aa r X i v : . [ m a t h . C V ] J u l ALGEBRAICITY OF THE BERGMAN KERNEL
PETER EBENFELT, MING XIAO, AND HANG XU
Abstract.
Our main result introduces a new way to characterize two-dimensional finite ball quo-tients by algebraicity of their Bergman kernels. This characterization is particular to dimensiontwo and fails in higher dimensions, as is illustrated by a counterexample in dimension three con-structed in this paper. As a corollary of our main theorem, we prove, e.g., that a smoothly boundedstrictly pseudoconvex domain G in C has rational Bergman kernel if and only if there is a rationalbiholomorphism from G to B . Introduction
The Bergman kernel, introduced by S. Bergman in [6, 7] for domains in C n and later cast indifferential geometric terms by S. Kobayashi [33], plays a fundamental role in several complexvariables and complex geometry. Its biholomorphic invariance properties and intimate connectionwith the CR geometry of the boundary make it an important tool in the study of open complexmanifolds. The use of the Bergman kernel, e.g., in the study of biholomorphic mappings andthe geometry of bounded strictly pseudoconvex domains in C n was pioneered by C. Fefferman[16, 17, 18], who developed a theory of Bergman kernels in such domains and initiated a nowfamous program to describe the boundary singularity in terms of the local invariant CR geometry;see also [1], [25] for further progress on Fefferman’s program.A broad and general problem of foundational importance is that of classifying complex manifolds,or more generally analytic spaces, in terms of their Bergman kernels or Bergman metrics. Forexample, a well-known result of Q. Lu [36] implies that if a relatively compact domain in an n -dimensional Kähler manifold has a complete Bergman metric with constant holomorphic sectionalcurvature, then the domain is biholomorphic to the unit ball B n in C n . Another example is theconjecture of S.-Y. Cheng [12], which states that the Bergman metric of a smoothly bounded stronglypseudoconvex domain in C n is Kähler–Einstein (i.e., has Ricci curvature equal to a constant multipleof the metric tensor) if and only if it is biholomorphic to the unit ball B n . This conjecture wasconfirmed by Fu-Wong [21] and Nemirovski–Shafikov [38] in the two dimensional case, and in thehigher dimensional case by X. Huang and the second author [32].In this paper, we introduce a new characterization of the two-dimensional unit ball B ⊂ C and,more generally, two-dimensional finite ball quotients B / Γ in terms of algebraicity of the Bergmankernel. It is interesting, and perhaps surprising then, to note that such a characterization fails inthe higher dimensional case. Indeed, in Section 6 below we construct a relatively compact domain G with smooth strongly pseudoconvex boundary in a three-dimensional algebraic variety V ⊂ C ,with an isolated normal singularity in the interior of G , such that the boundary ∂G is not sphericaland, furthermore, G is not biholomorphic to any finite ball quotient; recall that a CR hypersurface M of dimension n − is said to be spherical if near each point p ∈ M , it is locally CR diffeomorphic Mathematics Subject Classification. to an open piece of the unit sphere S n − ⊂ C n . Nevertheless, in two dimensions it turns out thatalgebraicity of the Bergman kernel does characterize finite ball quotients: Theorem 1.1.
Let V be a -dimensional algebraic variety in C N , and G a relatively compactdomain in V . Assume that every point in G is a smooth point of V except for finitely many isolatednormal singularities inside G , and that G has a smooth strongly pseudoconvex boundary. Then theBergman kernel form of G is algebraic if and only if there is an algebraic branched covering map F from B onto G , which realizes G as a ball quotient B / Γ where Γ is a finite unitary group with nofixed points on ∂ B . Remark 1.2.
We note that in addition to showing that Theorem 1.1 fails in dimension ≥ , ourexample in Section 6 also shows that the Ramanadov Conjecture for the Bergman kernel fails forhigher dimensional normal Stein spaces. Recall that the Ramadanov Conjecture (c.f., [40], [14,Question 3]) proposes that if the logarithmic term in Fefferman’s asymptotic expansion [17] of theBergman kernel vanishes to infinite order at the boundary of a normal reduced Stein space withcompact, smooth strongly pseudoconvex boundary, then the boundary is spherical. The RamadanovConjecture has been established in two dimensions by the work of D. Burns and R. C. Graham (see[22]). The normal reduced Stein space constructed in Section 6 gives a -dimensional counterexamplewith one isolated singularity. The counterexamples in [14] are smooth, but not Stein.Theorem 1.1 has two immediate consequences in the non-singular case: Corollary 1.3.
Let V be a -dimensional algebraic variety in C N , and let G be a relatively compactdomain in V with smooth strongly pseudoconvex boundary. Assume that every point in G is a smoothpoint of V . Then the Bergman kernel form of G is algebraic if and only if G is biholomorphic to B by an algebraic map. Corollary 1.4.
Let G be a bounded domain in C with smooth strongly pseudoconvex boundary.Then the Bergman kernel of G is rational (respectively, algebraic) if and only if there is a rational(respectively, algebraic) biholomorphic map from G to B . We remark that although Theorem 1.1 fails in higher dimension, Corollary 1.3 and 1.4 might stillbe true. For instance, it is clear from the proof below of Theorem 1.1 (see Remark 5.3) that if theRamadanov Conjecture is proved to hold for, e.g., strongly pseudoconvex bounded domains in C n ,which is still a possibility despite Remark 1.2 above, then Corollary 1.4 also holds in C n .We also remark that the rationality of the biholomorphic map G → B in Corollary 1.4, once itsexistence has been established, follows from the work of S. Bell [5]. For the reader’s convenience, aself-contained proof of the rationality is given in Section 5.As a final remark in this introduction, we note that, by Lempert’s algebraic approximation theorem[35], if G is a relatively compact domain in a reduced Stein space X with only isolated singularities,then there exist an affine algebraic variety V , a domain Ω ⊂ V , and a biholomorphism F from aneighborhood of G to a neighborhood of Ω with F (Ω) = G . We shall say such a domain Ω is an algebraic realization of G . Theorem 1.1 implies the following corollary. Corollary 1.5.
Let G be a relatively compact domain in a -dimensional reduced Stein space X with smooth strongly pseudoconvex boundary and only isolated normal singularities. If G has analgebraic realization with an algebraic Bergman kernel, then G is biholomorphic to a ball quotient B / Γ , where Γ is a finite unitary group with no fixed point on ∂ B . To prove the "only if" implication in Theorem 1.1, we use the asymptotic boundary behavior ofthe Bergman kernel to establish algebraicity and sphericity of the boundary of G . Fefferman’s asymptotic expansion [17] and the Riemann mapping type theorems due to X. Huang–S. Ji ([29])and X. Huang ([28]) play important roles in the proof. To prove the converse ("if") implication inthe theorem, we will need to compute the Bergman kernel forms of finite ball quotients. In order todo so, we shall establish a transformation formula for (possibly branched) covering maps of complexanalytic spaces. This formula generalizes a classical theorem of Bell ([3], [4]): Theorem 1.6.
Let M and M be two complex analytic sets. Let V ⊂ M and V ⊂ M be properanalytic subvarieties such that M − V , M − V are complex manifolds of the same dimension.Assume that f : M − V → M − V is a finite ( m − sheeted) holomorphic covering map. Let Γ be the deck transformation group for the covering map (with | Γ | = m ), and denote by K i ( z, ¯ w ) theBergman kernels of M i for i = 1 , . Then the Bergman kernel forms transform according to (1.1) X γ ∈ Γ ( γ, id) ∗ K = X γ ∈ Γ (id , γ ) ∗ K = ( f, f ) ∗ K on ( M − V ) × ( M − V ) , where id : M → M is the identity map. See Section 2 for the notation used in the formula in Theorem 1.6. We expect that this formula willbe useful in other applications as well. In an upcoming paper [13], the authors apply it to study thequestion of when the Bergman metric of a finite ball quotient is Kähler–Einstein. (This is alwaysthe case for finite disk quotients, i.e., one-dimensional ball quotients, by recent work of X. Huangand X. Li [31].)The paper is organized as follows. Section 2 gives some preliminaries on algebraic functions andBergman kernels of complex analytic spaces. Section 3 is devoted to establishing the transformationformula in Theorem 1.6. Then in Section 4 we apply it to show that every standard algebraicrealization (in particular, Cartan’s canonical realization) of a finite ball quotient must have algebraicBergman kernel, and thus prove the "if" implication in Theorem 1.1. Section 5 gives the proof ofthe "only if" implication in Theorem 1.1, as well as those of Corollaries 1.3 and 1.4. In Section 6and Appendix 7, we construct the counterexample mentioned above to the corresponding statementof Theorem 1.1 in higher dimensions.
