2. Momentum maps for K-representations and their projective compactifications

Let V be a finite-dimensional complex G-representation. Assume that we are given a K-invariant Hermitian product $\langle \cdot, \cdot \rangle$ on V, making the induced K-representation unitary. Let $\chi_V: V \to \mathbb{R}^{\geqslant 0}$ be defined by $v \mapsto \langle v,v \rangle$ and consider the Kähler form $-dd^c\chi_V$ on V. Note that the good quotient $\pi_V: V \to V/\hspace{-0.12cm}/ G = \mathrm{Spec}\left({\mathbb{C}}[V]^G \right)$ exists; it is endowed with a Kähler structure by Kählerian reduction with respect to the momentum map

\begin{equation*}\mu_V: V \to \mathfrak{k}^*, \; v \mapsto (\xi \mapsto 2\langle i\xi\cdot v, v\rangle) = d^c \chi_V (\xi_V)(v). \end{equation*}

These, however, are not the right Kähler form and momentum map to consider in our situation, since they do not globalize to the situation of a G-vector bundle over a non-trivial base manifold with typical fibre V; see Example 3.4.

Instead, we are going to compactify the situation. For this, we consider the G-representation space $W:=V\oplus{\mathbb{C}}$, where G acts trivially on ${\mathbb{C}}$, and embed V regularly and equivariantly into $\mathbb{P}(W) =\mathbb{P}(V \oplus {\mathbb{C}})$; explicitly, we consider

\begin{equation*}\theta: V \hookrightarrow \mathbb{P}(V \oplus {\mathbb{C}}),\; v \mapsto [v:1].\end{equation*}

We denote by $\mathbb{P}(W)^{ss}$ the Zariski-open subset of points that are semistable with respect to the linearization of the G-action on $\mathbb{P}(W)$ in W or equivalently the corresponding linearization in the line bundle $\mathcal{O}_{\mathbb{P}(W)}(1)$; the associated good quotient will be called $\pi_{\mathbb{P}(W)}\colon\mathbb{P}(W)^{ss} \to \mathbb{P}(W)^{ss}/\hspace{-0.12cm}/ G$. While not strictly needed for our subsequent arguments, the following observation regarding GIT is at least philosophically crucial.

Proof. This is almost tautological: if we choose linear coordinates (i.e., linear forms) $z_1, \dots, z_{\dim V}, z_0$ on $W=V \oplus {\mathbb{C}}$, then z ₀ is G-invariant, as the action of G on ${\mathbb{C}}$ is trivial. This implies that the associated section $\sigma = \sigma_{z_0} \in H^0\bigl(\mathbb{P}(W), \mathcal{O}_{\mathbb{P}(W)}(1)\bigr)$ is G-invariant. Since obviously

\begin{equation*}\theta (V) = \{[w] \in \mathbb{P}(W) \mid \sigma (w) \neq 0\} = \mathbb{P}(W)_{\sigma},\end{equation*}

every point in $\theta(V)$ is semistable. Moreover, for any $\tau \in H^0\bigl(\mathbb{P}(W), \mathcal{O}_{\mathbb{P}(W)}(1)\bigr)^G$, the corresponding open subset $\mathbb{P}(W)_\tau$ is $\pi_{\mathbb{P}(W)}$-saturated; see the proof of [Reference Mumford, Fogarty and Clare Kirwan18, Thm. 1.10].

Consequently, we will consider $V \subset \mathbb{P}(V \oplus {\mathbb{C}})$, suppressing θ. Using the linear coordinate z ₀ on ${\mathbb{C}}$ introduced in the previous proof, we endow the G-representation $W= V \oplus {\mathbb{C}}$ with the Hermitian form $\langle \cdot , \cdot \rangle$ corresponding to

\begin{equation*}\chi_W (v,z_0) := \chi_V(v) + |z_0|^2.\end{equation*}

This in turn yields a Fubini-Study form on $\mathbb{P}(W)$, which is K-invariant and makes the K-action Hamiltonian with momentum map given by

\begin{equation*}\mu^\xi ([w]) = \frac{2 \langle i\xi\cdot w, w \rangle}{\langle w, w \rangle}.\end{equation*}

The next observation is a momentum geometry counterpart of Lemma 2.1 and will be applied fibrewise in the bundle situation considered in $\S$ 3 below.

