _{\beta ,-\gamma }$ are the coefficients from Maurer–Cartan equation. Let $E_\alpha $ , $E_\beta $ , $E_{-\gamma }$ be the root vector of $\alpha $ , $\beta $ , $-\gamma $ satisfying (2.1), then

$$ \begin{align*} [E_\beta,E_{-\gamma}]=C^\alpha_{\beta,-\gamma} E_{\alpha.} \end{align*} $$

Proof Let $\{\omega _1, \dots , \omega _k\}$ be fundamental integral weights on X and $\lambda =\sum _{i}n_i\omega _i$ . By the definition of fundamental weights and our setting of $H_{\alpha _i}$ , we can evaluate $H_{\alpha _i}$ under $\lambda $ :

$$ \begin{align*} \lambda(H_{\alpha_i})=\sum_{j}n_j\omega_j(H_{\alpha_i}) =\frac{1}{2i}\sum_jn_j(\omega_j,\alpha_i) =-i\frac{|\alpha_i|^2}{4}&\sum_j\frac{2n_j(\omega_j,\alpha_i)}{(\alpha_i,\alpha_i)} =-i\frac{|\alpha_i|^2}{4} n_i. \end{align*} $$

By the definition of $L_\lambda $ , at distinguished point $q\in M$ with $\tilde {\pi }(e)=q$ , we have

(3.7) $$ \begin{align} \tilde{\pi}_*(H_{\alpha_i}) =\frac{d}{dt}\Big|_{t=0}\chi^\lambda(\exp (t H_{\alpha_{i}})) =\frac{d}{dt}\Big|_{t=0}\exp (t \lambda(H_{\alpha_{i}}) )=-\frac{|\alpha_i|^2}{4}n_i X_0. \end{align} $$

For each i,

$$ \begin{align*} \tilde{\pi}^*(\eta_0)(H_{\alpha_i})=\eta_0(\tilde{\pi}_*(H_{\alpha_i}))=-\frac{|\alpha_i|^2}{4}n_i \eta_0(X_0)=-\frac{|\alpha_i|^2}{4}n_i. \end{align*} $$

Since the previous calculation shows that $\omega _i(H_{\alpha _j})=-i|\alpha _j|^2\delta _{ij}/4$ , then the pullback of $\eta _0$ can be represented by $\tilde {\pi }^*\eta _0=-i\lambda $ . To get the formula (3.4), notice that

$$ \begin{align*} \tilde{\pi}^*d\eta_0(X_\alpha,Y_\alpha)= d\tilde{\pi}^*(\eta_0)(X_\alpha,Y_\alpha) =\tilde{\pi}^*\eta_0([X_\alpha,Y_\alpha]) =- i \lambda({H_\alpha}) =- \frac{1}{2} (\lambda,\alpha). \end{align*} $$

To get the formulas (3.5) and (3.6), if we pull back both sides of formulas by $\tilde {\pi }$ ,

$$ \begin{align*} \tilde{\pi}^*(d\eta_\alpha)(H_{\alpha_i},Y_\alpha) &=-\tilde{\pi}^*\eta_\alpha([H_{\alpha_i},Y_\alpha]) =-\frac{(\alpha,\alpha_i)}{2},\\ \frac{2(\alpha,\alpha_i)}{n_i|\alpha_i|^2}\tilde{\pi}^*(\eta_0\wedge\xi_\alpha)(H_{\alpha_i},Y_\alpha) &={-i}\frac{2(\alpha, \alpha_i)}{n_i|\alpha_i|^2}\sum_i \lambda\wedge \xi_{\alpha}(H_{\alpha_i},Y_\alpha) =-\frac{(\alpha_i, \alpha)}{2}. \end{align*} $$

Since $\tilde {\pi }^*$ is injective, $d\eta _\alpha $ admits the term $\displaystyle \frac {2(\alpha ,\alpha _i)}{n_i|\alpha _i|^2} \eta _0\wedge \xi _\alpha $ . We can show that the coefficient is independent of the index i and related to the factor l. Notice that $\lambda $ is proportional to $\alpha $ , $\alpha =-l\lambda $ , then

$$ \begin{align*} \frac{2(\alpha,\alpha_i)}{n_i |\alpha_i|^2} =\frac{2(-l\lambda,\alpha_i)}{n_i|\alpha_i|^2} =\frac{2(-l\sum_{j}n_j\omega_j,\alpha_i)}{n_i|\alpha_i|^2}=-l. \end{align*} $$

To compute the remaining cross terms of $d\eta _\alpha $ and $d\xi _\alpha $ , we shall understand the structure of Lie algebra. Notice that $d\eta _\alpha $ has a nonvanishing 2-form related with root $\beta ,\ \gamma $ only if (1) $\alpha =\beta +\gamma $ or (2) $\alpha =\beta -\gamma $ . Both in the case (1) and (2), we argue that $\beta ,\gamma \in D^+$ . For instance, in the case (1), assume that $\gamma \in \Delta (\mathfrak {t},\mathfrak {r})^+$ , then we have $\beta = \alpha -\gamma $ is a root, which contradicts the condition in Proposition 3.1.

