On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

In this subsection, we will briefly review some properties of the Bergman kernel and metric on a complex manifold. More details can be found in [21] and [22].

Let $M$ be an $n$-dimensional complex manifold. Let $L^{2}_{(n,0)}(M)$ denote the space of $L^{2}$-integrable $(n,0)$-forms on $M,$ equipped with the inner product

Define the Bergman space of $M$ to be

Assume $A^{2}_{(n,0)}(M)\neq\{0\}$. Then $A^{2}_{(n,0)}(M)$ is a separable Hilbert space. Taking any orthonormal basis $\{\varphi_{k}\}_{k=1}^{q}$ of $A^{2}_{(n,0)}(M)$ with $1\leq q\leq\infty$, we define the Bergman kernel (form) of $M$ to be

Then, $K_{M}(x,\bar{x})$ is a real-valued, real-analytic form of degree $(n,n)$ on $M$ and is independent of the choice of orthonormal basis.

Let $N_{1},N_{2}$ be two complex manifolds of dimension $n$. Let $\gamma:N_{1}\rightarrow M$ and $\tau:N_{2}\rightarrow M$ be holomorphic maps. The pullback of the Bergman kernel $K_{M}(x,\bar{y})$ of $M$ to $N_{1}\times N_{2}$ is defined in the standard way. That is, for any $z\in N_{1},w\in N_{2}$,

In terms of local coordinates, we may write the Bergman kernel form of $M$ as

where the function $\widetilde{K}_{M}(x,\bar{y})$ depends on the choice of local coordinates. We then have

where $J_{\gamma}$ and $J_{\tau}$ are the Jacobian determinants of the maps $\gamma$ and $\tau$, respectively. In particular, we observe that the kernel function $\widetilde{K}_{M}(x,\bar{y})$ transforms accordining to the usual biholomorphic invariance formula under changes of local coordinates.

Let $M$ be as in Definition 2.2. Assume $K_{M}(x,\bar{x})$ is non-vanishing (on the set of regular points of $M$, where it is defined). We define a Hermitian $(1,1)$-form on the regular part of $M$ by

The biholomorphic invariance of the Bergman kernel implies that this form is independent of the choice of local coordinates used to determine the function $\widetilde{K}_{M}(x,\bar{x})$. The Bergman metric on $M$ is the metric induced by $\omega_{M}$ (when it indeed induces a positive definite metric on the regular part of $M$).

We recall the Bergman kernel transformation formula in [11] for (possibly branched) covering maps of complex analytic spaces. This formula generalizes a classical theorem of Bell ([1], [2]; see also [7]):

In this subsection, we recall the canonical realization of a finite ball quotient due to H. Cartan [6]. Let $\mathbb{B}^{n}$ denote the unit ball in $\mathbb{C}^{n}$ and $\text{Aut}(\mathbb{B}^{n})$ its (biholomorphic) automorphism group. Let $\Gamma$ be a finite subgroup of $\text{Aut}(\mathbb{B}^{n})$. Assume $\Gamma$ is fixed point free; that is, assume no $\gamma\in\Gamma-\{\mathrm{id}\}$ has any fixed points on $\partial\mathbb{B}^{n}$. As the unitary group $U(n)$ is a maximal compact subgroup of $\operatorname{Aut}(\mathbb{B}^{n})$, by basic Lie group theory, there exists some $\psi\in\operatorname{Aut}(\mathbb{B}^{n})$ such that $\Gamma\subseteq\psi^{-1}\cdot U(n)\cdot\psi$. Thus without loss of generality, we can assume $\Gamma\subseteq U(n)$, i.e., $\Gamma$ is a finite unitary subgroup. The origin $0\in\mathbb{C}^{n}$ is then always a fixed point of every element in $\Gamma$. Moreover, the fixed point free condition on $\Gamma$ is equivalent to the assertion that every $\gamma\in\Gamma-\{\mathrm{id}\}$ has no other fixed point than $0$. We also note that, by the fixed point free condition, the action of $\Gamma$ on $\partial\mathbb{B}^{n}$ is properly discontinuous and $\partial\mathbb{B}^{n}/\Gamma$ is a smooth manifold.

