3 Bergman Function and Moduli Space of Complete Reinhardt Domains

Geometry

Stephen Yauand Huaiqing Zuo⁺

Mathematics Subject Classification (2010) 32V25, 32S25, 32T15, 32A07

1 Introduction

CR manifolds are abstract models of real hypersurfaces in complex spaces. The abstract definition of the boundary as a CR structure on a complex manifold is essentially in Cartan [7]. For more detail, see [24, 25]. Strongly pseudoconvex CR manifolds have rich geometric and analytic structures. Namely, there is an intrinsic pseudo conformed geometry for which complete local invariants have been obtained, see for example [8,14,41], as well as a deep analysis of the @b

complex, see for example [12,22,23,46]. The harmonic theory for the@bcomplex on compact strongly pseudoconvexCR manifolds was developed by Kohn [21].

Using this theory, Boutet de Monvel [4] proved that ifX is a compact strongly pseudoconvex CR manifold of dimension 2n 1, n 3, then there exist C¹

This work is supported by NSFC (No. 11531007)

+This work is supported by NSFC (No. 11401335 and No. 11531007) the Tsinghua University Initiative Scientific Research Program

S. Yau ()

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China e-mail:[email protected]

H. Zuo

Yau Mathematical Center, Tsinghua University, Beijing, 100084, P. R. China e-mail:[email protected]

© Springer International Publishing Switzerland 2015

J.E. Fornæss et al. (eds.),Complex Geometry and Dynamics, Abel Symposia 10, DOI 10.1007/978-3-319-20337-9_10

227

functionsf₁; : : : ;fN onX such that each@bfj D 0andf D .f₁; : : : ;fN/defines an embedding ofX in C^N. Thus, any compact strongly pseudoconvexCRmanifold of dimension 5can beCRembedded in some complex Euclidean space. On the other hand, 3-dimensional strongly pseudoconvex compact orientableCR-manifolds are not necessarily embeddable. The first example is due to Andreotti according to [37]. This example also appeared in the list of homogeneous structures of Cartan although the embeddability question was not addressed. Nirenberg [35] first proved that 3-dimensionalCRmanifolds might not be locally embeddable. Jacobowitz and Treves [19,20] showed that in fact non-embeddableCR structures are, in some sense, dense in the space of CR-structures over a 3-dimensional manifold. The theory of harmonic integrals on strongly pseudoconvexCR structures over small balls was due to Kuranishi [23]. Using this theory, Kuranishi [23] proved that any strongly pseudoconvexCRmanifold of dimension2n1withn5can be locally CRembedded as a real hypersurface in Cⁿ. Forn D 4, Akahori [1] proved that Kuranishi’s local embedding theorem is also true. However, the 5-dimensional case of local embeddability ofCRmanifolds remains open.

Throughout this paper, our CR manifolds are always assumed to be compact orientable and embeddable in someC^N. By a beautiful theorem of Harvey and Lawson [16,17], these CR manifolds are the boundaries of subvarieties in C^N. This allowed the first author [46] to relateCRgeometry and algebraic geometry of singularities for the first time. The purpose of this paper is to discuss the interplay betweenCRgeometry and algebraic geometry. Our paper is organized as follows.

In Sect.2, we shall recall the basic notion ofCRgeometry. In Sect.3, we show how to use the Bergman function of the first author to give canonical construction of moduli space for complete Reinhardt domains. In Sect.4, we use algebraic geometry to study the complex Plateau problem. In Sect.5, we study the minimal embedding dimension of compactCRmanifolds in complex Euclidean space. Finally in Sect.6, we study invariants of compact strongly pseudoconvexCRmanifolds arising from geometry of singularities.

2 Preliminary

Definition 2.1 LetXbe a connected orientable manifold of real dimension2n1.

ACR structure onX is an .n1/-dimensional subbundleS of the complexified tangent bundleCTXsuch that

(1) S\SD f0g

(2) IfL;L⁰are local sections ofS, then so isŒL;L⁰.

A manifold with aCRstructure is called aCRmanifold. There is a unique subbundle Hof the tangent bundleT.X/such thatCHDS˚S. Furthermore, there is a unique homomorphismJWH!Hsuch thatJ² D 1andSD fviJvWv2Hg. The pair.H;J/is called the real expression of theCRstructure.

Definition 2.2 LetL₁; : : : ;Ln1be a local frame ofS. ThenL₁; : : : ;Ln1is a local frame ofS and one may choose a local section N ofTX such that L₁; : : : ;Ln1, L₁; : : : ;Ln1;Nis a local frame ofCTX. The matrix.cij/defined by

ŒLi;LjD†a^kijLkC†b^kijLkCp 1cijN is Hermitian and is called the Levi form ofX.