In the case (1), notice that $\displaystyle [E_\beta , E\gamma ]= C^\alpha _{\beta , \gamma } E_\alpha $ and $\displaystyle [E_\beta , \overline {E_\gamma }]$ has no $E_\alpha $ terms. Combining with the relation (2.1), we have

$$ \begin{align*} \eta_\alpha([X_{\beta},X_{\gamma} ])&=-\eta_\alpha([Y_\beta,Y_\gamma])=\frac{C^\alpha_{\beta,\gamma}}{2},\\ \xi_{\alpha}([X_\beta, Y_\gamma])&=\xi_{\alpha}([Y_\beta, X_\gamma])=\frac{C^\alpha_{\beta,\gamma}}{2}. \end{align*} $$

The above equation implies that

$$ \begin{align*} \tilde{\pi}^*(d\eta_\alpha)(X_\beta, X_\gamma)&= - \pi^*\eta_\alpha([X_\beta,X_\gamma])= -\frac{C^\alpha_{\beta,\gamma}}{2},\\\tilde{\pi}^*(d\eta_\alpha)(Y_\beta, Y_\gamma)&= -\pi^*\eta_\alpha([Y_\beta,Y_\gamma])= \frac{C^\alpha_{\beta,\gamma}}{2}. \end{align*} $$

Hence, $d\eta _\alpha $ has the term $(C^\alpha _{\beta ,\gamma }/2)(-\eta _\beta \wedge \eta _\gamma + \xi _\beta \wedge \xi _\gamma )$ . Likewise, we can find the formulas for $d\eta _\alpha $ and $d\xi _\alpha $ as (3.5) and (3.6) and we completes the proof.

Fixing a line bundle related to a weight $\lambda $ , the Chern class of $L_\lambda $ can be represented by the form $-i \partial \overline {\partial } \log r^2$ , where r is the radius function induced by some Hermitian metric on $L_\lambda $ . Then we have

(3.8) $$ \begin{align} \frac{1}{2}dd^c\log r^2 = -d\big(J\frac{dr}{r}\big). \end{align} $$

Indeed, the 1-form $ \displaystyle -Jdr/r$ is induced by circle action along each fiber and more details will be discussed in the next section. Referring to Proposition 3.3, the curvature form can be represented as

(3.9) $$ \begin{align} -\frac{1}{2}dd^c \log h &= \frac{1}{2}\sum_{\alpha\in\Delta^+} (\lambda,\alpha) \eta_\alpha\wedge \xi_\alpha. \end{align} $$

Comparing with (2.17), the Kähler–Einstein metric is the curvature form associated with the anti-canonical bundle, $L_{\delta }$ .

3.3 Invariant $dd^c$ -lemma on $L_\lambda $

Consider the data $(X,L_\lambda )$ as in the previous section. According to Proposition 3.1, the space of left-invariant vector fields on M have two different cases (a) and (b). Suppose that $\lambda $ satisfies one of the following conditions:

• • The space of left-invariant vector fields on M satisfies case (a) in Proposition 3.1.

• • The space of left-invariant vector fields on M satisfies case (b) in Proposition 3.1. And $\lambda $ is proportional to some positive root $\alpha $ with $\lambda = -l \alpha $ , $l>0$ .

Then, we have the following invariant $dd^c$ lemma.

Proposition 3.4 Let $(X,L_\lambda )$ satisfy the conditions above. If $\omega $ is an G-invariant closed real $(1,1)$ -form on L, $[\omega ]=0$ , then there exists a G-invariant Kähler potential $\Phi \in \mathcal {C}^\infty (L_\lambda )$ such that

(3.10) $$ \begin{align} \omega=dd^c \Phi. \end{align} $$

Proof Since $\omega $ is exact, there exists an 1-form $\theta $ such that $d\theta =\omega $ . Moreover, $\theta $ can choose to be G invariant. Notice that $\omega = d (g^* \theta )$ , then, by taking integral over G, we obtain a G-invariant 1-form $\theta $ with $\omega =d \theta $ .

The main idea of proof is to represent $d\theta $ and $\partial \overline {\partial }\phi $ with respect to G-invariant coframe, then we can reduce the proof of Proposition 3.4 to solving a system of ODE.

Let $L_\lambda $ be the line bundle such that the space of left-invariant vector fields on M satisfies case (b). Suppose that $\lambda $ is proportional to $\alpha $ . The basis of left-invariant $1$ -form on level set M is $\{\eta _0,\eta _\alpha ,\xi _\alpha \}$ . Then, we can extend these invariant $1$ -forms in radian direction by rescaling $1/r$ on each level $M(r)$ . Precisely, there is a natural projection $p: L^\times \rightarrow M$ . By identifying $L^\times \cong G\times _{\lambda }\mathbb {C}^\times $ , the projection can be written explicitly,

$$ \begin{align*}p({(g, z)}) = {(g, |z|^{-1}z)}.\end{align*} $$

Thus, $p^*$ extends $\{\eta _0, \eta _\alpha , \xi _\alpha \}$ to $L^\times $ . By abusing the notion, we write $\{\eta _0, \eta _\alpha , \xi _\alpha \}$ as the extended vector fields over $L^{\times }$ . Also let $\mu = dr/r$ , hence $\{\mu ,\eta _0,\eta _\alpha ,\xi _\alpha \}$ forms a basis of invariant $1$ -forms over $L^\times $ . Then the invariant $1$ -form $\theta $ can be represented as