Acknowledgment.
The second author thanks Xiaojun Huang for many inspiring conversations onquotient singularities. 2.
Preliminaries
Algebraic Functions.
In this subsection, we will review some basic facts about algebraicfunctions. For more details, we refer the readers to [2, Chapter 5.4] and [30].
Definition 2.1 (Algebraic functions and maps) . Let K be the field R or C . Let U ⊂ K n be a domain.A K − analytic function f : U → K is said to be K − algebraic (i.e., real/complex-algebraic) on U ifthere is a non-trivial polynomial P ( x, y ) ∈ K [ x, y ] , with ( x, y ) ∈ K n × K , such that P ( x, f ( x )) = 0 for all x ∈ U . We say that a K − analytic map F : U → C N is K − algebraic if each of its componentsis so on U . Remark 2.2.
We make two remarks:(i) If f ( x ) is an K -analytic function in a domain U ⊂ K n , then f is K -algebraic if and only ifit is K -algebraic in some neighborhood of any point x ∈ U .(ii) If f ( x ) is an R -analytic function in a domain U ⊂ R n , then there is domain ˆ U ⊂ C n containing U ⊂ R n ⊂ C n and a C -analytic (i.e., holomorphic) function g ( x + iy ) in ˆ U such EBENFELT, XIAO, AND XU that f = g | U ; i.e., f ( x ) = g ( x ) for x ∈ U . Moreover, f is R -algebraic if and only if g is C -algebraic.We say a differential form on U ⊂ C n ∼ = R n is real-algebraic if each of its coefficient functions isso. We can also define real-algebraicity of a differential form on an affine (algebraic) variety. Definition 2.3.
Let V ⊂ C N be an affine variety and write Reg V for the set of its regular points.Let φ be a real analytic differential form on Reg V . We say φ is real-algebraic on V if for every point z ∈ Reg V , there exists a real-algebraic differential form ψ in a neighborhood U of z in C N ∼ = R N such that ψ | V = φ, on U ∩ V. Let T z V ∼ = T , z V be the complex tangent space of V at a smooth point z ∈ V considered as anaffine complex subspace in C n through z , and let ξ = ( ξ , · · · , ξ n ) be affine coordinates for T z V .Since V can be realized locally as a graph over T z V , the real and imaginary parts of ξ also serveas local real coordinates for V near z . We call such coordinates the canonical extrinsic coordinatesat z . Then the following statements are equivalent.(a) φ is real-algebraic on Reg V (in the sense of Definition 2.3).(b) For any z ∈ Reg V , φ is real-algebraic in canonical extrinsic coordinates at z .If in addition, there is a domain G ⊂ C n and a C -algebraic (i.e., holomorphic algebraic) immersion f : G → C N such that f ( G ) = Reg V , then (a) and (b) are further equivalent to(c) f ∗ φ is real-algebraic on G . Remark 2.4.
We can define complex-algebraicity of ( p, − forms, p > , on an complex affine(algebraic) variety in a similar manner as in Definition 2.3.2.2. The Bergman Kernel.
In this section, we will briefly review some properties of the Bergmankernel on a complex manifold. More details can be found in [34].Let M be an n-dimensional complex manifold. Write L n, ( M ) for the space of L -integrable ( n, forms on M, which is equipped with the following inner product:(2.1) ( ϕ, ψ ) L ( M ) := i n Z M ϕ ∧ ψ, ϕ, ψ ∈ L n, ( M ) , Define the
Bergman space of M to be(2.2) A n, ( M ) := (cid:8) ϕ ∈ L n, ( M ) : ϕ is a holomorphic ( n, form on M } . Assume A n, ( M ) = { } . Then A n, ( M ) is a separable Hilbert space. Taking any orthonormalbasis { ϕ k } qk =1 of A n, ( M ) (here ≤ q ≤ ∞ ), we define the Bergman kernel (form) of M to be K M ( x, ¯ y ) = i n q X k =1 ϕ k ( x ) ∧ ϕ k ( y ) . Then, K M ( x, ¯ x ) is a real-valued, real analytic form of degree ( n, n ) on M and is independent of thechoice of orthonormal basis. When M is also (the set of regular points on) an affine variety, we saythat the Bergman kernel of M is algebraic if K M ( x, ¯ x ) is real-algebraic in the sense of Definition2.3. The following definitions and facts are standard in literature. Definition 2.5 (Bergman projection) . Given g ∈ L n, ( M ) , we define for x ∈ MP g ( x ) = Z M g ( ζ ) ∧ K M ( x, ¯ ζ ) := i n q X k =1 (cid:16)Z M g ( ζ ) ∧ ϕ k ( ζ ) (cid:17) ϕ k ( x ) .P : L n, → A n, ( M ) is called the Bergman projection , and is the orthogonal projection to theBergman space A n, ( M ) .The Bergman kernel form remains unchanged if we remove a proper complex analytic subvariety.The following theorem is from [33]. Theorem 2.6 ([33]) . If M ′ is a domain in an n -dimensional complex manifold M and if M − M ′ is a complex analytic subvariety of M of complex dimension ≤ n − , then K M ( x, ¯ y ) = K M ′ ( x, ¯ y ) for any y ∈ M ′ . This theorem suggests the following generalization of the Bergman kernel form to complex analyticspaces.
Definition 2.7.
Let M be a reduced complex analytic space, and let V ⊂ M denote its set ofsingular points. The Bergman kernel form of M is defined as K M ( x, ¯ y ) = K M − V ( x, ¯ y ) for any x, y ∈ M − V, where K M − V denotes the Bergman kernel form of the complex manifold consisting of regular pointsof M . Let N , N be two complex manifolds of dimension n . Let γ : N → M and τ : N → M beholomorphic maps. The pullback of the Bergman kernel K M ( x, ¯ y ) of M to N × N is defined inthe standard way. That is, for any z ∈ N , w ∈ N , (cid:0) ( γ, τ ) ∗ K (cid:1) ( z, ¯ w ) = q X k =1 γ ∗ ϕ k ( z ) ∧ τ ∗ ϕ k ( w ) . In terms of local coordinates, writing the Bergman kernel form of M as K M ( x, ¯ y ) = e K ( x, ¯ y ) dx ∧ · · · dx n ∧ dy ∧ · · · ∧ dy n , we have (cid:0) ( γ, τ ) ∗ K M (cid:1) ( z, ¯ w ) = e K ( γ ( z ) , τ ( w )) J γ ( z ) J τ ( w ) dz ∧ · · · dz n ∧ dw ∧ · · · ∧ dw n , where J γ and J τ are the Jacobian determinants of the maps γ and τ , respectively.3. The transformation law for the Bergman kernel
In this section, we shall prove Theorem 1.6. For this, we shall adapt the ideas in [4] to our situation.More precisely, we shall first prove the following transformation law for the Bergman projections.Then (1.1) will follow readily by comparing the associated distributional kernels for the projectionoperators.
Proposition 3.1.