Proposition 2.1 The number of non-zero eigenvalues and the absolute value of the signature of the Levi form.cij/at each point are independent of the choice of L₁; : : : ;Ln1;N.

Definition 2.3 The CR manifoldXis called strongly pseudoconvex if the Levi form is definite at each point ofX.

Theorem 2.2 (Boutet de Monvel [4]) If X is a compact strongly pseudoconvex CR manifold of dimension.2n1/and n3, then X is CR embeddable inC^N.

Although there are non-embeddable compact 3-dimensionableCRmanifolds, in this paper allCRmanifolds are assumed to be embeddable in complex Euclidean space. The following theorem linksCRgeometry and algebraic geometry together.

Theorem 2.3 (Harvey-Lawson [16,17]) For any compact connected embeddable CR manifold X, there is a unique complex variety V inC^Nfor some N such that the boundary of V is X and V has only normal isolated singularities.

3 Bergman Function and Moduli Space of Complete Reinhardt Domains

Recall that a complex manifold M is called strictly pseudoconvex if there is a compact setBinM, and a continuous real valued functiononM, which is strictly plurisubharmonic outsideB and such that for each c 2 R, the set Mc D fx 2 MW.x/ <cgis relatively compact inM. Note that a strictly pseudoconvex complex manifold is a modification of a Stein space at a finite many points.

LetVbe a Stein variety of dimensionn>2inC^Nwith only irreducible isolated singularities. We assume that@V is a smoothCR manifold. LetWM ! V be a resolution of singularity withEas an exceptional set. We shall define thek-th order Bergman functionB^._M^k/.z/onMwhich is a biholomorphic invariant ofM.

Definition 3.1 LetF(respectively,Fk) be the set of allL²integrable holomorphic n-forms ‰ onM (respectively, vanishing at least thek-th order on the exception setE of M). Let fwjg (respectively,fw^._j^k/g) be a complete orthonormal basis of F(respectively,Fk). The Bergman kernel (respectively Bergman kernel vanishing

on the exceptional set of k-th order) is defined to be K.z/ D P

wj.z/^ wj.z/

(respectively,K^.k/.z/DP

w^._j^k/.z/^w^._j^k/.z/).

Lemma 3.1 F=Fkis a finite dimensional vector space.

Lemma 3.2 Bergman kernel vanishing on the exceptional set of k-th order K^.k/.z/ is independent of the choice of the complete orthonormal basis of Fkand K^.k/.z/is invariant under biholomorphic maps.

Definition 3.2 LetM be a resolution of a Stein varietyV of dimensionn > 2 in C^Nwith only irreducible isolated singularity at the origin. Thek-th order Bergman functionB^._M^k/onMis defined to beK_M^.k/=KM.

Theorem 3.1 B^._M^k/ is a global function defined on M which is invariant under biholomorphic maps. Moreover, B^._M^k/ is nowhere vanishing outside the exceptional set of M. If the canonical bundle is generated by its global sections in a neighbor- hood of the exceptional set, then the zero set of the k-th order Bergman function B^._M^k/ is precisely the exceptional set of M.

Theorem 3.2 Let M be a strictly pseudoconvex complex manifold of dimension n > 2 with exceptional set E. Let A be a compact submanifold contained in E.

LetWM₁ !M be the blow up of M along A. Then we have K^._M^k₁^/.z/D K_M^.k/.z/

and KM₁.z/DKM.z/. Consequently B^._M^k/₁.z/DB^._M^k/.z/.

LetiWMi!V,iD1; 2, be two resolutions of singularities ofV. By Hironaka’s theorem [18], there exists a resolutionQW QM !V of singularities ofVsuch thatMQ can be obtained fromMi,i D 1; 2, by successive blowing up along submanifolds in exceptional set. In view of Theorems3.1and3.2, the following definition is well defined if the canonical bundle is generated by its global sections in a neighborhood of the exceptional set.

Definition 3.3 Let V be a Stein variety in C^N with only irreducible isolated singularities. LetWM ! V be a resolution of singularities of V such that the canonical bundle is generated by its global sections in a neighborhood of the exceptional set. Define thek-th order Bergman functionB^._V^k/onV to be the push forward of thek-th order Bergman functionB^._M^k/by the map.

Theorem 3.3 Let V be a Stein variety in C^N with only irreducible isolated singularities. Assume that there exists a resolution M of singularities of V such that the canonical bundle is generated by its global sections in a neighborhood of the exceptional set. Then the k-th order Bergman function B^._V^k/on V is invariant under biholomorphic maps and B^._V^k/vanishes precisely on the singular set of V.

For the convenience of the readers, we recall the following two important theorems.

Theorem 3.4 ([13]) A biholomorphic mapping between two strictly pseudoconvex domains is smooth up to boundary and the induced boundary mapping gives a CR- equivalence between the boundaries.