Lemma 2.2. A potential for the restriction of the Fubini-Study Kähler form $\omega = \omega_{\mathbb{P}(W)}$ to $V \subset \mathbb{P}(W)$ is given by

\begin{equation*}\omega|_V = - d d^c \,\log (\chi_V +1 ) = 2i \partial \bar\partial \, \log (\chi_V +1 ).\end{equation*}

The function $\rho := \log (\chi_V +1) \in \mathcal{C}^\infty (V)$ is an exhaustion of V, and if $\xi_{\mathbb{P}(W)}$ is the vector field on $\mathbb{P}(W)$ induced by the K-action, then

\begin{equation*} V \ni v \mapsto d^c\rho (\xi_{\mathbb{P}(W)})([v:1]) = \mu^\xi([v:1]) \end{equation*}

defines a momentum map for the K-action on V with respect to $\omega|_V$.

Proof. The first part is well known, the exhaustion property is clear and the last part follows by direct computation that uses the obvious equality

\begin{equation*}\xi_W(v,1) = (\xi_V(v), 0)\end{equation*}

together with the fact that (being a G-equivariant isomorphism) θ ⁻¹ transforms the fundamental vector field $\xi_{\mathbb{P}(W)}$ into ξ _V .

Remark 2.3. In the following, we will only need that ρ is an exhaustive K-invariant potential for a Kähler form on V. The fact that $d^c\rho$ then yields a momentum map for the K-action can be seen as follows: since ρ and hence $d^c\rho$ is K-invariant, we have $\mathcal{L}_{\xi_X}(d^c\rho) = 0$, and therefore Cartan’s formula yields

\begin{equation*}d\bigl(\imath_{\xi_X} (d^c\rho) \bigr) = -\imath_{\xi_X}\bigl(dd^c(\rho)\bigr) = \imath_{\xi_X}\omega, \end{equation*}

so that indeed $\imath_{\xi_X} (d^c\rho)$ is the ξ-component of a momentum map with respect to ω, whose equivariance is easy to check as well.

3. Kähler forms on holomorphic principal bundles and vector bundles

Given a G-principal bundle $\pi\colon X \to Q$ over a compact Kählerian manifold $Q = X/G$, we will be looking for special Kähler structures on X. For this, the following construction will be useful.

Remark 3.1. As G is an affine algebraic G-variety (with algebraic action given by left-multiplication), it admits a closed (algebraic) embedding $\psi: G \hookrightarrow V$ into the vector space associated with a certain finite-dimensional G-representation $\varphi: G \to GL_{\mathbb{C}}(V)$, see for example [Reference Brion4, Prop. 2.2.5]. That is, there is a closed G-orbit $G \ \tiny{\bullet}\ v$ in V with $G_v =\{e\}$, and in particular, φ is injective. We may assume that V has a K-invariant Hermitian form h, so that $\varphi|_K\colon K \hookrightarrow SU(h)$. The associated vector bundle $\mathcal{V}:= X \times_G V$ has a natural holomorphic G-action (preserving each fibre), as well as a Hermitian metric that is induced by h and therefore invariant by the associated K-action. Slightly abusing notation, we will call the bundle metric h as well. Writing $X = X \times_G G$, we may produce a closed G-equivariant holomorphic embedding

of X into $\mathcal{V}$ over $Q = X/G$.

Let now $(B,\omega)$ be a compact Kähler manifold, and let $\pi\colon \mathcal{V}\to B$ be a holomorphic vector bundle.