Under the assumptions and notation in Theorem . , we denote by n the complexdimension of M − V and M − V . Let P i : L n, ( M i − V i ) → A n, ( M i − V i ) e the Bergmanprojection for i = 1 , . Then the Bergman projections transform according to (3.1) P ( f ∗ φ ) = f ∗ ( P φ ) for any φ ∈ L n, ( M − V ) . EBENFELT, XIAO, AND XU
We first check that f ∗ φ ∈ L n, ( M − V ) if φ ∈ L n, ( M − V ) in the next lemma. Recall that f is an m -sheeted covering map M − V → M − V . Lemma 3.2. k f ∗ φ k L ( M − V ) = m k φ k L ( M − V ) for any φ ∈ L n, ( M − V ) . Proof.
Let { U j } be a countable, locally finite open cover of M − V such that • each U j is relatively compact; • f − ( U j ) = ∪ mk =1 V j,k for some pairwise disjoint open sets { V j,k } mk =1 on M − V ; • f : V j,k → U j is a biholomorphsm for each j = 1 , , · · · m .Let { ρ j } be a partition of unity subordinate to the cover { U j } . Then i n Z M − V φ ∧ φ = X j i n Z U j ρ j φ ∧ φ = 1 m X j m X k =1 i n Z V j,k ( f ∗ ρ j ) f ∗ φ ∧ f ∗ φ. Note that { f ∗ ρ j } is a partition of unity subordinate to the countable, locally finite open cover {∪ mk =1 V j,k } of M − V . Thus, m X j m X k =1 i n Z V j,k ( f ∗ ρ j ) f ∗ φ ∧ f ∗ φ = 1 m X j i n Z ∪ mk =1 V j,k ( f ∗ ρ j ) f ∗ φ ∧ f ∗ φ = 1 m i n Z M − V f ∗ φ ∧ f ∗ φ. The result therefore follows immediately. (cid:3)
Let F , F , · · · , F m be the m local inverses to f defined locally on M − V . Note that P mk =1 F ∗ k isa well-defined operator on L n, ( M − V ) , though each individual F k is only locally defined. Lemma 3.3.
Let v ∈ L n, ( M − V ) and φ ∈ L n, ( M − V ) . Then P mk =1 F ∗ k ( v ) ∈ L n, ( M − V ) and (3.2) (cid:0) v, f ∗ φ (cid:1) L ( M − V ) = (cid:0) m X k =1 F ∗ k ( v ) , φ (cid:1) L ( M − V ) . Proof.
We first verify P mk =1 F ∗ k ( v ) ∈ L n, ( M − V ) . For that we note f ∗ m X k =1 F ∗ k ( v ) = X γ ∈ Γ γ ∗ v. By the same argument as in Lemma 3.2, we have(3.3) (cid:13)(cid:13)X γ ∈ Γ γ ∗ v (cid:13)(cid:13) L ( M − V ) = m (cid:13)(cid:13) m X k =1 F ∗ k ( v ) (cid:13)(cid:13) L ( M − V ) . Since each deck transformation γ : M − V → M − V is biholomorphic, it follows that (cid:13)(cid:13)X γ ∈ Γ γ ∗ v (cid:13)(cid:13) L ( M − V ) ≤ X γ ∈ Γ (cid:13)(cid:13) γ ∗ v (cid:13)(cid:13) L ( M − V ) = m k v k L ( M − V ) . Therefore by (3.3), P mk =1 F ∗ k ( v ) ∈ L n, ( M − V ) . Now we are ready to prove (3.2). Let { U j } , { V j,k } and { ρ j } be the open covers and partition ofunity as in Lemma 3.2. Then (cid:0) m X k =1 F ∗ k ( v ) , φ (cid:1) L ( M − V ) = X j i n Z U j ρ j m X k =1 F ∗ k ( v ) ∧ φ. Note that every F k : U j → V j,k is biholomorphic and the inverse of f : V j,k → U j . Thus, X j i n Z U j ρ j m X k =1 F ∗ k ( v ) ∧ φ = X j m X k =1 i n Z V j,k ( f ∗ ρ j ) v ∧ f ∗ φ = ( v, f ∗ φ ) L ( M − V ) . The last equality follows from the fact that { f ∗ ρ j } is a partition of unity subordinate to the count-able, locally finite open cover {∪ mk =1 V j,k } of M − V . This proves (3.2). (cid:3) We are now ready to prove Proposition 3.1.
Proof of Proposition 3.1. If φ ∈ A n, ( M − V ) , then f ∗ φ ∈ A n, ( M − V ) by Lemma 3.2, whence(3.1) holds trivially. It thus suffices to prove (3.1) for φ ∈ A n, ( M − V ) ⊥ . In this case, (3.1)reduces to P ( f ∗ φ ) = 0 for any φ ∈ A n, ( M − V ) ⊥ ; i.e., φ ∈ A n, ( M − V ) ⊥ implies that f ∗ φ ∈ A n, ( M − V ) ⊥ . To prove this, we note that for any v ∈ A n, ( M − V ) , we have by Lemma 3.3 (cid:0) v, f ∗ φ ) L ( M − V ) = (cid:0) m X k =1 F ∗ k ( v ) , φ (cid:1) L ( M − V ) = 0 . The last equality follows from the fact φ ∈ A n, ( M − V ) ⊥ . Thus, f ∗ φ ∈ A n, ( M − V ) ⊥ andthe proof is completed. (cid:3) We are now in a position to prove Theorem 1.6.
Proof of Theorem 1.6.
Let id M i be the identity map on M i for i = 1 , . Recall that { F k } mk =1 are localinverses of f . Note that P mk =1 (id M , F k ) ∗ K is a well-defined ( n, n ) form on ( M − V ) × ( M − V ) though each (id M , F k ) ∗ K is only locally defined.We shall write out the Bergman projection transformation law (3.1) in terms of integrals of theBergman kernel forms. For any φ ∈ L n, ( M − V ) , by Lemma 3.3 we have for any z ∈ M − V , P ( f ∗ φ )( z ) = Z M − V f ∗ φ ( η ) ∧ K ( z, η ) = Z M − V φ ( η ) ∧ m X k =1 (id M , F k ) ∗ K ( z, η ) . On the other hand, P ( φ )( ξ ) = Z M − V φ ( η ) ∧ K ( ξ, η ) for any ξ ∈ M − V . If we pull back the forms on both sides by f , then f ∗ P ( φ )( z ) = Z M − V φ ( η ) ∧ ( f, id M ) ∗ K ( z, η ) for any z ∈ M − V . EBENFELT, XIAO, AND XU
Therefore, the Bergman projection transformation law (3.1) translates to Z M − V φ ( η ) ∧ m X k =1 (id M , F k ) ∗ K ( z, η ) = Z M − V φ ( η ) ∧ ( f, id M ) ∗ K ( z, η ) . As this equality holds for any φ ∈ L n, ( M − V ) , it follows that for any z ∈ M − V and η ∈ M − V ,(3.4) m X k =1 (id M , F k ) ∗ K ( z, η ) = ( f, id M ) ∗ K ( z, η ) . If we further pull back the forms on both sides by (id M , f ) : ( M − V ) × ( M − V ) → ( M − V ) × ( M − V ) , then we obtain for z, w ∈ M − V , m X k =1 (id M , F k ◦ f ) ∗ K ( z, w ) = ( f, f ) ∗ K ( z, w ) . (3.5)By using the notation γ k for the deck transformation F k ◦ f , we may write this as m X k =1 (id M , γ k ) ∗ K ( z, w ) = ( f, f ) ∗ K ( z, w ) . (3.6)Note that m X k =1 (id M , γ k ) ∗ K ( z, w ) = m X k =1 ( γ k ◦ γ − k , γ k ◦ id M ) ∗ K ( z, w ) = m X k =1 ( γ − k , id M ) ∗ ( γ k , γ k ) ∗ K ( z, w ) . Since γ k is a biholomorphism on M − V , we have ( γ k , γ k ) ∗ K ( z, w ) = K ( z, w ) , and hence m X k =1 (id M , γ k ) ∗ K ( z, w ) = m X k =1 ( γ − k , id M ) ∗ K ( z, w ) = m X k =1 ( γ k , id M ) ∗ K ( z, w ) . Theorem 1.6 now follows by combining the above identity with (3.6). (cid:3) Proof of Theorem 1.1, part I: Bergman kernels of ball quotients