(3.11) $$ \begin{align} \theta=\varphi_r(r)\mu+\varphi_0 (r) \eta_0 + \varphi_\alpha(r) \eta_\alpha+ \phi_\alpha(r) \xi_\alpha. \end{align} $$

Applying Proposition 3.3, we can take derivative of $\theta $ in (3.11),

(3.12) $$ \begin{align} d\theta =& \ r\varphi_0'(r)\mu\wedge\eta_0 -\frac{\varphi_0(r)}{2}\sum_{\alpha \in D^+} (\lambda,\alpha)\eta_\alpha\wedge\xi_\alpha \end{align} $$

(3.13) $$ \begin{align} & +r\varphi_\alpha'(r)\mu\wedge \eta_\alpha+ r\phi_\alpha'(r) \mu\wedge \xi_\alpha -{l}\varphi_\alpha \eta_0\wedge\xi_\alpha +{l}\phi_\alpha \eta_0\wedge\eta_\alpha \end{align} $$

(3.14) $$ \begin{align} & -\varphi_\alpha \frac{C^\alpha_{\beta,-\gamma}}{2}\sum(\eta_\beta\wedge\eta_\gamma+\xi_\beta\wedge \xi_\gamma)+\phi_\alpha\frac{C^\alpha_{\beta,-\gamma}}{2} \sum(\eta_\beta\wedge \xi_\gamma-\xi_\beta\wedge \eta_\gamma) \end{align} $$

(3.15) $$ \begin{align} & -\varphi_\alpha \frac{C^\alpha_{\beta,\gamma}}{2}\sum(\eta_\beta\wedge\eta_\gamma-\xi_\beta\wedge \xi_\gamma) -\phi_\alpha\frac{C^\alpha_{\beta,\gamma}}{2} \sum(\eta_\beta\wedge \xi_\gamma+\xi_\beta\wedge \eta_\gamma). \end{align} $$

Let J be complex structure. Note that $d\theta $ is real $(1,1)$ -form if and only if $Jd\theta =d\theta $ . Notice that $J\eta _0=\mu $ , $J\xi _\alpha =\eta _\alpha $ , then

$$ \begin{align*} Jd\theta=& \ r \varphi^{\prime}_0(r)\mu\wedge \eta_0 - \frac{\varphi_0(r)}{2} \sum_{\alpha\in D^+}(\lambda,\alpha)\eta_\alpha\wedge\xi_\alpha \\ &+ r\varphi^{\prime}_\alpha\eta_0\wedge \xi_\alpha- r\phi^{\prime}_\alpha \eta_0\wedge \eta_\alpha -{l}\varphi_\alpha\mu\wedge \eta_\alpha-{l}\phi_\alpha\mu\wedge\xi_\alpha\\ & -\varphi_\alpha \frac{C^\alpha_{\beta,-\gamma}}{2}\sum(\eta_\beta\wedge\eta_\gamma+\xi_\beta\wedge \xi_\gamma)+\phi_\alpha\frac{C^\alpha_{\beta,-\gamma}}{2} \sum(\eta_\beta\wedge \xi_\gamma-\xi_\beta\wedge \eta_\gamma) \\ & +\varphi_\alpha \frac{C^\alpha_{\beta,\gamma}}{2}\sum(\eta_\beta\wedge\eta_\gamma-\xi_\beta\wedge \xi_\gamma) +\phi_\alpha\frac{C^\alpha_{\beta,\gamma}}{2} \sum(\eta_\beta\wedge \xi_\gamma+\xi_\beta\wedge \eta_\gamma). \end{align*} $$

Then, $Jd\theta =d\theta $ implies the following ODE:

(3.16) $$ \begin{align} r\varphi_\alpha'=-l\varphi_\alpha,\qquad r\phi^{\prime}_\alpha=-l\phi_\alpha \end{align} $$

and the terms of line (3.15) are vanishing. To ensure, the (3.15) vanishes

(3.17) $$ \begin{align} C^\alpha_{\beta, \gamma}=0 \qquad \text{ or }\qquad \varphi_\alpha=\phi_\alpha=0. \end{align} $$

Indeed, referring to (see [Reference Knapp24], Theorem 6.6), if $\alpha =\beta +\gamma $ , for some $\alpha ,\beta ,\gamma \in \Delta $ , then the corresponding constant $C^\alpha _{\beta ,\gamma }\ne 0$ . Assuming that there exist positive roots, $\beta $ and $\gamma $ , satisfying $\alpha =\beta +\gamma $ , by (3.17), we have $\varphi _\alpha =\phi _\varphi =0$ , which automatically satisfies (3.16); hence, if $d\theta $ is a real $(1,1)$ -form with some structure constants $C^\alpha _{\beta ,\gamma }$ nonvanishing, then $d\theta $ can be written as

$$ \begin{align*} d \theta= r\varphi_0'(r)\mu\wedge \eta_0 -\frac{\varphi_0(r)}{2}\sum_{\alpha\in D^+} (\lambda,\alpha)\eta_\alpha \wedge \xi_\alpha. \end{align*} $$