We will now analyse the construction of the Kähler form on $\mathcal{V} \subset \mathbb{P}(\mathcal{V} \oplus {\mathbb{C}})$ in more detail, with the aim of pointing out the specific features relevant to momentum map geometry.

Lemma 3.3. Let $\pi\colon \mathcal{V}\to B$ be a holomorphic vector bundle of rank n over a compact Kähler manifold $(B,\omega)$ on which the compact Lie group K acts holomorphically via fibre-preserving vector bundle automorphisms. Set $\mathfrak{k}:=\operatorname{Lie} (K)$. Let h be a K-invariant Hermitian metric on $\mathcal{V}$, with associated length function

\begin{equation*}\chi_h: \mathcal{V} \to \mathbb{R}^{\geqslant 0}, \; v \mapsto h(v,v).\end{equation*}

Then, for all positive real numbers $c \gg 0$, the $(1,1)$-form

\begin{equation*}\omega_\mathcal{V} = 2i \partial \bar\partial \log(\chi_h + 1) + c \cdot \pi^*(\omega) \end{equation*}

has the following properties:

1. (1) $\omega_\mathcal{V}$ is Kähler,

2. (2) the K-action on $\mathcal{V}$ admits a momentum map $\mu\colon \mathcal{V} \to \mathfrak{k}^*$ with respect to $\omega_\mathcal{V}$,

3. (3) every point $x \in B$ has an open neighbourhood U such that on $\pi^{-1}(U)$ we may write $\omega = 2i \partial\bar\partial \rho$ for some function $\rho \in \mathcal{C}^\infty(\pi^{-1}(U))^{K}$ that has the following properties:

   1. (a) ρ is an exhaustion when restricted to $\mathcal{V}_y$ for all $y \in U$,

   2. (b) for all $\xi \in \mathfrak{k}$ and for all $v \in \pi^{-1}(U)$, we have

      (3.1)\begin{equation} \mu^\xi (v) = d^c\rho(\xi_\mathcal{V}(v)) = d\rho(J\xi_\mathcal{V}(v)), \end{equation}
      where as usual $J \in \mathrm{End}(T\mathcal{V})$ denotes the (almost) complex structure of the complex manifold $\mathcal{V}$ and $\xi_\mathcal{V}$ is the vector field on $\mathcal{V}$ associated with ξ via the K-action.

Proof. We consider $\mathcal{W} = \mathcal{V} \oplus {\mathbb{C}}$ the direct sum of $\mathcal{V}$ with the trivial line bundle, and its projectivization $\mathbb{P}(\mathcal{W}) = (\mathcal{W} \setminus z_{\mathcal{W}})/{\mathbb{C}}^*$, which contains $\mathbb{P}(\mathcal{V})$ as a co-dimension one sub-bundle with (open) complement isomorphic to $\mathcal{V}$. Endow $\mathcal{W}$ with the Hermitian metric

\begin{equation*}h_\mathcal{W}(v \oplus z) := \chi_h(v) + |z|^2.\end{equation*}

Using the section of $\mathcal{O}_{\mathbb{P}(\mathcal{W})}(1)$ corresponding to the divisor $\mathbb{P}(\mathcal{V}) \subset \mathbb{P}(\mathcal{W})$, one computes that the relative Fubini-Study form $\omega_{FS}^{\mathcal{W}} $, that is, the curvature form of the natural Hermitian metric in $\mathcal{O}_{\mathbb{P}(\mathcal{W})}(1)$ induced by h, when restricted to $\mathcal{V} = \mathbb{P}(\mathcal{W}) \setminus \mathbb{P}(\mathcal{V})$, is given by

\begin{equation*}c\cdot \omega_{FS}^{\mathcal{W}}|_{\mathcal{V}} = 2i\partial \bar \partial\, \log(\chi_h +1) = - d d^c\, \log(\chi_h + 1)\qquad \text{for some }c\in \mathbb{R}^{ \gt 0};\end{equation*}

see for example [Reference Demailly6, Chap. V, § 15.C; p. 282], but notice the different convention regarding projectivizations of vector bundles. Restricting to any fibre $\mathcal{V}_y$, $y \in B$, we recover the Kähler form associated with the Hermitian scalar product h _y discussed in Lemma 2.2. Using compactness of B, Part (1) is now proven with the argument of [Reference Voisin21, p. 78].

