Области Бергмана-Гартогса и их автоморфизмы

Аннотация

Для областей Картана-Гартогса, а также для областей Бергмана-Гартогса находятся их группы автоморфизмов - соответственно для случаев, когда база есть произвольная ограниченная симметрическая область и общая ограниченная однородная область.

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Cartan-Hartogs domains (see definition below) are in general non homogeneous, but their automorphism group acts transitively on the real hypersurfaces of a one parameter family. The exact automorphism group has been determined by Ahn Heungju, Byun Jisoo, Park Jong-do [1] when the base of the Cartan-Hartogs domain is a bounded symmetric domain of classical type. Their method, using the Wong-Rosay theorem, may be extended to the case where the base is any bounded symmetric domain. The result holds also for BERGMAN-HARTOGS DOMAINS AND THEIR AUTOMORPHISMS 317 “Bergman-Hartogs domains” which are defined in the same way as Cartan-Hartogs domains, with a base which is a general bounded homogeneous domain. 1. Definitions and notations 1.1. Bergman kernel. Let be a bounded domain in a complex space V of dimension d . Let V be oriented by a translation invariant volume form ! . Let H( ) = f 2 O( ) j kfk2 := Z jf(z)j2 !(z) < +1 be the Bergman space of . Then H( ) , with the scalar product (u j v) := Z u(z) v(z) !(z) is a Hilbert space of holomorphic functions (that is, H( ) is a Hilbert space and the inclusion H( ) ,! O( ) is continuous). For z 2 , let K ;z 2 H( ) such that f(z) = (f j K ;z) for all f 2 H( ) . The Bergman kernel of is the reproducing kernel K(z; t) = K (z; t) = K ;z(t) of H( ) . Denote K(z) = K (z) := K (z; z) = kK ;zk2 (which is also called Bergman kernel of ). If g : ! is a holomorphic automorphism of , then K (gz) = K (z) jJg(z)j2 ; where Jg(z) is the complex Jacobian of g at z . 1.2. Cartan domains. Let be an irreducible complex symmetric domain of non compact type (“Cartan domain”), realized as the spectral unit ball of a simple Hermitian positive Jordan triple V . We denote by (a; b; r) the numerical invariants of V ; by the genus of V : = 2 + a(r 1) + b and by N(x; y) the generic norm of V (which is an irreducible polynomial of bidegree (r; r) ). The Bergman kernel of is then K (z) = K (0)N(z; z) : 318 G. Roos 1.3. Cartan-Hartogs domains. D e f i n i t i o n 1. For a real number > 0 and an integer N > 0 , let e be the Hartogs type domain defined by e = e ( ;N) := (z;Z) 2 - CN j kZk2 < N(z; z) : The domain e ( ;N) is called Cartan-Hartogs domain. Cartan-Hartogs domains have been introduced by Weiping Yin and G. Roos in 1998. They generalize various domains like complex ellipsoids (Thullen domains). 1.4. Bergman-Hartogs domains. Let be a bounded complex domain. Let c > 0 be a positive real number and N > 0 an integer. D e f i n i t i o n 2. The Bergman-Hartogs domain b (c;N) is b (c;N) := (z;Z) 2 - CN j kZk2 < K (z) c ; where CN is endowed with the standard Hermitian structure. The Cartan-Hartogs domain e ( ;N) is linearly equivalent to the Bergman-Hartogs domain: e ( ;N) ' b ( = ;N): 1.5. Example: Thullen domains. Let V = Cn be the standard Hermitian vector space, with scalar product (z j t) = Pn j=1 zjtj and Hermitian norm kzk2 = (z j z) . The