When it comes to the cases that the corresponding weight $\lambda $ is proportional to a simple root $\alpha $ , i.e., $\alpha $ cannot be written as the sum of two positive roots, $d\theta $ should satisfy the equation (3.16) and its solution is given by $\varphi =C/r^l$ with C an arbitrary constant. In the sequel, it suffices to show that the constant C in the expression of solution $\varphi $ should equal zero. To prove this, we need to apply the condition that the form, $d\theta $ , is well-defined across the zero level of line bundle. Firstly, we take a reference metric $\omega _0$ near the zero level of line bundle L. Let h be the canonical invariant Hermitian metric defined as before and r be the radial function related to h. Also, given a bundle coordinate u, we have

$$ \begin{align*} \omega_\epsilon&= \pi^*\omega_{KE}+\epsilon\cdot dd^c r^{{2}}\\&= \pi^*\omega_{KE} +{\epsilon}r^{{2}}\cdot dd^c \log h + {\epsilon}h\cdot i du\wedge d\overline{u}. \end{align*} $$

It is easy to see that $\omega _\epsilon $ is positive around zero level. To simplify calculation, let $\epsilon $ tends to $0$ , then we obtain a semi-positive form $\omega _0=\pi ^* \omega _{KE}$ . Consider the following integration:

(3.18) $$ \begin{align} \int_K \omega_0^{n-1}\wedge d\theta\wedge d\theta, \end{align} $$

where the K is a compact neighborhood of zero level, which is defined as $K=\{x\in L, r(x) \leq \delta \}$ . In the sequel, we write 3.12, 3.13, and 3.14 as $\Theta _1$ , $\Theta _2$ , $\Theta _3$ . Noticing that all crossing terms $\Theta _i\wedge \Theta _j\wedge \omega _0^{n-1}=0$ , ( $i\ne j$ ), and $\Theta _3\wedge \Theta _3\wedge \omega _0^{n-1}=0$ . Only two nonvanishing terms of the integration (3.18) are the following:

(3.19) $$ \begin{align} \int_K\omega_0^{n-1}\wedge d\theta\wedge d\theta =\int_K \omega_0^{n-1} \wedge\Theta_1\wedge \Theta_1 +\int_K \omega_0^{n-1} \wedge \Theta_2\wedge \Theta_2. \end{align} $$

Inserting the solution of ODE (3.16),

$$ \begin{align*} \Theta_2= -C_1 l r^{-l} \mu \wedge \eta_\alpha -C_2 l r^{-l} \mu \wedge \xi_\alpha -C_1 l r^{-l} \eta_0 \wedge \xi_\alpha +C_2 l r^{-l} \eta_0 \wedge \eta_\alpha, \end{align*} $$

then we can compute the two terms in (3.19) separately,

$$ \begin{align*} \int_K \Theta_2 \wedge \Theta_2 \wedge \omega_0^{n-1} & = -\int_K (C_1^2 +C_2^2) l^2 r^{-2l} \mu \wedge \eta_0 \wedge \eta_\alpha \wedge \xi_\alpha \wedge \omega_0^{n-1}\\ & = -(C_1^2 +C_2^2 ) l^2 \int_0^\delta r^{-2l-1} dr \int_{M_1}\eta_0 \wedge \eta_\alpha \wedge \xi_\alpha \wedge \omega_0^{n-1} = -\infty \end{align*} $$

and assume $\lambda $ is proportional to a positive root by a negative constant,

$$ \begin{align*} \int_K \Theta_1 \wedge \Theta_1 \wedge \omega_0^{n-1} &=-\frac{1}{2}\sum_{\alpha\in \Delta^+} \int_K r\varphi_0'(r) \varphi_0(r) (\lambda,\alpha) \eta_\alpha \wedge \xi_\alpha \wedge \mu \wedge \eta_0 \wedge \omega_0^{n-1} \\ &=-\frac{1}{2}\sum_{\alpha\in \Delta^+}(\lambda, \alpha)\int^{\delta}_0 \varphi_0'(r) \varphi_0(r) dr \int_{M_1} \eta_0 \wedge \eta_\alpha \wedge \xi_\alpha \wedge \omega_0^{n-1}\\ &=C\lim_{\epsilon\rightarrow 0} (\varphi_0(r))^2\Big|_{\epsilon}^\delta< C', \quad (C, C' \text{ are nonnegative constants}). \end{align*} $$

Hence, $\displaystyle \int _K d\theta \wedge d\theta \wedge \omega _0^{n-1}=-\infty $ , which leads to a contradiction. We obtain that $\varphi _\alpha =\phi _\alpha =0$ . According to the discussion of two cases, we have $\varphi _\alpha =\phi _{\alpha }=0$ and $d\theta $ can be represented as

(3.20) $$ \begin{align} d\theta= r\varphi_0'(r)\mu\wedge \eta_0 - \frac{\varphi_0(r)}{2}\sum_{\alpha\in \Delta^+}(\lambda, \alpha)\eta_\alpha \wedge \xi_\alpha. \end{align} $$

Take a function $\Phi (r)$ such that $\Phi '(r)=\varphi _0(r)/r$ , we compute $i\partial \overline {\partial } \Phi $ ,

$$ \begin{align*} dd^c \Phi =-d\cdot Jd \Phi =-d\cdot J(\varphi_0(r)\mu) =d(\varphi_0(r)\eta_0) = r \varphi^{\prime}_0(r) \mu \wedge \eta_0+\varphi_0 (r) d\eta_0. \end{align*} $$