The statements made in Parts (2) and (3) will be proven simultaneously. Let $\{U_k\}_{k \in I}$ be a finite open covering of B so that $\omega|_{U_k}$ is given by a potential φ _k; that is, $\omega|_{U_k} = - d d^c \varphi_k$. It is clear that $\omega_\mathcal{V}|_{\pi^{-1}(U_k)}$ has K-invariant potential

(3.2)\begin{equation}\rho_k := \log(\chi_h + 1)|_{\pi^{-1}(U_k)} + c\cdot\pi^*(\varphi_k). \end{equation}

On any of the open sets $\pi^{-1}(U_k)$, a corresponding momentum map for the K-action is given by $\mu^\xi_k:=\iota_{\xi_\mathcal{V}}d^c\rho_k$, cf. Remark 2.3. Setting $\eta:= \log(\chi_h + 1)$ in order to shorten notation, still on $\pi^{-1}(U_k)$, we compute

(3.3)\begin{align} \iota_{\xi_\mathcal{V}}d^c\bigl(\eta+c\cdot\pi^*\varphi_k\bigr)&=\iota_{ \xi_\mathcal{V}}d^c\eta + c\cdot \iota_{\xi_\mathcal{V}}d^c\pi^*\varphi_k \nonumber\\ &=\iota_{\xi_\mathcal{V}}d^c\eta + c\cdot\iota_{\xi_\mathcal{V}}\pi^*d^c\varphi_k\\ & =\iota_{\xi_\mathcal{V}}d^c\eta. \nonumber \end{align}

Here, we used the fact that $\xi_\mathcal{V}$ is tangential to the fibres of π, whereas the forms $\pi^*d^c\varphi_k$ vanish in fibre direction. Consequently, the maps µ _k on $\pi^{-1}(U_k)$ glue together to a well-defined momentum map $\mu_\mathcal{V}\colon \mathcal{V}\to{\mathfrak{k}}^*$ with respect to $\omega_\mathcal{V}$; as the computation Equation (3.3) shows, it is given by the map formally associated with the (in general only fibrewise strictly plurisubharmonic) function $\eta = \log (\chi_h +1 )$ defined on the whole of $\mathcal{V}$. This shows the statements made in Part (2). The properties listed in Part (3) then follow from the definition of ρ _k, see Equation (3.2), and the fibrewise properties established in Lemma 2.2.

The following example explains the choice of the potential $\log(\chi_h+1)$ and the remark at the end of the first paragraph of $\S$ 2.

Example 3.4. The total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}_1$ can be explicitly realized as $\mathbb{P}_2\setminus\left\{[0:0:1]\right\}$ with bundle projection $\pi\left([z_0:z_1:z_2]\right)=[z_0:z_1]$. Let us consider the bundle metric on $\mathcal{O}(1)$ with associated length function

\begin{equation*} \chi_h\left([z_0:z_1:z_2]\right)=\frac{\lvert z_2\rvert^2}{\lvert z_0\rvert^2+ \lvert z_1\rvert^2}. \end{equation*}

