Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit II

Abstract

Podestà and Spiro (Osaka J Math 36(4):805–833, 1999) introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kähler structure (g, J) (“standard G-manifolds”) and studied invariant Kähler and Kähler–Einstein metrics on M. In the first part of this paper, we gave a combinatoric description of the standard non-compact G-manifolds as the total space $M_{\varphi }$ of the homogeneous vector bundle $M = G\times _H V \rightarrow S_0 =G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kähler–Einstein metric g on such manifolds M in terms of the existence of an interval in the T-Weyl chamber of the flag manifold $F = G \times _H PV$ which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group $G = SU_n, Sp_n. Spin_n$ and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold $S_0 = G/H$. If this condition is fulfilled, the explicit construction of the Kähler–Einstein metric reduces to the calculation of the inverse function to a given function of one variable.

Appendix: Basic facts on flag manifolds

Let $F = G/K = \mathrm {Ad}_G Z$, where $Z \in {\mathfrak {g}}$, be a flag manifold, i.e., an adjoint orbit of a compact semisimple Lie group G with the B-orthogonal

(where B is the Killing form) reductive decomposition

$$\begin{aligned} {\mathfrak {g}} = \mathfrak {k}+ \mathfrak {m}= C_{\mathfrak {g}}(Z) + \mathfrak {m}. \end{aligned}$$

We can decompose $\mathfrak {k}$ as

$$\begin{aligned} \mathfrak {k}= Z(\mathfrak {k}) \oplus \mathfrak {k}', \end{aligned}$$

where $\mathfrak {k}'$ is the semisimple part and $Z(\mathfrak {k})$ is the center. We fix a Cartan subalgebra $\mathfrak {c}$ of $\mathfrak {k}$ (hence also of $\mathfrak {g}$) and denote by R the root system of the complex Lie algebra $\mathfrak {g}^{{\mathbb {C}}}$ w.r.t. the Cartan subalgebra $\mathfrak {c}^{{\mathbb {C}}}$. We set

$$\begin{aligned} R_{{\mathfrak {k}}} :=\{ \alpha \in R, \, \alpha (Z(\mathfrak {k})) = 0 \},\,R_{{\mathfrak {m}}} := R {\setminus } R_{{\mathfrak {k}}}. \end{aligned}$$

Then,

$$\begin{aligned} \mathfrak {k}= \mathfrak {c}+ \mathfrak {g}(R_{{\mathfrak {k}}})^{\tau },\, \mathfrak {m}= \mathfrak {g}(R_{{\mathfrak {m}}})^{\tau }, \end{aligned}$$

where for a subset $P \subset R$, we set

$$\begin{aligned} \mathfrak {g}(P) = \sum _{\alpha \in P} \mathfrak {g}_{\alpha }, \end{aligned}$$

being $\mathfrak {g}_{\alpha }$ the root space with root $\alpha$ and $V^{\tau }$ means the fix point set in $V\subset \mathfrak {g}^{{{\mathbb {C}}}}$ of the complex conjugation $\tau$. Recall that the Killing form induces an Euclidean metric in the real vector space $i \mathfrak {c}$ and roots are identified with real linear forms on $i \mathfrak {c}$. We set $\mathfrak {t}:= i Z(\mathfrak {k}) \subset i \mathfrak {c}$ and denote by

$$\begin{aligned} \rho : R \rightarrow R|_{\mathfrak {t}},\,\, \alpha \mapsto {\bar{\alpha }} := \alpha |_{\mathfrak {t}} \end{aligned}$$

the restriction map.

Definition 7

The set $R_T =\rho (R_{{\mathfrak {m}}})= R_{{\mathfrak {m}}}|{\mathfrak {t}}$ of linear forms on $\mathfrak {t}$ which are restriction of roots from $R_{{\mathfrak {m}}}$ is called the system of T-roots and connected components C of the set $\mathfrak {t}{\setminus } \{ \mathrm {ker\;}{\bar{\alpha }} ,\, {\bar{\alpha }} \in R_T \}$ are called T-Weyl chambers.