By a theorem of H. Cartan [6], the quotient $\mathbb{C}^{n}/\Gamma$ can be realized as a normal algebraic subvariety $V$ in some $\mathbb{C}^{N}$. To be more precise, we write $\mathcal{A}$ for the algebra of $\Gamma$-invariant holomorphic polynomials, that is,

By Hilbert’s basis theorem, $\mathcal{A}$ is finitely generated. Moreover, we can find a minimal set of homogeneous polynomials $\{p_{1},\cdots,p_{N}\}\subseteq\mathcal{A}$ such that every $p\in\mathcal{A}$ can be expressed in the form

where $q$ is some holomorphic polynomial in $\mathbb{C}^{N}$. The map $Q:=(p_{1},\cdots,p_{N}):\mathbb{C}^{n}\rightarrow\mathbb{C}^{N}$ is proper and induces a homeomorphism of $\mathbb{C}^{n}/\Gamma$ onto $V:=Q(\mathbb{C}^{n})$. As $Q$ is a proper holomorphic polynomial map, $V$ is an algebraic variety. The restriction of $Q$ to the unit ball $\mathbb{B}^{n}$ maps $\mathbb{B}^{n}$ properly onto a relatively compact domain $\Omega\subseteq V$. In this way, $\mathbb{B}^{n}/\Gamma$ is realized as $\Omega$ by $Q$. Following [28], we call such $Q$ the basic map associated to $\Gamma$. The ball quotient $\Omega=\mathbb{B}^{n}/\Gamma$ is nonsingular if and only if the group $\Gamma$ is generated by reflections, i.e., elements of finite order in $U(n)$ that fix a complex subspace of dimension $n-1$ in $\mathbb{C}^{n}$ (see [28]); thus, if $\Gamma$ is fixed point free and nontrivial, then $\Omega=\mathbb{B}^{n}/\Gamma$ must have singularities. Moreover, $\Omega$ has smooth boundary if and only if $\Gamma$ is fixed point free (see [14] for more results along this line).

The implication (i) $\implies$ (ii) in Theorem 1.2 is trivial. We therefore only need to prove the converse. Let $G$ be as in Theorem 1.2 and assume the conditions in (ii) hold. To prove (i), assuming that Theorem 1.4 has been proved, we proceed in three steps.

Step 1. It follows from the assumption that the boundary $\partial G$ is strongly pseudoconvex. We first prove that the boundary $\partial G$ is indeed spherical. Recall that a CR hypersurface $M$ of dimension $2n-1$ is said to be spherical if it is locally CR diffeomorphic, near every point, to an open piece of the unit sphere $S^{2n-1}\subseteq\mathbb{C}^{n}$. To prove that $\partial G$ is spherical near a given boundary point $q\in\partial G$, one first uses the Kähler-Einstein assumption and the localization of the Bergman kernel (see [12], [5] and also see Proposition 3.1 in [17] for a detailed and nice proof) near $q$ to study the coefficients in Fefferman’s expansion of the Bergman kernel of a smaller domain in $V$, which shares an open piece of its boundary with $G$ and is biholomorphic to a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^{n}$. In the two-dimensional case ($n=2$), one applies the argument in [15] (see Section 2 in [15]) to prove that the coefficient of the logarithmic term in Fefferman’s expansion of the Bergman kernel vanishes to infinite order at in an open neighborhood of $q$. Using the (local) resolution of the Ramadanov Conjecture in $\mathbb{C}^{2}$, as in [15], one deduces that $\partial G$ is spherical. In the higher dimensional case ($n\geq 3$), one uses the argument in [19] (see the proof of Theorem 1.1 in [19]) to study the coefficient of the principle term (strong singularity) in Fefferman’s expansion of the Bergman kernel and prove that every boundary point of $G$ is CR-umbilical, which implies that $\partial G$ is spherical. The detailed proof for step 1 is contained in [17] (see Theorem 1.1 in [17]). We will omit the proof here.