Combining with Proposition 3.3, we obtain

(3.21) $$ \begin{align} dd^c \Phi = r \varphi_0'(r)\mu \wedge \eta_0 - \frac{\varphi_0 (r)}{2}\sum_{\alpha \in D^+}(\lambda, \alpha)\eta_\alpha \wedge \xi_\alpha. \end{align} $$

Now, we find $\Phi \in C^\infty (L^\times )$ such that $\omega = dd^c \Phi $ . It suffices to prove that $\Phi $ can extend smoothly across the zero level of L. To prove this, we need to apply a basic fact from complex functions: Let $f: [0,\infty ) \rightarrow \mathbb {R}$ be a smooth function, then, $g(z)=f(|z|)$ is smooth in $\mathbb {C}$ if and only if there exits a smooth function $h: [0,\infty ) \rightarrow \mathbb {R}$ such that $f(r)=h(r^2)$ . Since $\omega $ is defined in the whole line bundle L, on each fiber, the terms of $\omega $ ,

(3.22) $$ \begin{align} r\varphi^{\prime}_0(r) \mu\wedge\eta_0, \qquad \varphi_0(r) \eta_\alpha\wedge\xi_\alpha \end{align} $$

can extend smoothly across the zero level. Notice that $\mu \wedge \eta _0 = Cr^{-2} du \wedge d\overline {u} $ , where u denotes the fiber coordinate of L. Then, it is easy to see that the terms in (3.22) is smooth on each fiber if and only if $r^{-1}\varphi '(r)$ and $\varphi (r)$ are smooth on $\mathbb {C}$ if and only if there exists a smooth function $h: [0,\infty ) \rightarrow \mathbb {R}$ such that $h(r^2)=\varphi (r)$ . According to the definition of $\Phi $ , we expand $\Phi $ near $0$ :

$$ \begin{align*} \Phi'(r) = h(r^2) /r =C_{-1}/r +C_1 r + C_2 r^3 + \ldots, \end{align*} $$

we have

$$ \begin{align*} \Phi(r) = C_{-1} \log r + C_0 +C_1 r^2 +C_2 r^4 +\ldots. \end{align*} $$

We claim that $C_{-1}$ is vanishing. Otherwise, $\varphi (0)\ne 0$ , which implies that $\varphi _0 (r) \eta _0$ is not well-defined on the zero level. Recalling the expression of $\theta $ , this contradicts against the fact that $\theta $ is well-defined on L. Therefore, we find a global Kähler potential for $\omega $ , which completes the proof.

Remark 3.5 The invariant $\partial \overline {\partial }$ -lemma does not hold for all line bundles. For instance, let $\alpha $ be a simple root with $\alpha =2 \lambda $ , for instance, the line bundle $\mathcal {O}(1)\rightarrow \mathbb {C}\mathbb {P}^1$ . Consider the following invariant 1-form:

$$ \begin{align*} \theta = r^2 \eta_0 + r^2 \eta_\alpha +r^2 \xi_\alpha. \end{align*} $$

Then, we can check that $d\theta $ is an invariant exact real (1,1) form. However, comparing with (3.21), there is no potential function for this form.

3.4 Proof of Theorem A

Firstly, it is easy see that the invariant $dd^c$ lemma can be applied when L is a negative line bundle over X. Notice that the homogeneous line bundle L can shrink to the base manifold X. Furthermore, we can require the shrinking process to be G-equivalent. In particular, we have the following isomorphism between cohomology groups:

$$ \begin{align*}p^*: \ H^{1,1}_{G}(X) \ \cong \ H^{1,1}_{G} (L),\end{align*} $$

which implies that all invariant Kähler classes of L arise from the invariant Kähler classes of X. In each invariant Kähler class of X, there exists exactly one invariant Kähler form, which follows directly from $dd^c$ -lemma on X. Given an invariant Kähler form $\omega $ on L, by previous discussion, there exists an invariant Kähler form $\omega _X$ on X such that $[\omega ]=p^*[\omega _X]$ then, by Propositions 3.4, there exists a smooth function $\Phi $ defined on L such that

$$ \begin{align*} \omega= p^* \omega_X + dd^c \Phi. \end{align*} $$

Hence, we complete the proof of Theorem A.

4.1 Momentum construction of Calabi ansatz

Let $(X,\omega _X)$ be a Kähler manifold and $\pi :(L,h)\rightarrow (X, \omega _X)$ be a holomorphic line bundle of M with Hermitian metric h. Let t be the logarithm of fiber norm function related to h; i.e., given a local line bundle coordinate chart $(u,v)$ , where u represents the fiber coordinate, $t=\log [r(u)^2]:=\log h(u,\overline {u})$ . Then, Kähler metrics arise from Calabi ansatz is given by

(4.1) $$ \begin{align} \omega=\pi^*\omega_X+\frac{1}{2}dd^c f(t), \end{align} $$

where f is a smooth function of one real variable. According to Theorem A, all invariant Kähler metrics comes from Calabi ansatz (4.1).

It is well-known that the problem of prescribed scalar curvature is equivalent to a fourth-order PDE of potential function, specially, in this case, a fourth-order ODE. However, Hwang and Singer comes up the method of momentum profile in [Reference Hwang and Singer22] by which it can be reduced to be a second-order ODE, as the curvature formula related to momentum profile is of second order. In the following, we study the momentum profile associated with the given original data.