Sets of T-roots $\xi$ bijectively correspond to irreducible $\mathfrak {k}$-submodules $\mathfrak {m}(\xi ):= \mathfrak {g}(\rho ^{-1}(\xi ))$ of the complexified isotropy module $\mathfrak {m}^{{\mathbb {C}}}$ of the flag manifold $F =G/K$.

So a decomposition of the $\mathfrak {k}$-modules $\mathfrak {m}^{{\mathbb {C}}}$ and $\mathfrak {m}$ into irreducible submodules can be written as

$$\begin{aligned} \mathfrak {m}^{{\mathbb {C}}} = \sum _{\xi \in R_T} \mathfrak {m}(\xi ),\,\, \mathfrak {m}= \sum _{\xi \in R^+_T}[ \mathfrak {m}(\xi ) + \mathfrak {m}(-\xi ) ]^{\tau }, \end{aligned}$$

where $R^+_T :=\rho (R_{{\mathfrak {m}}}^+)$ is the system of positive T-roots associated with a system of positive roots $R^+$, see [1, 4].

We fix a system of simple roots $\Pi _W$ of $R_{{\mathfrak {k}}}$ and denote by $\Pi = \Pi _W \cup \Pi _B$ its extension to a system of simple roots of R. Let $R^+ = R^+(\Pi )$ be the associated system of positive roots and $R^+_{{\mathfrak {m}}}:= R^+ \cap R_{{\mathfrak {m}}}$. The set $R^+_T:= \rho (R^+_{{\mathfrak {m}}})$ is called positive T-root set.

We need the following

Theorem 8

[4] There exists a one-to-one correspondence between extensions $\Pi = \Pi _W \cup \Pi _B$ of the system $\Pi _W$ of simple system of $R_{{\mathfrak {k}}}$, T-Weyl chambers $C\subset \mathfrak {t}$ and invariant complex structures (ICS) J on $F = G/K$. If $\Pi _B = \{ \beta _1, \dots ,\beta _k \}$, then the corresponding T-Weyl chamber is defined by $C = \{ {\bar{\beta }}_1>0, \dots ,{\bar{\beta }}_k >0 \}$ where ${\bar{\beta }} = \rho (\beta )$ and the complex structure is defined by $\pm i$-eigenspace decomposition

$$\begin{aligned} \mathfrak {m}^{{{\mathbb {C}}}} = \mathfrak {m}^+ + \mathfrak {m}^- = \mathfrak {g}(R^+_{{\mathfrak {m}}}) + \mathfrak {g}(-R^+_{{\mathfrak {m}}}) \end{aligned}$$

(61)

of the complexified tangent space $\mathfrak {m}^{{{\mathbb {C}}}} = T_{eK}(G/K)$.

The extension $\Pi = \Pi _W \cup \Pi _B$ can be graphically described by a painted Dynkin diagram, i.e., the Dynkin diagram which represents the system $\Pi$ with the nodes representing $\Pi _B$ painted in black. Such a diagram, which we sometimes identify with the pair $(\Pi _W, \Pi _B)$, allows to reconstruct the flag manifold $F= G/K$ with invariant complex structure $J^F$ as follows: The semisimple part $\mathfrak {k}'$ of the (connected) stability subalgebra $\mathfrak {k}$ is defined as the regular semisimple subalgebra associated with the closed subsystem $R_{{\mathfrak {k}}}= R \cap \mathrm {span}(\Pi _W)$ and the vectors $ih_{j}$ defined by condition

$$\begin{aligned} \beta _k(h_j) = \delta _{kj}, \, \alpha _i(h_j)=0,\, \beta _j \in \Pi _B, \alpha _i \in \Pi _W \end{aligned}$$

form a basis of the center $Z({\mathfrak {k}})$. The complex structure is defined by (61).

Now, an element $Z \in {\mathfrak {t}}$ is called K-regular if its centralizer $C_G(Z) = K$ or, equivalently, any T-root has a nonzero value on Z. Then, we have the following.