Step 2. In this step, we prove that $G$ is biholomorphic to a ball quotient $\mathbb{B}^{n}/\Gamma$ for some finite fixed point free subgroup $\Gamma\subseteq U(n)$. Since we know $\partial G$ is spherical from Step 1 and $\partial G$ is contained in a real algebraic hypersurface in $\mathbb{C}^{N}$, it follows from Corollary 3.3 in [16] that $\partial G$ is CR equivalent to CR spherical space form $S^{2n-1}/\Gamma$ where $\Gamma\subseteq U(n)$ is as above. More precisely, there is an algebraic CR map $F:S^{2n-1}\rightarrow\partial G$, which is a finite covering map. From this one can further prove that $G$ is biholomorphic to $\mathbb{B}^{n}/\Gamma$. The proof of this is identical with Step 3 in Section 5 of [11]. The general setting of [11] is in dimension $n=2$, but, as pointed out in Remark 5.4 in [11], this argument works for all dimensions. The argument shows that $F$ extends to a proper, holomorphic branched covering map from $\mathbb{B}^{n}$ onto $G$, which realizes $G$ as the ball quotient $\mathbb{B}^{n}/\Gamma$. In particular, $G$ is biholomorphic to $\mathbb{B}^{n}/\Gamma$ as claimed. Since $\Gamma\subseteq U(n)$ is fixed point free, either $G$ has one unique singular point at $F(0)$ when $\Gamma\neq\{\mathrm{id}\}$ or $G$ is smooth when $\Gamma=\{\mathrm{id}\}$. In the former case, $F:\mathbb{B}^{n}-\{0\}\rightarrow G-\{F(0)\}$ is a smooth covering map whose group of deck transformations is $\Gamma$, and in the latter case, $F$ extends as a biholomorphism $\mathbb{B}^{n}\to G$.

Step 3. By the conclusion in Step 2, the fundamental group of the regular part of $G$ is isomorphic to $\Gamma$. By assumption in (ii), $\Gamma$ is abelian. Moreover, the biholomorphism between $G$ and $\mathbb{B}^{n}/\Gamma$ gives an isometry between the Bergman metrics of $G$ and $\mathbb{B}^{n}/\Gamma$. By assumption in (ii) again, the Bergman metric of $\mathbb{B}^{n}/\Gamma$ is Kähler-Einstein. Thus, by Theorem 1.4, $\Gamma$ must then be the trivial group $\{\mathrm{id}\}$. Hence $G$ is biholomorphic to $\mathbb{B}^{n}$. ∎

We now prove Theorem 1.1, under the assumption that Theorem 1.4 has been proved. The "if" implication is trivial, and we only need to prove the converse. Thus, we assume that $G$ is as in Theorem 1.1 and the Bergman metric of $G$ is Kähler-Einstein, and we shall prove $G$ is biholomorphic to $\mathbb{B}^{n}$. By copying the argument in Step 1 and Step 2 in Section 3.1, we conclude that there is an algebraic CR map $F$ from $S^{2n-1}$ to $\partial G\subseteq\partial\mathbb{B}^{N}=S^{2N-1}$, which is a finite covering map. In particular, the map $F$ induces a smooth, nonconstant CR map from the spherical space form $S^{2n-1}/\Gamma$, for some finite fixed point free subgroup $\Gamma\subseteq\operatorname{Aut}(\mathbb{B}^{n})$, to $S^{2N-1}$ (see [26], [10] and [9]). Since $\Gamma$ is a finite subgroup of $\operatorname{Aut}(\mathbb{B}^{n})$, by basic Lie group theory as above, $\Gamma$ is contained in a conjugate of the unitary group $U(n)$. By Theorem 8 in [10], $\Gamma$ is conjugate to one in a short list of special cyclic subgroups of $U(n)$. In particular, the finite subgroup $\Gamma$, as well as the fundamental group of the regular part of $G$ then, is abelian. Now, Theorem 1.1 follows from Theorem 1.2. ∎