The Kähler metric, $\omega $ , arising from Calabi ansatz (4.1) may not exist in the whole line bundle L. For instance, in some cases, it might be blowing up in a finite domain with respect to fiber coordinate. We use $L'$ to denote the possible existence region of L for $\omega $ . Notice that Kähler metric $\omega $ constructed in (4.1) admits a natural Killing field $X_0$ , which generates a circle action on each fiber and can be written in local bundle coordinates, $X_0=r J({\partial }/{\partial r}).$ In aspects of symplectic geometry, $X_0$ generates a Hamiltonian action on $(L,\omega )$ by

(4.2) $$ \begin{align}i_{X_0}\omega=-d\tau.\end{align} $$

At each point on $p \in X$ , there exists a coordinate chart around p such that $\partial \log h |_p=\overline {\partial } \log h|_p=0$ . By computing $\omega $ at each point in the chart,

(4.3) $$ \begin{align} \omega=\pi^*\omega_X+f'(t)\frac{1}{2}dd^c\log h+f"(t)i\frac{du\wedge d\overline{u}}{|u|^2} \end{align} $$

and inserting (4.3) into (4.2), we have $\tau =f'(t)$ . Let the interval I be the image of moment map $\tau $ . Noting that $||X_0||_\omega $ is a constant along each level of $\tau $ , we can define the function $\varphi : I\rightarrow \mathbb {R}_{\geq 0}$ by factoring through $\tau $ ,

$$ \begin{align*}\varphi(\tau)=\frac{1}{2}||X_0(\tau)||^2_{\omega}.\end{align*} $$

The interval I together with the function $\varphi $ is called momentum profile related to $(L',\omega )$ . The essential relation between $\varphi (\tau )$ and the potential f is given by

(4.4) $$ \begin{align}\varphi(\tau)=\frac{1}{2}\omega(X_0, JX_0)=-\frac{1}{2}JX_0(f'(t))=-\frac{1}{2}f"(t) \cdot JX_0(\log r^2)=f"(t).\end{align} $$

Also by observing (4.3), the positivity of Kähler form implies the following two things: f is a convex function; hence the moment map $\tau =f'(t)$ induces a Legendre transformation from t to $\tau $ . Moreover, if we denote $\gamma =-i\partial \overline {\partial } \log h$ , the positivity of $\omega $ also requires $\omega -\tau \gamma $ to be positive. An interval, I, is defined to be a momentum interval if for all $\tau \in I$ , $\omega (\tau )=\omega -\tau \gamma $ is positive. In the following, we shall reconstruct $(L',\omega )$ by momentum profile $(I,\varphi )$ with a momentum interval I and $\varphi : I\rightarrow \mathbb {R}_{\geq 0}$ .

Based on the inverse Legendre transformation, we can rebuild the Kähler metric $\omega $ explicitly by momentum profile $(I, \varphi )$ as follows. Let $(a,b)$ be the interior of I with $-\infty \leq a<b\leq \infty $ and fix $\tau _0\in I$ .

1. (a) Fiber domain: Let T be the defining domain of $f(t)$ with $T^\circ =(t_1,t_2)$ , then,

   $$ \begin{align*} t_1=\lim_{\tau\rightarrow a+}\int_{\tau_0}^\tau \frac{dx}{\varphi(x)}\quad \text{ and }\quad t_2=\lim_{\tau\rightarrow b-}\int^\tau_{\tau_0}\frac{dx}{\varphi(x)}. \end{align*} $$

2. (b) Potential function: $f(t)$ is given by data $(I,\varphi )$

   $$ \begin{align*} f(t)=\int^{\tau(t)}_{\tau_0}\frac{xdx}{\varphi(x)}. \end{align*} $$

3. (c) Fiber metric: The metric $\omega $ induces the metric on each fiber in terms of coordinate u, $\displaystyle g_{\text {fiber}}$ and the $\omega $ -distance between $\tau _0$ level and $\tau (t)$ level, $s(t)$

   $$ \begin{align*} g_{\text{fiber}}=\varphi(\tau)\bigg|\frac{du}{u}\bigg|^2, \qquad s(t)=\int_{\tau_0}^{\tau(t)}\frac{dx}{2\sqrt{\varphi(x)}}, \end{align*} $$

where the formulas in (b) and (c) can be obtained by change the variable though Legendre transformation.

The next step is to work out the curvature formula in terms of momentum profiles. To make curvature formula fit in the momentum profile, define $(\tau ,\pi ): L'\rightarrow I\times X$ . Here are some notations that will be used in the following:

• • Let $\omega _\varphi $ represent the Kähler metric constructed by momentum profile $(\varphi ,I)$ , and we can rewrite (4.3) in terms of $\tau $ ,

  (4.5) $$ \begin{align} \omega_\varphi =p^*\omega_X(\tau)+\varphi(\tau)\frac{idu\wedge d\overline{u}}{|u|^2}, \end{align} $$
  where $\omega _X(\tau ) = \omega _X -\tau \gamma $ .