An explicit computation shows that there does not exist any c > 0 such that $i\partial\overline{\partial}\chi_h+c\cdot\pi^*\omega_{FS}$ is positive. Namely, in the affine coordinates (z, w) on $\mathbb{P}_2\setminus\left\{[0:0:1]\right\}\cap\{z_0\not=0\}\cong {\mathbb{C}}^2\setminus\{0\}$, the Hermitian form of $i\partial\overline{\partial}\chi_h+c\cdot\pi^*\omega_{FS}$ is given by

\begin{equation*} \frac{1}{\left(\lvert z\rvert^2+\lvert w\rvert^2\right)^3} \begin{pmatrix} \lvert z\rvert^2\left(1+c\lvert w\rvert^2\right)-\lvert w\rvert^2+c\lvert w\rvert^4 & \left(2-c\left(\lvert z\rvert^2+\lvert w\rvert^2\right)\right)z\overline{w}\\ \left(2-c\left(\lvert z\rvert^2+\lvert w\rvert^2\right)\right)\overline{z}w & \left(1+c\lvert z\rvert^2\right)\lvert w\rvert^2-\lvert z\rvert^2+c\lvert z\rvert^4 \end{pmatrix}, \end{equation*}

which is never positive definite.

4. Proof of Theorem 1.1

Before we start the proof proper, we observe that in case that G is not connected with identity component G ^∘, the quotient $Q^\circ := X / G^\circ$ is a finite topological covering of $X/G$ and therefore also compact. Moreover, for $l \in \{a,b,c\}$, statement (l) of Theorem 1.1 holds if and only if the corresponding statement (l ^∘) holds for the action of G ^∘, the maximal compact group $K^\circ := K \cap G^\circ$ of G ^∘, the equivariant compactification $\overline{G^\circ} \subset \overline{G}$, and the quotient Q ^∘. For this observation, the main point to notice is that $\overline{G} = G \times_{G^\circ} \overline{G^\circ}$, and hence

\begin{equation*}\overline{X} = X \times_G \overline{G} = X \times_G G \times_{G^\circ} \overline{G^\circ} \cong X \times_{G^\circ} \overline{G^\circ}.\end{equation*}

We therefore assume for the remainder of the proof that $G=G^\circ$ is connected.

4.1. Implication ‘(b) $\Rightarrow$ (a)’

We quickly summarize the fundamental results in the Kählerian version of the theory of Symplectic Reduction: Let $G=K^{\mathbb{C}}$ be a complex-reductive group and let X be a holomorphic G-manifold. Let ω be a K-invariant Kähler form on X such that the K-action on X is Hamiltonian with equivariant momentum map $\mu\colon X\to{\mathfrak{k}}^*$. If the zero fibre $\mu^{-1}(0)$ is non-empty, then the set

\begin{equation*} X^{ss}(\mu):=\bigl\{x\in X \mid \overline{G\ \tiny{\bullet}\ x}\cap\mu^{-1}(0)\not=\emptyset\bigr\} \end{equation*}

of semistable points is a non-empty open G-invariant of X admitting an analytic Hilbert quotient $q\colon X^{ss}(\mu)\to X^{ss}(\mu)/\hspace{-0.12cm}/ G$; that is, q is a surjective G-invariant, (locally) Stein map to a complex space $X^{ss}(\mu)/\hspace{-0.12cm}/ G$ fulfilling

\begin{equation*}q_*(\mathcal{O}_{X^{ss}(\mu)})^G = \mathcal{O}_{X^{ss}(\mu)/\hspace{-0.12cm}/ G}.\end{equation*}

In particular, q is universal with respect to G-invariant holomorpic maps from $X^{ss}(\mu)$ to complex spaces. Moreover, every fibre of q contains a unique G-orbit that is closed in $X^{ss}(\mu)$; this orbit is the only one in $\pi^{-1}(0)$ that intersects $\mu^{-1}(0)$, and the intersection is a uniquely determined K-orbit. With more work, one can show that $X^{ss}(\mu)/\hspace{-0.12cm}/ G$ is homeomorphic to the symplectic reduction $\mu^{-1}(0)/K$ and via this homeomorphism inherits a Kähler structure induced by ω, see [Reference Heinzner and Loose15, Reference Heinzner, Huckleberry and Loose16, Reference Sjamaar20] and the detailed survey [Reference Heinzner and Huckleberry14].