In this section, we will apply the transformation law in Theorem 1.6 to study the Bergman kernelform of a finite ball quotient and prove the "if" implication in Theorem 1.1. For this part, therestriction of the dimension of the algebraic variety to two is not needed, and we shall thereforeconsider the situation in an arbitrary dimension n .Let B n denote the unit ball in C n and Aut ( B n ) its (biholomorphic) automorphism group. Let Γ be a finite subgroup of Aut ( B n ) . As the unitary group U ( n ) is a maximal compact subgroup of Aut( B n ) , by basic Lie group theory, there exists some ψ ∈ Aut( B n ) such that Γ ⊂ ψ − · U ( n ) · ψ .Thus without loss of generality, we can assume Γ ⊂ U ( n ) , i.e., Γ is a finite unitary group. Notethat the origin ∈ C n is always a fixed point of every element in Γ . We say Γ is fixed point free if every γ ∈ Γ − { id } has no other fixed point, or equivalently, if every γ ∈ Γ − { id } has no fixedpoint on ∂ B n . In this case, the action of Γ on ∂ B n is properly discontinuous and ∂ B n / Γ is a smoothmanifold. By a theorem of Cartan [11], the quotient C n / Γ can be realized as a normal algebraic subvariety V in some C N . To be more precise, we write A for the algebra of Γ invariant holomorphic polynomials,that is, A := (cid:8) p ∈ C [ z , · · · , z n ] : p ◦ γ = p for all γ ∈ Γ (cid:9) . By Hilbert’s basis theorem, A is finitely generated. Moreover, we can find a minimal set of homo-geneous polynomials { p , · · · , p N } such that every p ∈ A can be expressed in the form p ( z ) = q ( p ( z ) , · · · , p N ( z )) for z ∈ C n , where q is some holomorphic polynomial in C N . The map Q := ( p , · · · , p N ) : C n → C N is properand induces a homeomorphism of C n / Γ onto V := Q ( C n ) . By Remmert’s proper mapping theorem(see [24]), V is an analytic variety. As Q is a polynomial holomorphic map, V is furthermorean algebraic variety. The restriction of Q to the unit ball B n maps B n properly onto a relativelycompact domain G ⊂ V . In this way, B n / Γ is realized as G by Q . Following [41], we call such Q the basic map associated to Γ . The ball quotient G = B n / Γ is nonsingular if and only if the group Γ is generated by reflections , i.e., elements of finite order in U ( n ) that fix a complex subspace ofdimension n − in C n (see [41]); thus, if Γ is fixed point free and nontrivial, then G = B n / Γ musthave singularities. Moreover, G has smooth boundary if and only if Γ is fixed point free (see [19]for more results along this line).We are now in a position to state the following theorem, which implies the "if" implication inTheorem 1.1. Theorem 4.1.
Let G be a domain in an algebraic variety V in C N and Γ ⊂ U ( n ) a finite unitarysubgroup with | Γ | = m . Suppose there exist proper complex analytic varieties V ⊂ B n , V ⊂ G and F : B n − V → G − V such that F is an m-sheeted covering map with deck transformation group Γ .If F is algebraic, then the Bergman kernel form of G is algebraic.Proof. Note that the Bergman kernel form of G coincides with that of e G := G − V by Theorem2.6, and likewise the Bergman kernel form K B n of B n coincides with that of e B := B n − V . By thetransformation law in Theorem 1.6, we have X γ ∈ Γ (id B n , γ ) ∗ K B n = ( F, F ) ∗ K G on e B × e B. Since all γ ∈ Γ and K B n are rational, so is the right hand side of the equation. This implies that K G is algebraic (see the equivalent condition (c) of algebraicity in §2.1). (cid:3) Theorem 4.1 applies in particular to Cartan’s canonical realization of ball quotient.
Corollary 4.2.
Let Γ ⊂ U ( n ) be a finite unitary group. Suppose Q : C n → C N is the basic mapassociated to Γ . Let G = Q ( B n ) , which is a relatively compact domain in the algebraic variety V = Q ( C n ) . Then the Bergman kernel form of G is algebraic.Proof. We let Z = { z ∈ C n : the Jacobian of Q at z is not full rank } . Clearly, Z is a proper complex analytic variety in C n . By Remmert’s proper mapping theorem, Q ( Z ) ⊂ V is a proper complex analytic variety. Moreover, Q : B n − Z → G − Q ( Z ) is a covering mapwith m sheets, where m = | Γ | , and Γ is its deck transformation group (Note that Q − ( Q ( Z )) = Z ;see [11]). The conclusion now follows from Theorem 4.1. (cid:3) Remark 4.3.
Note that the "if" implication in Theorem 1.1 in fact holds under a much weakerassumption than that stipulated in the theorem. In Theorem 4.1 we do not assume n = 2 nor thatthe group Γ is fixed point free. We remark that the formula for the Bergman kernel of the finiteball quotient is also obtained by Huang-Li [31].5. Proof of Theorem 1.1, part II
In this section, we prove one of the main results of the paper—the "only if" implication in Theorem1.1. We also prove Corollary 1.3 and 1.4.
Proof of the "only if" implication in Theorem 1.1.
Let V and G be as in Theorem 1.1 and assumethat G has algebraic Bergman kernel. We shall prove that G is a finite ball quotient. We proceedin several steps. Step 1.
In this step, we prove ∂G is real analytic, and furthermore, real algebraic. For this step,we do not need to assume that the dimension of V is two. Proposition 5.1.
Let G be a relatively compact domain in an n -dimensional ( n ≥ ) algebraicvariety V ⊂ C N with smooth strongly pseudoconvex boundary. If the Bergman kernel K G of G isalgebraic, then the boundary ∂G of G is Nash algebraic, i.e., ∂G is locally defined by a real algebraicfunction.Proof. Fix a point p ∈ ∂G . Then there exists a neighborhood U of p in V with canonical extrinsiccoordinates z = ( z , · · · , z n ) on U (see Section 2). Write the Bergman kernel form K G of G as K G = K ( z, ¯ z ) dz ∧ dz on U ∩ G, where dz = dz ∧ · · · ∧ dz n , dz = dz ∧ · · · ∧ dz n and K ( z, ¯ z ) is a real algebraic function on U ∩ G .As K is real algebraic, there exist real-valued polynomials a ( z, ¯ z ) , · · · , a q ( z, ¯ z ) in C n ∼ = R n with a q = 0 such that(5.1) a q ( z, ¯ z ) K ( z, ¯ z ) q + · · · + a ( z, ¯ z ) K ( z, ¯ z ) + a ( z, ¯ z ) = 0 , on U ∩ G. Note that when z → ∂G , we have K ( z, ¯ z ) → ∞ as ∂G is strictly pseudoconvex. We divide bothsides of (5.1) by K ( z, ¯ z ) q and let z → ∂G to obtain a q ( z, ¯ z ) = 0 , on U ∩ ∂G. Write z k = x k + iy k for ≤ k ≤ n , z ′ = ( z , · · · , z n − ) and x ′ = ( x , y , · · · , x n − , y n − , x n ) . Byrotation, we can assume that ∂G near p is locally defined by y n = ϕ ( x ′ ) , where ϕ is a smooth function. We then have a q (cid:0) z ′ , x n + iϕ ( x ′ ) , z ′ , x n − iϕ ( x ′ ) (cid:1) = 0 . By Malgrange’s theorem (see [37] and references therein), ϕ is real analytic and thus, since a q is apolynomial, also real algebraic. Hence, ∂G is Nash algebraic. (cid:3) Step 2.