• • Let B denote the endomorphism $\omega _X^{-1}\gamma $ , $\rho _X$ be the Ricci curvature form of X, define the following functions on $I\times X$ :

  $$ \begin{align*} Q(\tau)&=\det(I-\tau B),\\ R(\tau)&=\operatorname{tr}[(I-\tau B)^{-1}(\omega^{-1}_X\rho_X)]. \end{align*} $$

Then, the Ricci curvature, Laplacian and scalar curvature have the following representation in terms of momentum profile and notations above:

• • The Ricci form of $\omega _\varphi $ ,

  (4.6) $$ \begin{align} \rho_\varphi=p^*\rho_X-i\partial\overline{\partial}\log \varphi Q(\tau). \end{align} $$

• • Scalar curvature $S_\varphi $ ,

  (4.7) $$ \begin{align} S_\varphi =R(\tau)-\Delta_{\omega_{X(\tau)}}\log Q(\tau)-\frac{1}{Q}\frac{\partial^2}{\partial \tau^2}(\varphi Q)(\tau). \end{align} $$

In the case of G-invariance, $ Q(\tau )$ is a polynomial in $\tau $ and $\Delta _{X(\tau )}\log Q(\tau )=0$ . Then, we can assume that $R(\tau )=P(\tau )/Q(\tau )$ for some polynomial P in $\tau $ . Therefore, we reduce the problem of prescribed scalar curvature to a second-order ODE

(4.8) $$ \begin{align} (\varphi Q)"+QS_{\varphi}=P. \end{align} $$

4.2 Proof of Theorem B

In the following, we compute the explicit formula of the polynomials $Q(\tau )$ , $P(\tau )$ in terms of $\omega _X$ and corresponding weight $\lambda $ . Recall the formulas (2.14) and (3.9) and $\omega _X$ , $\gamma $ can be expressed in terms of $dz_\alpha =\eta _\alpha +i \xi _\alpha $ and $d\overline {z}_\alpha = \eta _\alpha - i\xi _\alpha $ ,

$$ \begin{align*} \omega_X = \frac{i}{2} C_{\alpha,S}\ dz_\alpha \wedge d\overline{z}_\alpha,\qquad \gamma=-i\partial\overline{\partial} \log h= \frac{i}{4}\sum_{\alpha\in D^+}( \lambda , \alpha )\ dz_\alpha \wedge d\overline{z}_\alpha, \end{align*} $$

where $S\in \mathfrak {s}$ such that $C_{\alpha ,S}>0$ . Then, the matrix B is diagonal, and $Q(\tau )$ has the following expression:

$$ \begin{align*} Q(\tau)=\det(I-\tau B)=\prod_{\alpha \in D^+}\bigg[1-{\tau} \frac{(\lambda, \alpha)}{2 C_{\alpha,S}} \bigg]. \end{align*} $$

Since the Ricci curvature $\rho _X$ has the expression

$$ \begin{align*} \rho_X = \frac{i}{4} \sum_{\alpha \in D^+} (\alpha, \delta)\ dz_\alpha \wedge d\overline{z}_\alpha. \end{align*} $$

Then,

$$ \begin{align*} R(\tau)= \operatorname{tr} \big[(I-\tau B)^{-1}\ \omega_X^{-1} \rho_X \big]= \sum_{\alpha \in D^+ } \frac{(\alpha,\delta)}{2C_{\alpha,S}-\tau(\lambda,\alpha)}. \end{align*} $$

Hence, the ODE (4.8) can be rewrite as follows:

(4.9) $$ \begin{align} \Big( \varphi \prod_{\alpha \in D^+} \big( 2 C_{\alpha,S} - \tau (\lambda,\alpha) \big) & \Big)" + S_\varphi \prod_{\alpha\in D^+}\big( 2 C_{\alpha,S}- \tau (\lambda, \alpha) \big) \nonumber \\ &= \prod_{\alpha \in D^+} \big( 2 C_{\alpha,S}- \tau (\lambda,\alpha) \big) \sum_{\beta \in D^+}\frac{(\beta,\delta)}{2 C_{\beta,S}-\tau (\lambda, \beta)}. \end{align} $$

To determine the initial data of the ODE (4.9), we need to apply the following completeness proposition in [Reference Hwang and Singer22, Propositions 2.2 and 2.3].

And the corresponding $(L',\omega )$ behaves differently under different decay conditions provided in Proposition 4.1. We conclude the corresponding relations in the Table 1, where we consider finite ends at $\tau =0$ and infinite ends as $\tau \rightarrow \infty $ . The proof of these bundle behaviors directly follows from the reconstruction of the data $(L',\omega )$ by moment profile $(\varphi ,I)$ , (a)–(c).

Table 1:

Behaviors of $(L',\omega )$ .

To fit in our cases, we define the momentum interval $I=[0,b)$ with $b\leq +\infty $ . The reason we take the left ends to be $0$ is to ensure $\omega |_{X}=\omega _{S}$ . Assume that the corresponding scalar curvature of $\omega _\varphi $ is constant. Combining with Proposition 4.1 and Table 1, we shall solve the ODE with initial condition $\varphi (0)=0$ , $\varphi '(0)=1$ , and $S_\varphi =0$ . It is obvious that there is a unique solution $\varphi $ satisfies (4.9) and the initial condition.