associated symmetric domain is the Hermitian unit ball = Bn of V . The genus of is g = n + 1 . The generic norm is N(z; t) = 1 (z j t) : The Cartan-Hartogs domain e ( ;N) is then e ( ;N) = n (z;Z) 2 V - CN j kzk2 + kZk2= < 1 o : These domains are called Thullen domains and also known as complex ellipsoids, or complex ovals, or egg domains. Let = Bn be the Hermitian unit ball of V = Cn . For = 1 , e ( ;N) is the Hermitian unit ball Bn+N of Cn+N and is homogeneous. P r o p o s i t i o n 1. The Thullen domain e ( ;N) is biholomorphic to Bn+N if and only if = 1 . Proof. Let f : Bn+N ! e ( ;N) be a biholomorphism. By composing f with a suitable automorphism of Bn+N , we may assume that f(0) = 0 . As Bn+N is a bounded circled domain and e ( ;N) is bounded, a lemma of H. Cartan implies that f is linear. It is then easy to check that the image of the boundary of Bn+N by f is the boundary of e ( ;N) if and only if = 1 . BERGMAN-HARTOGS DOMAINS AND THEIR AUTOMORPHISMS 319 2. Boundary and automorphisms 2.1. Strictly pseudoconvex boundary points. Let be a bounded complex domain. Let c > 0 be a positive real number and N > 0 an integer. Let X : - CN ! (0;+1) be defined by X(z;Z) := K (z)c kZk2 : P r o p o s i t i o n 2. The points of @0 b (c;N) := (z;Z) 2 - CN j kZk2 = K (z) c are strictly pseudoconvex boundary points of b (c;N) . This property has been noticed by Ahn Heungju, Byun Jisoo, Park Jong-do [1] when is a bounded symmetric domain of classical type, and proved by them case-by-case for symmetric domains of classical type. Proof. Consider the function lnX(z;Z) = c lnK (z) + ln kZk2 : Its Levi form at (z;Z) is L(z;Z)((w1;W1); (w2;W2)) = @(w1;W1)@(w2;W2) lnX(z;Z) = c @w1@w2 lnK (z) + @W1 @W2 ln kZk2 : Then @w1@w2 lnK (z) is the Bergman metric hz(w1;w2) of at z and @W @W ln kZk2 = kZk2 kWk2 j(W j Z)j2 kZk4 : The complex tangent hyperplane H(z;Z) to @0 b (c;N) = flnX(z;Z) = 0g at (z;Z) is H(z;Z) = (w;W) j c h@ lnK (z);wi + (W j Z) kZk2 = 0 : For (w;W) 2 H(z;Z) , L(z;Z) ((w;W); (w;W)) = hz(w;w) + kZk2 kWk2 j(W j Z)j2 kZk4 > 0: If L(z;Z)((w;W); (w;W)) = 0 , then w = 0 , which implies (W j Z) = 0 , hence L(z;Z)((w;W); (w;W)) = kZk 2 kWk2 and W = 0 . 2.2. Automorphisms of Cartan-Hartogs domains. Let be a bounded irreducible circled symmetric domain in V , with generic norm N , genus and Bergman kernel K(z; t) . 320 G. Roos Let e be the Cartan-Hartogs domain ( > 0 , N > 1 ) e = e ( ;N) = (z;Z) 2 - Cm j kZk2 < N(z; z) : Define X : e ! [0; 1) X(z;Z) = kZk2 N(z; z) : 2.2.1. Boundary of Cartan-Hartogs domains. The boundary of the Cartan domain is a disjoint union of locally closed manifolds @ = ar j=1 @j : The boundary of the Cartan-Hartogs domain e = e ( ;N) is @e = @0 e t (@ - f0g) = ar j=0 @j e ; with @0e = (z;Z) 2 - CN j kZk2 = N(z; z) ; @j e = @j - f0g (1 6 j 6 r): The points of @0e are strictly pseudoconvex boundary points. 2.2.2. Restricted automorphisms of Cartan-Hartogs domains. Denote by Aut0e the subgroup of automorphisms of e which leave X invariant. P r o p o s i t i o n 3. The group Aut0e consists of all : (z;Z) 7! ( (z); (z)U(Z)) , where 2 Aut , U : CN ! CN is special unitary and satisfies j (z)j2 = N( z; z) N(z; z) : For 2 Aut , let z0 = 1(0) ; then the functions satisfying this condition are the functions (z) = ei N (z0; z0) =2 N (z; z0) : The orbits of Aut0e are the level sets = fX = j 2 [0; 1)g . See [3]. 