We now return to the case where V is two-dimensional. We shall prove that ∂G is spherical,where G is as in Theorem 1.1. Fix p ∈ ∂G , and a canonical extrinsic coordinates chart ( U, z ) of V at p , where z = ( z , z ) . We again write K G ( z, ¯ z ) = K ( z, ¯ z ) dz ∧ dz on U ∩ G, where dz = dz ∧ dz and dz = dz ∧ dz . Choose a strongly pseudoconvex domain D ⋐ U ∩ G suchthat B ( p, δ ) ∩ D = B ( p, δ ) ∩ G for some small δ > . Here B ( p, δ ) = { z ∈ U : k z − p k < δ } is the ball centered at p with radius δ with respect to thecoordinates ( U, z ) . Write K D for the Bergman kernel of D , which is now considered as a function.Then K D − K extends smoothly across B ( p, δ ) ∩ ∂D (see [16, 9], see also [31] for a nice and detailedproof of this fact). Consequently, K D ( z, ¯ z ) = K ( z, ¯ z ) + h ( z, ¯ z ) on D, where h ( z, ¯ z ) is real analytic in D and extends smoothly across B ( p, δ ) ∩ ∂D . Let r be a Feffermandefining function of D and express the Fefferman asymptotic expansion of K D as K D ( z, ¯ z ) = φ ( z, ¯ z ) r ( z ) + ψ ( z, ¯ z ) log r ( z ) on D, where φ and ψ are smooth functions on D that extend smoothly across B ( p, δ ) ∩ ∂D ; see [17]. Thus,(5.2) K ( z, ¯ z ) = φ ( z, ¯ z ) − h ( z, ¯ z ) r ( z ) r ( z ) + ψ ( z, ¯ z ) log r ( z ) on D. As in Step 1, there exist real-valued polynomials a ( z, ¯ z ) , · · · , a q ( z, ¯ z ) in C ∼ = R with a q = 0 forsome q ≥ , such that a q K q + · · · + a K + a = 0 on D. If we substitute (5.2) into the above equation and multiply both sides by r q , then(5.3) a q ψ q r q (log r ) q + q − X j =0 b j (log r ) j = 0 on D, where all b j for ≤ j ≤ q − are smooth on D and extend smoothly across B ( p, δ ) ∩ ∂D . We recallthe following lemma from [21]. Lemma 5.2 ([21]) . Let f ( t ) , · · · , f q ( t ) ∈ C ∞ ( − ε, ε ) for ε > . If f ( t ) + f ( t ) log t + · · · + f q ( t )(log t ) q = 0 for all t ∈ (0 , ε ) , then each f j ( t ) for ≤ j ≤ q , vanishes to infinite order at . It follows from the above lemma and (5.3) that the coefficient ψ of of the logarithmic term vanishesto infinite order at ∂G near p . Since G is two-dimensional, it follows that G is locally spherical near p by [23] (see page 129 where the result is credited to Burns) and [8] (see page 23). Remark 5.3.
Recall from the introduction that the sphericity near p above follows from the affir-mation of the Ramadanov Conjecture in two dimensions. This is also the only place where the factthat G is two dimensional is essentially used. Step 3.
In this step, we will prove there is an algebraic branched covering map F : B → G withfinitely many sheets. Since we have already shown that ∂G is a Nash algebraic and spherical CRsubmanifold in C N , by a theorem of Huang (see Corollary 3.3 in [28]), it follows that ∂G is CRequivalent to a CR spherical space form ∂ B / Γ with Γ ⊂ U ( n ) a finite group with no fixed pointson ∂ B . In particular, there is a CR covering map f : ∂ B → ∂G (see the proof of Theorem 3.1in [28] and also [29]). By Hartogs’s extension theorem, f extends as a smooth map F : B → V ,holomorphic in B and sending ∂ B onto ∂G . The latter implies that F is moreover algebraic by X.Huang’s algebraicity theorem [26]. It is not difficult to see that F sends B into G . Since F maps ∂ B to ∂G , we conclude that F is a proper algebraic mapping B → G . Claim 1. F : B → G is surjective. Proof of Claim 1.
By the properness of F , F ( B ) is closed in G . Let us denote by Z := { z ∈ B : F is not full rank at z } . Since F is a local biholomorphic map at every point of ∂ B , Z is a finite set. We also note that if p ∈ B − Z , then F ( p ) is a smooth point of V , and F ( p ) is an interior point of F ( B ) . Assume,in order to reach a contradiction, that F ( B ) = G . Since F ( B ) is closed in G , its complement G \ F ( B ) is then a non-empty open subset of G . Note that any boundary point of F ( B ) in G canonly be in F ( Z ) . But F ( Z ) is a finite set, which cannot separate the (non-empty) interior of F ( B ) and the (non-empty) open complement G \ F ( B ) in the domain G . This is the desired contradictionand, hence, F ( B ) = G . (cid:3) Now, we let T := F − ( F ( Z )) ⊃ Z . Then T is a compact analytic subvariety of B and thus is afinite set. Consider the restriction of F : F | B − T : B − T → G − F ( Z ) , still denoted by F . Clearly, F is a proper surjective map. Since F is also a local biholomorphism, F is a finite covering map.Note that B − T is simply connected. It follows that the deck transformation group e Γ = { e γ k } mk =1 of the covering map F : B − T → G − f ( Z ) acts transitively on each fiber. Since each e γ k is abiholomorphism from B − T to B − T , it extends to an automorphism of B . Consequently, e Γ = (cid:8)e γ ∈ Aut( B ) : F ◦ e γ = F on B (cid:9) . Recall that Γ is the deck transformation group of the original covering map f : ∂ B → ∂G . Fromthis, it is clear that we can identify Γ with e Γ . From now on, we will simply use the notation Γ foreither group.Note that Z and T are both closed under the action of Γ , and ( B − T ) / Γ is biholomorphic to G − f ( Z ) . Claim 2: If z, w ∈ B satisfy F ( z ) = F ( w ) , then w = γ ( z ) for some γ ∈ Γ . Consequently, T = Z . Proof of Claim 2.
We only need to prove the first assertion. If both z, w are in B − T, then theconclusion is clear as Γ acts transitively on each fiber of the covering map F : B − T → G − f ( Z ) .Next we assume one of z and w is in T . Seeking a contradiction, suppose w = γ ( z ) for every γ ∈ Γ . Writing q := F ( z ) = F ( w ) , there are then points in distinct orbits of Γ that are mappedto q. Writing t for the number of orbits of Γ that are mapped to q , we must then have t ≥ . Pick p , · · · , p t from these t distinct orbits of Γ . Since T is a finite set, we can choose, for each ≤ i ≤ t, some disjoint neighborhoods U i of p i such that U i ∩ T ⊆ { p i } . Moreover, we can make γ ( U i ) ∩ U j = ∅ for all γ ∈ Γ if i = j. Consequently, F ( U i − { p i } ) ∩ F ( U j − { p j } ) = ∅ . Note thereis a small open subset W containing q such that W ⊆ ∪ ti =1 F ( ∪ γ ∈ Γ γ ( U i )) = ∪ ti =1 F ( U i ) . Thus W − { q } ⊆ ∪ ti =1 F ( U i − { p i } ) . But the sets F ( U i − p i ) ∩ ( W − { q } ) are open and disjoint, and we canchoose W − { q } to be connected. This is a contradiction. Thus we must have t = 1 and w = γ ( z ) for some γ ∈ Γ . (cid:3) Recall that is assumed to be the only fixed point for elements in Γ . We write q := F (0) and provethat q is the only possible singularity in G . Also, recall that all singularities of G are assumed tobe isolated and normal. Claim 3. G can only have a singularity at q . Proof of Claim 3.