Since any two realizations of $\mathbb{B}^{n}/\Gamma$ are biholomorphic, we can use H. Cartan’s canonical realization of $\mathbb{B}^{n}/\Gamma$, which was discussed Section 2.2. Thus, let $Q:\mathbb{C}^{n}\rightarrow\mathbb{C}^{N}$ be the basic map realizing $\mathbb{B}/\Gamma$ as a domain $\Omega:=Q(\mathbb{B}^{n})$ in the $n$-dimensional algebraic variety $Q(\mathbb{C}^{n})$, as explained in Section 2.2. Set

Note that in fact $Z=\{0\}$ by the fixed point free condition and nontriviality of $\Gamma$ (see [6] and [11]). We denote by $K_{\Omega}$ and $K_{\mathbb{B}^{n}}$ the Bergman kernel forms of $\Omega$ and $\mathbb{B}^{n}$ respectively. By the transformation formula in Theorem 2.3, they are related by

where $\mathrm{id}:\mathbb{B}^{n}\rightarrow\mathbb{B}^{n}$ is the identity map. We note that

Furthermore, we also note that the Kähler-Einstein condition is a local property and that $Q$ is a local biholomorphism (on $\mathbb{B}^{n}-Z$). It follows that the Bergman metric of $\Omega$ is Kähler-Einstein if and only if the logarithm of the left hand side of (4.1), restricted to the diagonal $w=z$, gives the potential function of a Kähler-Einstein metric on $\mathbb{B}^{n}-Z.$

Recall the notation $\langle u,v\rangle=\sum_{i=1}^{n}u_{i}v_{i}$ for two column vectors $u=(u_{1},\cdots,u_{n})^{\intercal},v=(v_{1},\cdots,v_{n})^{\intercal}$. Set $d_{\gamma}:=\det\gamma$ for $\gamma\in U(n)$. The left hand side of (4.1), in the standard coordinates $z,w$ of $\mathbb{C}^{n}$, equals

where $z,w\in\mathbb{B}^{n}$ are regarded as column vectors and the elements of $\Gamma$ as unitary matrices. We introduce the function

and note that $\varphi(z,\overline{z})$ is real analytic on $\mathbb{B}^{n}.$ By the preceding discussion, we conclude that the Bergman metric of $\Omega$ is Kähler-Einstein if and only if $\varphi=\varphi(z,\bar{z})$ is the potential function of a Kähler-Einstein metric, i.e., for $z\in\mathbb{B}^{n}-Z$ and some constant $c_{1}\in\mathbb{R}$,

where $G=\det(g_{i\overline{j}})$ with $g_{i\overline{j}}=\partial_{z_{i}}\partial_{\overline{z_{j}}}\log\varphi$. (We remark that one can use the result of Klembeck [20] to find the value of $c_{1}$, but this value will also come out directly from our arguments below.) The equation (4.2) is equivalent to the statement that $\log G-c_{1}\log\varphi$ is pluriharmonic on $\mathbb{B}^{n}-Z$. Consequently, since $Z=\{0\}$ and $n\geq 2$ so that $\mathbb{B}^{n}-Z$ is simply connected, there exists some holomorphic function $h$ on $\mathbb{B}^{n}-Z$ such that

We define the Monge-Ampère type operator $J$ as follows (note that it differs by a sign from the standard operator introduced by Fefferman [13]):