Suppose from now on that in addition to the assumptions made above the G-action on X is free and proper. In this situation, since all G-orbits are closed in X, we have

\begin{equation*}X^{ss}(\mu)=G\cdot\mu^{-1}(0),\end{equation*}

and hence $X^{ss}(\mu)/\hspace{-0.12cm}/ G=X^{ss}(\mu)/G =q(X^{ss}(\mu))$ is an open subset of $X/G$ that is homeomorphic to $\mu^{-1}(0)/K$ and possesses a Kähler form, whose construction is much easier in this simple case; for example, see [Reference Hitchin, Karlhede, Lindström and Roček17, Thm. 3.1] and cf. [Reference Guillemin and Sternberg12, Thm. 3.5] or [Reference Heinzner and Huckleberry14, Prop. 2.4.6]. Under the assumption made in (b), the non-empty subset $q(X^{ss}(\mu)) \simeq \mu^{-1}(0)/K$ is not only open but also compact. It therefore coincides with $X/G=Q$, which is hence Kähler.

4.2. Implication ‘(a) $\Rightarrow$ (b)’

Choose a G-equivariant closed holomorphic embedding $\Psi\colon X \hookrightarrow \mathcal{V}$ into the total space of a holomorphic G-vector bundle over Q, as constructed in Remark 3.1, and let h be a corresponding K-invariant Hermitian metric on $\mathcal{V}$. Next, we apply Lemma 3.3 to obtain a constant $c \gg 0$ such that

\begin{equation*}\omega = 2i \partial \bar\partial \log(\chi_h + 1) + c \cdot \pi^*(\omega)\end{equation*}

is a Kähler form with properties (2) and (3) stated in the Lemma. We will show that the restriction of ω to the closed submanifold $X \subset \mathcal{V}$, which we continue to denote by ω, has the properties claimed in Theorem 1.1(b).

First, we notice that the K-action on X is Hamiltonian with respect to ω, with momentum map being given by the restriction of the momentum map from $\mathcal{V}$ to X, which we continue to denote by µ. Now, let $q\in Q$ be any point, let $x \in X$ a point in the fibre over q, and let U and ρ be as in part (3) of Lemma 3.3. Then, by (3.a) the restriction of ρ to $X_q = G\ \tiny{\bullet}\ x \subset \mathcal{V}_q$ continues to be a strictly plurisubharmonic exhaustion function. Therefore, $\rho|_{X_q}$ has a minimum and in particular a critical value, say at $x_0 \in G\ \tiny{\bullet}\ x$. As $J\xi_\mathcal{V}(x_0) = J\xi_{X_q}(x_0) = J\xi_{G\ \tiny{\bullet}\ x_0} (x_0)$ is obviously tangent to the (complex) orbit $G\ \tiny{\bullet}\ x_0 = G\ \tiny{\bullet}\ x$, Formula (3.1) together with the fact that x ₀ is critical implies that

\begin{equation*}\mu^\xi(x_0) = d\rho(J \xi_{\mathcal{V}}(x_0)) = d\rho(J \xi_{X_q}(x_0)) = 0 \qquad \text{for all }\xi \in \mathfrak{k},\end{equation*}

so that $x_0 \in \mu^{-1}(0)$. In other words, every fibre of $\pi\colon X \to Q$ intersects $\mu^{-1}(0)$, so that

\begin{equation*}\pi|_{\mu^{-1}(0)}\colon \mu^{-1}(0) \to Q\end{equation*}

is a K-principal bundle over the compact manifold Q. In particular, $\mu^{-1}(0) \subset X$ is compact.

4.3. Implication ‘(a) $\Rightarrow$ (c)’

As a normal projective G-variety for the connected group G, the completion $\overline{G}$ admits a G-equivariant embedding into the projectivization $\mathbb{P}(W)$ of a finite-dimensional complex G-representation W; see [Reference Brion4, Prop. 5.2.1, Prop. 3.2.6]. Using the associated bundle construction, this leads to a closed holomorphic embedding

into the projectivization of the holomorphic vector bundle $\mathcal{W} = X \times_G W$ over Q. By assumption, Q is compact Kähler, so Remark 3.2 yields the claim.