Proposition 9

[4, 8] There exists a natural one-to-one correspondence between elements $Z \in {\mathfrak {t}}$ and closed invariant 2-form $\omega _Z$ on G / K, given by

$$\begin{aligned} Z \leftrightarrow \omega _Z|_o = i \, d(B \circ Z), \end{aligned}$$

where d is the exterior differential in the Lie algebra ${\mathfrak {g}}$ defined by $d\alpha (X,Y) = - 1/2\alpha ([X,Y])$ and $o = eK \in G/K$.

Moreover, regular elements $Z \in C$ from a T-Weyl chamber C correspond to the Kähler forms $\omega _Z$ with respect to the complex structure J(C) associated with C, that is they define an invariant Kähler structure $(\omega _Z, J(C))$. The 2-form $\frac{1}{2\pi }\omega _Z$ is integral if the 1-form $B\circ Z$ has integer coordinates with respect to the fundamental weights $\pi _i$ associated with the system of black simple roots $\beta _i \in \Pi _B$.

Recall that if $\Pi _W = \{ \alpha _1, \dots , \alpha _m\}$ (resp. $\Pi _B = \{ \beta _1, \dots , \beta _k\}$) is the set of white (resp. black) simple roots, then the fundamental weight $\pi _i$ associated with $\beta _i$, $i = 1, \dots , k$, is the linear form defined by

$$\begin{aligned} \frac{2\langle \pi _i, \beta _j \rangle }{\Vert \beta _j \Vert ^2} = \delta _{ij}, \ \ \langle \pi _i, \alpha _j \rangle = 0, \end{aligned}$$

(62)

where $<.,.>$ is the scalar product in $i\mathfrak {c}^* = \mathrm {span}(R)$ induced by the Killing form.

The B-dual to $\pi _i$ vectors $h_i$ form a basis of $\mathfrak {t}$.

Let $E_{\alpha } \in \mathfrak {g}_{\alpha }, \, \alpha \in R$, be the Chevalley basis of ${\mathfrak {g}}(R)$ such that $B(E_{\alpha }, E_{- \alpha }) = \frac{2}{<\alpha , \alpha >}$ We denote by $\omega _{\alpha } = B\circ E_{\alpha }$ the dual basis of 1-forms. Then, for $Z \in \mathfrak {t}$

$$\begin{aligned} \omega _Z = -i\sum _{\alpha \in R_{{\mathfrak {m}}}^+} \frac{2 \alpha (Z)}{<\alpha , \alpha >} \omega _{\alpha } \wedge \omega _{-\alpha }. \end{aligned}$$

(63)

Indeed,

$$\begin{aligned} \begin{array}{ll} i \, d(B\circ Z)(E_{\alpha }, E_{- \alpha }) &{}= -\frac{i}{2} B(Z, [E_{\alpha }, E_{-\alpha }])\\ &{}=-\frac{i}{2}B([Z,E_{\alpha }],E_{-\alpha })\\ &{}= - \frac{i}{2}\alpha (Z)B(E_{\alpha }, E_{-\alpha })\\ &{}=- \frac{i\alpha (Z)}{<\alpha , \alpha>}\\ &{}= -2i\frac{\alpha (Z)}{<\alpha , \alpha >} \omega _{\alpha } \wedge \omega _{-\alpha }(E_{\alpha }, E_{-\alpha }). \end{array} \end{aligned}$$

Definition 10

The 1-form

$$\begin{aligned} \sigma = \sum _{\beta \in R_{{\mathfrak {m}}}^+} \beta \in \mathfrak {t}^* \subset i\mathfrak {c}^* \end{aligned}$$

is called the Koszul form and the dual vector $Z^{Kos}:= B^{-1} \circ \sigma$ is called the Koszul vector.

Proposition 11

[4] The Koszul vector$Z^{Kos}$ defines the invariant Kähler–Einstein structure $(\omega _{Z^{Kos}}, J(C))$ on $F=G/K$, where J(C) is the invariant complex structure associated with the T-Weyl chamber C which is defined by $\Pi _B$.