Suppose q is a (normal) singular point in G and q = q . Since F is onto, thereexists some p ∈ B such that f ( p ) = q .First, note that we can find a small neighborhood U of p , and a small neighborhood W of q in G such that(i) U ∩ T = { p } ; (ii) F is injective on U ; (iii) W ⊆ F ( U ) and W ∩ F ( Z ) = { q } . It is easy to see that we can make (i) and (ii) hold. It is guaranteed by Claim 2 (see its proof)that we can find W ⊆ F ( U ) ; the second condition in (iii) is then easy to satisfy, since F ( Z ) isa finite set. Now, we let U := U ∩ F − ( W ) , which is an open subset of B containing p . Then F : U − { p } → W − { q } is a biholomorphism. We let g : W − { q } → U − { p } denote its inverse.By the normality of q , we can assume that g is the restriction of some holomorphic map b g definedon some open set c W ⊂ C N , where c W contains W . Since g ◦ F | U −{ p } equals the identity map, b g ◦ F equals identity on U by continuity. Similarly F ◦ ( b g | W ) equals the identity on W . Therefore, q cannot be a singular point. (cid:3) By Claim 2 and Claim 3, we also see that T = Z = { } or ∅ . Therefore, F gives a holomorphicalgebraic branched covering map from B to G with a possible branch point at . This completesthe proof of the "only if" implication in Theorem 1.1. (cid:3) Remark 5.4.
We reiterate (see Remark 5.3 above) that in the above proof, the condition that dim V = 2 is only used in the second step where we apply the affirmative solution of the Ramadanovconjecture in C ([23], [8]).We shall now prove Corollaries 1.3 and 1.4. Proof of Corollary 1.3.
By Theorem 1.1, it follows that G can be realized as a finite ball quotient B / Γ by an algebraic map for some finite unitary group Γ with no fixed point on ∂ B . We mustprove that Γ = { id } . Suppose not. But, then G must have a singular point (see [41]), which is acontradiction. (cid:3) Proof of Corollary 1.4.
The algebraic case follows immediately from Corollary 1.3. Thus, we onlyneed to consider the rational case. First, as a consequence of the algebraic case, there exists analgebraic biholomorphic map f : G → B . It remains to establish that f is in fact rational. Thisfollows immediately from a result by Bell [5]. For the convenience of the readers, however, wesketch an independent proof here. Denote by K G and K B the Bergman kernels (now considered asfunctions) of G and B , respectively. By the transformation law, they are related as K G ( z, w ) = det (cid:0) J f ( z ) (cid:1) · K B (cid:0) f ( z ) , f ( w ) (cid:1) · det (cid:0) J f ( w ) (cid:1) = 2! π det (cid:0) J f ( z ) (cid:1) · det (cid:0) J f ( w ) (cid:1) · (cid:0) − f ( z ) · f ( w ) (cid:1) . (5.4)We may assume ∈ G by translating G if necessary and, by composing f with an automorphismof B , we may also assume f (0) = 0 . Thus, at w = 0 , we have K G ( z,
0) = 2! π det (cid:0) J f ( z ) (cid:1) · det (cid:0) J f (0) (cid:1) . It follows that(5.5) det (cid:0)
J f ( z ) (cid:1) = det (cid:0) J f (0) (cid:1) K G ( z, K G (0 , . In particular, this implies that K G ( z, = 0 for any z ∈ D . We evaluate (5.4) on the diagonal w = z and use (5.5) to obtain K G ( z, z ) = 2! π (cid:12)(cid:12) det (cid:0) J f (0) (cid:1)(cid:12)(cid:12) | K G ( z, | | K G (0 , | (cid:0) − k f ( z ) k (cid:1) . Taking the logarithm of both sides yields log K G ( z, z ) + 3 log (cid:0) − k f ( z ) k (cid:1) = log 2! π + log (cid:12)(cid:12) det (cid:0) J f (0) (cid:1)(cid:12)(cid:12) + log | K G ( z, | − log | K G (0 , | . For j = 1 , , we apply the derivative ∂∂z j to both sides and obtain K G ( z, z ) ∂K G ( z, z ) ∂z j − − k f ( z ) k X i =1 ∂f i ( z ) ∂z j f i ( z ) = 1 K G ( z, ∂K G ( z, ∂z j . Complexifying the above equation and evaluating it at w = 0 , after rearrangement, we obtain X i =1 ∂f i ∂z j (0) f i ( z ) = 13 K G ( z, ∂K G ∂z j ( z, − K G (0 , ∂K G ∂z j (0 , ! . Note this is a linear system for f ( z ) = ( f ( z ) , f ( z )) and the coefficient matrix J f (0) is non-singular.By solving this linear system for f , it is immediately clear that the rationality of K G implies thatof f . (cid:3) Remark 5.5.
Corollary 1.3 implies, in particular, that the Burns–Shnider domains in C (see page244 in [10]) cannot have algebraic Bergman kernels. In fact, this holds for any Burns–Shniderdomain in C n for n ≥ , which can be seen as follows. By Proposition 5.1, if the Bergman kernelwere algebraic, then the boundary would be Nash algebraic. While this can be seen to not be soby inspection, a contradiction would also be reached by the Huang–Ji Riemann mapping theorem[30] since the boundary of a Burns–Shnider domain is spherical while the domain itself is notbiholomorphic to the unit ball.6. Counterexample in higher dimension
In this section, we construct a − dimensional reduced Stein space G with only one normal singularityand compact, smooth strongly pseudoconvex boundary, realized as a relatively compact domain ina complex algebraic variety V in C . We will show that its Bergman kernel is algebraic, while G isnot biholomorphic to any finite ball quotient B n / Γ , which shows that Theorem 1.1 cannot hold inhigher dimensions.Let G be defined as G = (cid:8) w = ( w , w , w , w ) ∈ C : | w | + | w | + | w | + | w | < , w w = w w (cid:9) . Then G is a relatively compact domain in the complex algebraic variety(6.1) V = (cid:8) w ∈ C : w w = w w (cid:9) . Since G is a closed algebraic subvariety of B ⊂ C , G is a reduced Stein space. Note that isthe only singularity of V . Moreover it is a normal singularity as it is a hypersurface singularity ofcodimension 3 ( > ; see [42]). It is also easy to verify that G has smooth strongly pseudoconvexboundary in V . Proposition 6.1.
The boundary M = ∂G of G is homogeneous and non-spherical.Proof. Consider the product complex manifold CP × CP . For j = 1 , , let π j : CP × CP → CP be the projection map to the j -th component and let ( L , h ) → CP be the tautological line bundle L with its standard Hermitian metric h . We set the Hermitian line bundle ( L, h ) over CP × CP to be: ( L, h ) := π ∗ ( L , h ) ⊗ π ∗ ( L , h ) . We begin the proof with the following claim.
Claim 1.
Let ( L, h ) → CP × CP be as above and let S ( L ) → CP × CP be its unit circle bundle.Then M is CR diffeomorphic to S ( L ) by the restriction of biholomorphic map. Proof of Claim 1.
Note that the circle bundle S ( L ) → CP × CP can be written as(6.2) S ( L ) = (cid:26)(cid:16) λ ( ζ , z ) ⊗ ( ζ , z ) , [ ζ , z ] , [ ζ , z ] (cid:17) : [ ζ , z ] ∈ CP , [ ζ , z ] ∈ CP | λ | ( | ζ | + | z | )( | ζ | + | z | ) = 1 (cid:27) . Define F : L → C as(6.3) F (cid:16) λ ( ζ , z ) ⊗ ( ζ , z ) , [ ζ , z ] , [ ζ , z ] (cid:17) = (cid:16) λζ ζ , λz ζ , λζ z , λz z (cid:17) . Then the map F gives a biholomorphism that sends a neighborhood of S ( L ) in L to a neighborhoodof M in V ⊂ C . This proves the claim. (cid:3) Note that S ( L ) is homogeneous (see [14]) and non-spherical by Theorem 12 in [43]. Thus, M ishomogeneous and non-spherical. (cid:3) Proposition 6.2.