5. Examples

The following example shows that even the ‘Kähler’ statement of the implication $(a)\Longrightarrow(b)$ of Theorem 1.1 does not hold for non-compact Q.

Example 5.1. The homogeneous fibration

\begin{equation*} {\rm{SL}}(2,{\mathbb{C}})/ \left(\begin{smallmatrix}1&{\mathbb{Z}}\\0&1\end{smallmatrix}\right)\to {\rm{SL}}(2,{\mathbb{C}})/ \left(\begin{smallmatrix}1&{\mathbb{C}}\\0&1\end{smallmatrix}\right) \end{equation*}

is a ${\mathbb{C}}^*$-principal bundle over the non-compact Kähler base ${\mathbb{C}}^2\setminus\{0\}$. Owing to [Reference Berteloot and Oeljeklaus2, Theorem 3.1], however, the total space ${\rm{SL}}(2,{\mathbb{C}})/\left(\begin{smallmatrix}1&{\mathbb{Z}}\\0&1 \end{smallmatrix}\right)$ is not Kähler.

A slight modification of Example 5.1 yields an example showing that an analogue of Lemma 3.3 does not hold if we replace the vector bundle $\pi\colon \mathcal{V}\to X$ by an arbitrary holomorphic fibre bundle having Stein fibres, even though these kind of fibres also admit strictly plurisubharmonic exhaustion functions similar to (the logarithm of) the norm square function used in the proof of Lemma 3.3.

Next, we give an example (originally due to Lescure) which explains that even for $K^{\mathbb{C}}$-actions on Kählerian manifolds X, further conditions are needed to ensure that the quotient $X/G$ is Kähler.

Example 5.3. Let us consider the Hopf surface $Y=({\mathbb{C}}^2\setminus\{0\})/{\mathbb{Z}}$ with respect to the ${\mathbb{Z}}$-action given by $m\ \tiny{\bullet}\ v=2^mv$. Let $p\colon {\mathbb{C}}^2\setminus\{0\}\to Y$ be the universal covering. The group $G={\rm{GL}}(2, {\mathbb{C}})$ acts transitively on Y, and a direct calculation shows that the isotropy group of $y_0:=p(1,0)\in Y$ is

\begin{equation*} G_{y_0}=\left\{ \begin{pmatrix} 2^m&b\\0&a \end{pmatrix};\ m\in{\mathbb{Z}}, b\in{\mathbb{C}}, a\in{\mathbb{C}}^*\right\}. \end{equation*}

Let us consider the closed subgroups

\begin{equation*} H:=\left\{ \begin{pmatrix} 2^m&b\\0&1 \end{pmatrix};\ m\in{\mathbb{Z}}, b\in{\mathbb{C}}\right\}\text{and } T:=\left\{ \begin{pmatrix} 1&0\\0&a \end{pmatrix};\ a\in{\mathbb{C}}^*\right\} \end{equation*}

of $G_{y_0}$. One verifies directly that $T\cong{\mathbb{C}}^*$ normalizes H and thus that $G_{y_0}=TH\cong T \lt imes H$. Hence, we have the holomorphic ${\mathbb{C}}^*$-principal bundle $X:=G/H\to Y=G/(HT)$.

We claim that X is a Kähler manifold. For this, let $S:={\rm{SL}}(2,{\mathbb{C}})$ and note that

1. (1) $S\cap H=\left(\begin{smallmatrix}1&{\mathbb{C}}\\0&1\end{smallmatrix}\right)$ is an algebraic subgroup of S, as well as that

2. (2) $SH=\bigl\{2^{m/2}g;\ m\in{\mathbb{Z}}, g\in S\bigr\}$ is a closed subgroup of G. (In fact, $G/(SH)$ is an elliptic curve.)