2.2.3. The automorphism group of a Cartan-Hartogs domain. The following result is proved by Ahn Heungju, Byun Jisoo, Park Jong-do [1] when is a symmetric domain of classical type. BERGMAN-HARTOGS DOMAINS AND THEIR AUTOMORPHISMS 321 Theorem 1. (1) The Cartan-Hartogs domain e ( ;N) is homogeneous if and only if is of type I1;n (that is, an Hermitian ball of dimension n ) and = 1 . Then e (1;N) is symmetric of type I1;n+m . (2) If e = e m( ) is not homogeneous, then Aut e = Aut0 e . The proof relies on the Wong-Rosay theorem: Theorem. [2] Let D be a bounded complex domain and 0 a strictly pseudoconvex C2 boundary point of D. If there exist an interior point x 2 D and a sequence (Tk) of holomorphic automorphisms of D, such that Tk(x) ! 0 , then D is biholomorphic to an Hermitian ball. The proof of Ahn-Byun-Park relies on the strict pseudoconvexity of @0e ( ;N) , so this proof is valid for any irreducible symmetric domain . Proof. Let 2 Aut e ( ;N); zj 2 ! 2 @ : There exist gj 2 Aut such that gj(0) = zj ; egj 2 Aut e ( ;N) such that egj(0; 0) = (zj ; 0): Then ( (zj ; 0)) = (Tj(0; 0)) ; Tj = egj 2 Aut e ( ;N): The main steps of the proof are then If (zj) has a subsequence such that ( (zj ; 0)) converges to a point 0 2 @0e ( ;N) , then e ( ;N) is biholomorphic to an Hermitian ball by the Wong-Rosay theorem. e ( ;N) is biholomorphic to an Hermitian ball if and only if is an Hermitian ball and = 1 . If e ( ;N) is not an Hermitian ball, then ( - f0g) = - f0g for all 2 Aut e ( ;N) . Let 2 Aut e ( ;N) . If ( - f0g) = - f0g , then 2 Aut0 e ( ;N) . 2.3. Bergman-Hartogs domains. From now on, we assume that is a bounded homogeneous domain. Let G denote its automorphism group. 2.3.1. Restricted automorphisms. For g 2 G, let eg 2 Aut b (c;N) be defined by eg(z;Z) := (gz; Jg(z)cZ) : 322 G. Roos Note that the function z 7! Jg(z)c is in general not unique and is defined up to multiplication by a power of exp (2i c) . The group eG = feg j eg(z;Z) = (gz; Jg(z)cZ) ; g 2 Gg is a covering of G and a subgroup of Aut b (c;N) . D e f i n i t i o n 3. The restricted automorphism group of b (c;N) is Aut0 b (c;N) = n 2 Aut b (c;N) j X = X o ; where X(z;Z) := K (z)c kZk2 . P r o p o s i t i o n 4. Let 2 Aut b (c;N) . The following properties are equivalent: 1. 2 Aut0 b (c;N) ; 2. ( - f0g) = - f0g ; 3. there exist g 2 G and U 2 U(N) such that (z;Z) = (gz; Jg(z)cUZ) . 2.3.2. The automorphism group of a Bergman-Hartogs domain. Theorem 2. (1) The Bergman-Hartogs domain b (c;N) is homogeneous if and only if is an Hermitian ball of dimension n and c = 1 n + 1 . Then b 1 n + 1 ;N is an Hermitian ball of dimension n + N . (2) In all other cases, Aut b (c;N) = Aut0 b (c;N) . The main steps of the proof are the same than for Cartan-Hartogs domains: If (zj) has a subsequence such that ( (zj ; 0)) converges to a point 0 2 @0 b (c;N) , then b (c;N) is biholomorphic to an Hermitian ball by the Wong-Rosay theorem. b (c;N) is biholomorphic to an Hermitian ball if and only if is an Hermitian ball and c = 1 n + 1 : If b (c;N) is not an Hermitian ball, then ( - f0g) = - f0g for all 2 Aut b (c;N) . Let 2 Aut b (c;N) . If ( - f0g) = - f0g , then 2 Aut0 b (c;N) .