The Bergman kernel form K G of G is algebraic.Proof. Set(6.4)
Ω := (cid:8) ( λ, z ) = ( λ, z , z ) ∈ C : | λ | (1 + | z | )(1 + | z | ) < (cid:9) . Note that Ω is an unbounded domain with smooth boundary in C . Moreover, Ω has a rationalBergman kernel form K Ω (see Appendix 7 for a proof of this fact). Define the map F : C → C as F ( λ, z , z ) := ( λ, λz , λz , λz z ) , We note that F ( C ) is contained in V as defined by (6.1). And F is a holomorphic embedding on C − { λ = 0 } . Moreover, F (Ω) ⊂ G and F : e Ω := Ω − { λ = 0 } → e G := G − { w = 0 } is a biholomorphism. By Theorem 2.6, the Bergman kernel form K e Ω of e Ω is the restriction (pullback)of K Ω to e Ω . Thus, K e Ω is rational. By the transformation law (1.1), we have K e Ω = ( F, F ) ∗ K e G . This implies that K e G is algebraic (see the equivalent condition (c) in §2.1), and thus K G is alsoalgebraic by Theorem 2.6. (cid:3) Before we prove G is not biholomorphic to any finite ball quotient, we pause to study the followingbounded domain U in C : U := n ( w , w , w ) ∈ C : | w | + | w | ( | w | + | w | ) + | w w | < | w | o . Proposition 6.3.
The domain U has algebraic Bergman kernel and its boundary is non-sphericalat every smooth boundary point. Proof.
Let π : C → C be the projection map defined by π ( w , w , w , w ) := ( w , w , w ) . Let G be the closure of G in C . Then the image of G under the projection π is b U := π ( G ) = (cid:8) ( w , w , w ) ∈ C : | w | + | w | ( | w | + | w | ) + | w w | ≤ | w | , w = 0 (cid:9) ∪ (cid:8) (0 , w , w ) ∈ C : | w | + | w | ≤ , w w = 0 (cid:9) = (cid:8) ( w , w , w ) ∈ C : | w | + | w | ( | w | + | w | ) + | w w | ≤ | w | , | w | + | w | ≤ (cid:9) . Note that b U o = U and b U = U , where b U o denotes the interior of b U . But U = π ( G ) . On theother hand if we remove the variety { w = 0 } , then the projection map π : G − { w = 0 } → U is an algebraic biholomorphism. Consequently, by Theorem 2.6 the Bergman kernel form K U of U is algebraic. This proves the first part of the proposition.To prove the second part of the proposition (i.e., the non-sphericity), we observe that the boundary ∂U of U is given by ∂U = (cid:8) ( w , w , w ) ∈ C : | w | + | w | ( | w | + | w | ) + | w w | = | w | , w = 0 (cid:9) ∪ (cid:8) (0 , w , w ) ∈ C : | w | + | w | ≤ , w w = 0 (cid:9) = (cid:8) ( w , w , w ) ∈ C : | w | + | w | ( | w | + | w | ) + | w w | = | w | , | w | + | w | ≤ (cid:9) . Write ∂U = (cid:0) ∂U ∩ { w = 0 } (cid:1) ∪ (cid:0) ∂U ∩ { w = 0 } (cid:1) . Since the projection map π is a biholomorphism from G − { w = 0 } to b U − { w = 0 } , every point p ∈ ∂U ∩ { w = 0 } is a smooth point of ∂U , and, moreover, ∂U is strictly pseudoconvex andnon-spherical at p . We note that a defining function for ∂U near p is given by ρ = | w | + | w | ( | w | + | w | ) + | w w | − | w | . Furthermore, it is easy to verify that every other point q ∈ ∂U ∩ { w = 0 } is not a smooth boundarypoint of U . This proves the second part of the assertion. (cid:3) We are now ready to show that G is indeed a counterexample to the conclusion of Theorem 1.1 inthree dimensions. Proposition 6.4. G is not biholomorphic to any finite ball quotient.Proof. Seeking a contradiction, we suppose G is biholomorphic to a finite ball quotient B / Γ , where Γ ⊂ U ( n ) is a finite unitary group. We realize B / Γ as the image G ⊂ C N of B under the basicmap Q associated to Γ , where Q = ( p , · · · , p N ) : C → C N gives a proper map from B to G . Let F be a biholomorphism from G ∼ = B / Γ to G . Then there is an analytic variety W ⊂ G suchthat F : G − W → G − { w = 0 } is a biholomorphism . There also exists an analytic variety W such that W = Q − ( W ) and thus Q : B − W → G − W is proper and onto. Set f := π ◦ F ◦ Q : B − W → U = π ( G − { w = 0 } ) , where π is the projection defined in the proof of Proposition 6.3 and is a biholomorphism from G − { w = 0 } to U . Note that f is proper. Since U ⊂ C , we can write f as ( f , f , f ) . Claim 2.
There is a sequence { ζ i } ⊂ B − W with ζ i → ζ ∗ ∈ ∂ B − W such that f ( ζ i ) → p ∗ ∈ ∂U ∩ { w = 0 } . Proof of Claim 2.
Suppose not. Then for any { ζ i } ⊂ B n − W with ζ i → ζ ∗ ∈ ∂ B n − W , everyconvergent subsequence of f ( ζ i ) converges to some point in ∂U ∩ { w = 0 } . That is to say, if f ( ζ i k ) is convergent, then f ( ζ i k ) → . Note that U is bounded. Thus, f ( ζ i ) → for any { ζ i } ⊂ B − W with ζ i → ζ ∗ ∈ ∂ B − W . By a standard argument using analytic disks attached to ∂ B − W , wesee that f = 0 on B − W. This is a contradiction. (cid:3)
Let ζ i , ζ ∗ and p ∗ be as in Claim 2. Note that ζ ∗ is a smooth strictly pseudoconvex boundarypoint of B − W , and p ∗ is a smooth strictly pseudoconvex boundary point of U (see the proof ofProposition 6.3). It follows from [20] (see page 239) that f extends to a Hölder- continuous mapon a neighborhood of ζ ∗ in B . Since f is proper B − W → U , its extension to the boundaryis a (Hölder- ) continuous, nonconstant CR map sending a piece of ∂ B containing ζ ∗ to a pieceof ∂U containing p ∗ . By [39], f extends holomorphically to a neighborhood of ζ ∗ , since bothboundaries are real analytic (in fact, real-algebraic). Now, since f is non-constant and sends astrongly pseudoconvex hypersurface to another, it must be a CR diffeomorphism, which wouldmean that ∂U is locally spherical near p ∗ . This contradicts Proposition 6.1. (cid:3) We conclude this section by a couple of remarks.
Remark 6.5.
Since the Bergman kernel forms of G and U are algebraic, it follows from the proofof Theorem 1.1 in Section 5 (see Step 2) that the coefficients of the logarithmic term in Fefferman’sexpansions of K G and K V both vanish to infinite order at every smooth boundary point. Thereduced normal Stein space G gives the counterexample mentioned in Remark 1.2. The domain U ⊂ C establishes the following fact, which implies that the Ramadanov conjecture fails for non-smooth domains in higher dimension. There exists a bounded domain in C with smooth, real-algebraic boundary away from a 1-dimensionalcomplex curve such that every smooth boundary point is strongly pseudoconvex and non-spherical,while the coefficient of the logarithmic term in Fefferman’s asymptotic expansion of the Bergmankernel vanishes to infinite order at every smooth boundary point. Remark 6.6.