Now we can apply [Reference Gilligan, Miebach and Oeljeklaus9, Theorem 5.1] to deduce that X is Kähler.

However, there is no momentum map for the S ¹-action on $X=G/H$ with respect to any S ¹-invariant Kähler form: Indeed, since the subgroup H has infinitely many connected components and is therefore not algebraic, the action of ${\rm{U}}(2)$ on X is not Hamiltonian, see [Reference Gilligan, Miebach and Oeljeklaus9, Theorem 4.11]. Since ${\rm{U}}(2)={\rm{SU}}(2)S^1$, this implies that X is not a Hamiltonian S ¹-manifold, either.

In Example 5.3 above, while X is Kähler, the action of a maximal compact group K is not Hamiltonian with respect to any Kähler form. Therefore, one might wonder whether the existence of a momentum map for the K-action is actually sufficient to conclude that the base of a Kählerian principal bundle is Kähler. Candidates of counterexamples to this statement can be obtained by the Cox construction, which realizes toric varieties associated with simplicial fans and without torus factors as geometric quotients of certain domains in ${\mathbb{C}}^N$ by a linear action of a complex torus T, see [Reference Cox, Little and Schenck5, Theorem 5.1.11]. The following concrete example, which is in some sense minimal, consists of a smooth complete non-projective toric threefold for which Cox’ geometric quotient is a T-principal bundle.

Example 5.4. Let Σ be the simplicial fan from [Reference Fujino and Payne8, Example 2]. It is shown that the associated toric threefold $X_\Sigma$ is smooth, complete, and non-projective, hence non-Kähler by Moishezon’s Theorem, [Reference Grauert, Peternell and Remmert10, Chap. VII, Thm. 6.23]. Since $\bigl\lvert \Sigma(1)\bigr\rvert=8$ and $X_\Sigma$ is smooth, we have

\begin{equation*} \operatorname{Cl}(X_\Sigma)=\operatorname{Pic}(X_\Sigma)\cong{\mathbb{Z}}^{\lvert \Sigma(1)\rvert-\dim X(\Sigma)} ={\mathbb{Z}}^5, \end{equation*}

see [Reference Cox, Little and Schenck5, Theorem 4.1.3 and Proposition 4.2.6]. Moreover, since $X_\Sigma$ has no torus factors, the relevant group is $T=\operatorname{Hom}_{\mathbb{Z}}({\mathbb{Z}}^5,{\mathbb{C}}^*) \cong({\mathbb{C}}^*)^5$, see [Reference Cox, Little and Schenck5, Lemma 5.1.1]. Consequently, an application of [Reference Cox, Little and Schenck5, Theorem 5.1.11] implies that there exists a geometric T-quotient $\pi\colon{\mathbb{C}}^8\setminus Z(\Sigma)\to X_\Sigma$ where T acts linearly on ${\mathbb{C}}^8$. In particular, there is a momentum map for the action of any compact real form of T on ${\mathbb{C}}^8\setminus Z(\Sigma)$, described explicitly in $\S$ 2 above.

Using again the fact that $X_\Sigma$ is smooth, we may apply [Reference Arzhantsev, Derenthal, Hausen and Laface1, Proposition 2.1.4.6] to see that T acts freely on ${\mathbb{C}}^8\setminus Z(\Sigma)$. Since the orbit space $\bigl({\mathbb{C}}^8\setminus Z(\Sigma)\bigr)/T\cong X_\Sigma$ is Hausdorff and since the holomorphic slice theorem for Hamiltonian actions, for example see [Reference Heinzner and Huckleberry14, Thm. 4.1.], yields local slices for the T-action, it follows that T acts properly on ${\mathbb{C}}^8\setminus Z(\Sigma)$, see [Reference Palais19, Theorem 1.2.9] ; that is, π is indeed a T-principal bundle.