4.2.1 The asymptotic behavior of scalar-flat Kähler metrics

In the scalar-flat cases, $\varphi $ satisfies

$$ \begin{align*} \Big( \varphi \prod_{\alpha \in D^+} \big( &2 C_{\alpha,S} - \tau (\lambda,\alpha) \big) \Big)'= \\ &\int_0^{\tau}\prod_{\alpha \in D^+} \big( 2 C_{\alpha,S}- t (\lambda,\alpha) \big) \sum_{\beta\in D^+}\frac{(\beta,\delta)}{2 C_{\beta,S}-t (\lambda, \beta)} dt+ \prod_{\alpha\in D^+} 2 C_{\alpha,S}. \end{align*} $$

Assuming that $\lambda $ is negative, $\varphi $ is a strictly increasing function with initial value ${\varphi (0)=0}$ with degree one. According to Table 1, $\omega _\varphi $ is well-defined over the whole bundle $L_\lambda $ . By solving the ODE (4.9), the leading coefficient of the solution is given as follows:

$$ \begin{align*} i_{\lambda, X} = \frac{1}{(n-1)n} \sum_{\alpha \in D^+} \frac{(\alpha, \delta)}{(\alpha,-\lambda)}. \end{align*} $$

We call $i_{\lambda , X} $ the metric index of line bundle of $(L_\lambda , X)$ . Recall the Kähler metric associated with $\varphi $ is given as in (4.5). Let $g_X(\tau )$ , $g_X$ , $g_\gamma $ be the metric corresponding to $\omega _X(\tau ) = \omega _X - \tau \gamma $ , $\omega _X$ , $-\gamma $ , respectively, then

$$ \begin{align*} g_\varphi =g_X + \tau g_\gamma +2 \varphi(\tau) \eta_0^2+ 2\varphi(\tau)\mu^2, \end{align*} $$

where $\mu $ and $\eta _0$ are the dual of $r\partial /\partial r$ and $X_0$ , respectively. Then, on each level set of L, $M(\tau )$ , there is a metric induced by $g_\varphi $ , denoted by $g(\tau )$ ,

$$ \begin{align*} g_{M(\tau)}= g_X + \tau g_\gamma +2 \varphi(\tau) \eta_0^2. \end{align*} $$

Let $C_\lambda $ be the cone associated with $L_\lambda $ by collapsing the base manifold X. Then, we should determine the radial function l of cone $C_\lambda $ such that the scalar-flat Kähler metric is asymptotically conical to $A i\partial \overline {\partial } l^2$ , where A is the constant coefficient and can be canceled by rescaling. Based on the discussion in Section 4.1, we have the following relationship between $\tau $ and t:

$$ \begin{align*} t=\int_{\tau_0}^{\tau(t)} \frac{dx}{\varphi(x)} = \int_{\tau_0}^{\tau(t)} \frac{dx}{i_{\lambda, X} x}+\frac{a_1 dx}{x^2}+\ldots = a_0+\frac{1}{i_{\lambda, X}}\log \tau - \frac{a_1}{\tau}+\ldots, \end{align*} $$

where the second equality is just the Taylor expansion of $1/\varphi (x)$ . Taking exponential and solve for $\tau $ , we can see that $\tau $ admits the following expansion at infinity:

(4.10) $$ \begin{align} \tau= b_1 r^{2i_{\lambda, X}} + b_0 + b_{-1} r^{-2 i_{\lambda, X}}+\ldots. \end{align} $$

Now, let $l=r^{i_{\lambda ,X}}$ , then the model Kähler metric over $C_\lambda $ is defined by the radial function l,

$$ \begin{align*} \omega_{\text{mod}}= i\partial\overline{\partial} l^2= - i_{\lambda, X} l^2 \gamma + 2 i_{\lambda, X}^2 l^2 \mu \wedge \eta_0. \end{align*} $$

Rewrite the model Kähler form in terms of metric,

$$ \begin{align*} g_{\text{mod}} &= i_{\lambda, X} l^2 g_\gamma+ 2 i_{\lambda, X}^2 l^2 \eta_0^2 + 2 i_{\lambda, X}^2 l^2 \mu^2\\ &= l^2 (i_{\lambda, X} g_\gamma+ 2 i_{\lambda, X}^2 \eta_0^2 ) + 2 dl^2. \end{align*} $$

And the metric can be represented by $dl$ as follows:

$$ \begin{align*} g_\varphi & = l^2\Big( \frac{1}{l^2} g_X + \frac{\tau}{l^2} g_\gamma +\frac{2 \varphi(\tau)}{l^2} \eta^2_0 \Big) + \frac{2\varphi(\tau)}{i_{\lambda, X}^2 l^2} dl^2\\ & = {b_1} \big[ l^2 (i_{\lambda, X} g_\gamma+ 2 i^2_{\lambda, X} \eta^2_0) + dl^2 \big]+ O(l^{-2})\\ & = b_1 g_{\text{mod}} +O(l^{-2}). \end{align*} $$

Therefore, all scalar-flat Kähler metrics on $L_\lambda $ are asymptotically conical to $(C_\lambda , g_{\text {mod}})$ , which complete the proof of Theorem B.

In general, $g_\varphi $ decays to $g_{\text {mod}}$ by order $-2$ , which can be improved in some special cases. For instance, let the base metric $\omega _X $ be equal to the curvature form $\gamma $ , then, the similar calculation shows that the metric $g_\varphi $ with scalar-flat curvature decays to order $-2n+2$ .