Using the same idea as in the above example, we can actually construct significantlymore general examples of higher dimensional domains in affine algebraic varieties V ⊂ C N withsimilar properties. Indeed, let X be a compact Hermitian symmetric space of rank at least . Write X = X × · · · × X t , t ≥ , where X , · · · , X t are the irreducible factors of X . Fix a Kähler -Einstein metric ω j on X j and let ( b L j , b h j ) be the top exterior product Λ n T , of the holomorphic tangent bundle over X j with themetric induced from ω j . Then there is a homogeneous line bundle ( L j , h j ) with a Hermitian metric h j such that its p j -th tensor power gives ( b L j , b h j ) , where p j is the genus of X j . (see [14] for moredetails).Let π j be the projection from X onto the j -th factor X j for ≤ j ≤ t . Define the line bundle L over X with a Hermitian metric h to be: ( L, h ) := π ∗ ( L , h ) ⊗ · · · ⊗ π ∗ t ( L t , h t ) . Let ( L ∗ , h ∗ ) be the dual line bundle of ( L, h ) . Write D ( L ∗ ) and S ( L ∗ ) for the associated unit discand unit circle bundle. The specific example above is the special case t = 2 and X = X = CP .Proceeding as in that example, one finds that there is a canonical way to map L ∗ to C N , for some N , induced by the minimal embedding of X into some complex projective space (see [15]). If wedenote this map L ∗ → C N by F (in the example above, the map F is as given by (6.3)), then F sends the zero section of L ∗ to the point and is a holomorphic embedding away from the zerosection. It follows that the image of D ( L ∗ ) under the map F is a domain G with a singular pointat . The boundary of G is given by the image of S ( L ∗ ) . It is not spherical since S ( L ∗ ) is not by[43]. Moreover, as the Bergman kernel form of D ( L ∗ ) is algebraic by [14], the Bergman kernel formof G is also algebraic by Theorem 2.6. 7. Appendix
In this section, we will prove the claim that the Bergman kernel of the domain Ω in C as defined in(6.4) is rational. This fact actually follows from a general theorem in [14] (see Theorem 3.3 in [14]and its proof). We include here a proof in this particular example for the convenience of readersand self-containedness of this paper. In fact, we shall compute the Bergman kernel of Ω explicitly(Theorem 7.3 below).Recall that(7.1) Ω := (cid:8) ( z, λ ) = ( z , z , λ ) ∈ C : | λ | (1 + | z | )(1 + | z | ) < (cid:9) . We let h ( z ) := (1 + | z | )(1 + | z | ) , and denote the defining function by ρ ( z, λ ) := | λ | (1 + | z | )(1 + | z | ) − . We recall that the Bergman space on Ω is defined as(7.2) A (Ω) := (cid:8) f ( z, λ ) is holomorphic in Ω : i Z Ω | f ( z, λ ) | dz ∧ dλ ∧ dz ∧ dλ < ∞ (cid:9) , and let(7.3) A m (Ω) := (cid:8) f ( z ) is holomrphic in C : λ m f ( z ) ∈ A (Ω) (cid:9) . Note that the L norm of λ m f ( z ) is given by k λ m f ( z ) k = i Z Ω | λ | m | f ( z ) | dz ∧ dλ ∧ dz ∧ dλ = Z z ∈ C (cid:16)Z | λ | Let f ( z, λ ) ∈ A (Ω) . If we fix z ∈ C , then λ is contained in the disc { λ ∈ C , | λ | < h ( z ) − } .By taking the Taylor expansion at λ = 0 , we obtain f ( z, λ ) = ∞ X j =0 a j ( z ) λ j , for | λ | < h ( z ) − , where each a j ( z ) is holomorphic on C . We shall first write k f ( z, λ ) k in terms of { a j ( z ) } ∞ j =0 . Wehave k f ( z, λ ) k = Z z ∈ C Z | λ | Let m ≥ . The reproducing kernel of A m (Ω) is (7.6) K ∗ m ( z, λ, w, τ ) = ( m + 1) m (2 π ) λ m τ m (1 + z w ) m − (1 + z w ) m − , where ( z, λ ) , ( w, τ ) are points in Ω .Proof. Denote z α = z α z α , for any multi-index α = ( α , α ) ∈ Z ≥ . By (7.5), since Ω is Reinhardt, it is easy to see that (cid:8) λ m z α : z α ∈ A ( C , h − ( m +1) ) (cid:9) forms an orthogonal basis of A m (Ω) . We shall compute the norm for each λ m z α . Using (7.4), wehave k λ m z α k = 2 πm + 1 Z C | z α | (1 + | z | ) − ( m +1) (1 + | z | ) − ( m +1) dz ∧ dz = 2 πm + 1 Z C | z | α (1 + | z | ) − ( m +1) i dz ∧ dz · Z C | z | α (1 + | z | ) − ( m +1) i dz ∧ dz = (2 π ) m + 1 Z ∞ r α (1 + r ) − ( m +1) dr · Z ∞ r α (1 + r ) − ( m +1) dr . By the elementary integral identity(7.7) Z ∞ r p r ) q dr = ( q − p − p !( q − , for any nonnegative integers p, q with q ≥ p + 2 , we get k λ m z α k = (2 π ) m + 1 ( m − α − m − α − α ! m ! if α , α ≤ m − , + ∞ otherwise . Thus, (cid:8) λ m z α k λ m z α k : 0 ≤ α , α ≤ m − (cid:9) is an orthonormal basis of A m (Ω) , and the reproducing kernelof A m (Ω) is given by K ∗ m ( z, λ, w, τ ) = X ≤ α ,α ≤ m − z α λ m w α τ m k z α λ m k = ( m + 1) m (2 π ) λ m τ m m − X α =0 (cid:18) m − α (cid:19) z α w α m − X α =0 (cid:18) m − α (cid:19) z α w α = ( m + 1) m (2 π ) λ m τ m (1 + z w ) m − (1 + z w ) m − . (cid:3) Now we are ready to compute the Bergman kernel form of Ω . Theorem 7.3. The Bergman kernel form of the domain Ω ⊂ C in (7.1) is given by K Ω ( z, λ, w, τ ) = iK ∗ ( z, λ, w, τ ) dz ∧ dλ ∧ dw ∧ dτ , where K ∗ ( z, λ, w, τ ) = ∞ X m =1 ( m + 1) m (2 π ) λ m τ m (1 + z w ) m − (1 + z w ) m − . It can be written in terms of the complexified defining function ρ ( z, λ, w, τ ) = λτ (1 + z w )(1 + z w ) − as K ∗ ( z, λ, w, τ ) = 1(2 π ) (cid:16) λτρ ( z, λ, w, τ ) + 6 λτρ ( z, λ, w, τ ) (cid:17) . Proof. By Lemma 7.1, we immediately get the reproducing kernel of A (Ω) by adding up thereproducing kernels of A m (Ω) for all m . Since A (Ω) = { } , we obtain K ∗ ( z, λ, w, τ ) = ∞ X m =1 K ∗ m ( z, λ, w, τ )= ∞ X m =0 ( m + 2)( m + 1) (2 π ) λ m +1 τ m +1 (1 + z w ) m (1 + z w ) m . It remains to write K ∗ ( z, λ, w, τ ) in terms of the defining function ρ ( z, λ, w, τ ) . We use the Taylorexpansion of / (1 − x ) j +1 for ≤ j ≤ to obtain − ρ ( z, λ, w, τ )) j +1 = 1(1 − (1 + z w )(1 + z w ) λτ ) j +1 = ∞ X m =0 (cid:18) m + jj (cid:19) (1+ z w ) m (1+ z w ) m λ m τ m . Note that ( m + 2)( m + 1) is a polynomial in m of degree . Since { (cid:0) m + jj (cid:1) } j =0 is a basis ofpolynomials in m with degree ≤ , we can write ( m + 2)( m + 1) = X j =0 a j (cid:18) m + jj (cid:19) . One can check that the coefficients are given by a = a = 0 , a = − and a = 6 . Therefore, K ∗ ( z, λ, w, τ ) = 1(2 π ) ∞ X m =0 3 X j =0 a j (cid:18) m + jj (cid:19) λ m +1 τ m +1 (1 + z w ) m (1 + z w ) m = 1(2 π ) X j =0 a j λτ ( − ρ ( z, λ, w, τ )) j +1 , and the result follows. (cid:3) References [1] Toby N. Bailey, Michael G. Eastwood, and C. Robin Graham. Invariant theory for conformal and CR geometry. Ann. of Math. (2) , 139(3):491–552, 1994.[2] M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild. 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