We use Lemma 4.1 and the well-known formula $G=J(\varphi)/\varphi^{n+1}$ to further simplify (4.2) into

for $z\in\mathbb{B}^{n}-Z.$ Since both sides of (4.8) are in fact real-analytic in $\mathbb{B}^{n},$ we see (4.8) holds on $\mathbb{B}^{n}$ by continuity. We pause here to observe that if $\Gamma$ is such that $\varphi(0,0)\neq 0$, then it follows that $\log\varphi$ extends as the potential of a Kähler-Einstein metric in the whole unit ball $\mathbb{B}^{n}$, which by uniqueness of the Cheng-Yau metric can be used to directly conclude that $\Gamma=\{\mathrm{id}\}$; this was previously observed in [17, Corollary 5.4]. Now, let us compute $J(\varphi)$. Clearly, we have

where $(\gamma z)_{i}$ denotes the $i$-th entry of the column vector $\gamma z$ and similarly $(z^{\intercal}\overline{\gamma})_{j}$ denotes the $j$-th entry of the row vector $z^{\intercal}\overline{\gamma}$. By differentiating both sides one more time, we obtain

where $\gamma_{ij}$ is the $(i,j)$ component of the matrix $\gamma$.

For each $\gamma\in\Gamma,0\leq j\leq n$, we define a column vector-valued function $\xi_{j}(\gamma):\mathbb{B}^{n}\rightarrow\mathbb{C}^{n+1}$ in the variables $(z,\overline{z})$ as follows:

where $(\gamma)_{j}$ is the $j$-th column vector of the matrix $\gamma$. Given any $(n+1)$ (possibly repeated) elements $\gamma_{0},\cdots,\gamma_{n}$ in $\Gamma$, we define a matrix-valued function $A(\gamma_{0},\cdots,\gamma_{n}):\mathbb{B}^{n}\rightarrow\mathbb{C}^{(n+1)^{2}}$ as follows:

We emphasize that the map $A(\gamma_{0},\cdots,\gamma_{n})$ sends a point $z\in\mathbb{B}^{n}$ to an $(n+1)\times(n+1)$ matrix. We then expand the determinant in (4.7) by multi-linearity with respect to columns. We obtain the following formula:

By the proof of Lemma 4.2, we can assume $\Gamma\subseteq U(1)\times\cdots\times U(1)$. As $\Gamma$ is actually cyclic, we can write

for some generator

Here $m\geq 2$ by the nontriviality of $\Gamma.$ Since $\Gamma$ is fixed point free, $\varepsilon_{1},\cdots,\varepsilon_{n}$ are primitive $m$-th roots of unity. By setting $\varepsilon:=\varepsilon_{1}$, for $1\leq j\leq n$ we can write $\varepsilon_{j}$ in the form of

Without loss of generality, we can assume

For any $\gamma\in\Gamma\subseteq U(1)\times\cdots\times U(1)$, note that $\gamma^{-1}=\overline{\gamma}^{\intercal}=\overline{\gamma}$. Hence, we can replace all $\overline{\gamma_{j}}$ by $\gamma_{j}$ in the sum in $J(\varphi)$ and obtain

Write $\gamma_{j}=\gamma^{k_{j}}$ for some $0\leq k_{j}\leq m-1$. Then $d_{\gamma_{j}}=\det\gamma_{j}=\varepsilon^{k_{j}(\sum_{i=1}^{n}t_{i})}.$ Choose $z=z^{*}:=(z_{1},0,\cdots,0)^{\intercal}$ with $|z_{1}|<1$ and set $x=z^{*}\cdot\overline{z^{*}}=z_{1}\overline{z_{1}}<1$. Then at $z^{*},$ we have

In the following, we use the notation

• •

  $T=(t_{1},\cdots,t_{n})$, where $t_{1}=1$.

• •

  $K=(k_{0},k_{1},\cdots,k_{n})$ and $K^{\prime}=(k_{1},\cdots,k_{n})$.

• •

  $|T|=\sum_{j=1}^{n}t_{j}$ and $|K^{\prime}|=\sum_{j=1}^{n}k_{j}$.

Using these notations, we have, at $z=z^{*}$,

If we set