Theorem 3.5 ([40]) Two n-dimensional bounded Reinhardt domains D₁ and D₂ are mutually equivalent if and only if there exists a transformation W Cⁿ !Cⁿ given by zi7!riz_.i/.ri> 0;iD1; ;n andbeing a permutation of the indices i) such that.D₁/DD₂.

The following Proposition 3.1 tells us how to use singularity structures to distinguishCRstructures.

Proposition 3.1 ([50]) Let X₁, X₂ be two strictly pseudoconvex CR manifolds of dimension2n1which bound varieties V₁, V₂respectively inC^Nwith only isolated normal singularities. IfˆWX₁ !X₂is a CR-isomorphism, thenˆcan be extended to a biholomorphic map from V₁to V₂.

In view of the above Proposition3.1, ifX₁andX₂are two strictly pseudoconvex CRmanifolds which bound varietiesV₁ andV₂respectively with non-isomorphic singularities, then X₁ and X₂ are not CR equivalent. Therefore to study the CR equivalence of two strictly pseudoconvexCRmanifolds X₁ andX₂, it remains to consider the case when X₁ and X₂ are lying on the same varietyV. It is known that the global invariant Bergman function ofk-th order can be used to study the CRequivalence problem of smooth CR manifolds lying on the same variety. As an example, we shall show explicitly howCRmanifolds varies in the An-variety VQnD f.x;y;z/2C³Wf.x;y;z/Dxyz^nC1D0g. An explicit resolutionW QQ Mn! QVn

can be given in terms of coordinate charts and transition functions as follows:

Coordinate charts: WQkDC²D f.uk; vk/g; kD0; 1; ;n:

Transition functions:

8<

:

ukC1D 1 vk

vkC1Dukvk2 or 8<

:

ukDukC12vkC1

vkD 1 ukC1

Resolution map: .uQ k; vk/D

u^kC_k ¹v^kk;u^nk_k v^nCk ^1k;ukvk or .x;y;z/D.u₀;uⁿ₀v₀^nC1;u₀v0/D D.unnC1vnⁿ; vn;unvn/ Exceptional set: ED Q¹.0/DCkD fuk1D0g [ fvkD0g;

kD1; ;n:

From now on, we suppose V to be a bounded complete Reinhardt domain in VQn (cf. Definition 3.5). Then let M D Q¹.V/ D [ⁿ_kD0Wk, where Wk D Q¹.V/\ QWk; k D 0; 1; ;n:Observe that under WD QjMWM ! V,W₀nC₁ is mapped biholomorphically ontoVny-axis. In particularMnW₀is of measure zero

in the obvious sense. Hence, we may compute integrals onM using the .u0; v0/ coordinate on the chartW₀alone.

The following proposition is a general consequence of the proof of Proposi- tion 3.2 of [50].

Proposition 3.2 ([10]) In the above notations, let˛ˇ D u^˛₀v^ˇ₀du₀^dv0,˛; ˇ D 0; 1; 2; : : :. Then

n

k˛ˇkM˛ˇ : ˛ > _nCⁿ₁ˇo

is a complete orthonormal base of F and n

k˛ˇ˛ˇkMW˛> _nCⁿ₁ˇand˛>k o

is a complete orthonormal base of Fk. Therefore the Bergman kernel vanishing on the exceptional set of k-th order K_M^.k/and the Bergman kernel KMare given respectively by:

K_M^.k/.u₀; v0/D‚^.M^k/du₀^dv0^du₀^dv0

where

‚^._M^k/D X

˛>nC1ⁿ ˇ

˛>k

ju₀j^2˛jv0j^2ˇ k˛ˇk²_M ;

and

KM.u₀; v0/D 0 BB

@ 1

k00k²_M C X

˛>nC1ⁿ ˇ 16˛6k1

ju₀j^2˛jv0j^2ˇ k˛ˇk²_M C‚^.M^k/

1 CC

Adu₀^dv0^du₀^dv0:

The following results generalize Theorem 3.3 in [50].

Theorem 3.6 ([10]) In the above notations, the k-th order Bergman function for the strongly pseudoconvex complex manifold M is given by

B^._M^k/.u0; v0/D 0 ‚^.M^k/

BB

@ 1

k₀₀k²_M C X

˛>nC1ⁿ ˇ

˛>1

ju₀j^2˛jv0j^2ˇ k_˛ˇk²_M

1 CC A

The k-th order Bergman function for the variety is given by

B^._V^k/.x;y/D ‚^._V^k/ 1

k₀₀k²_M C‚^.1/V

;

where

‚^.V^k/D X

˛>nC1ⁿ ˇ

˛>k

jxj^2˛n2C1nˇjyj^n2ˇC1 k˛ˇk²_M

Definition 3.4 An open subset D Cⁿ is a complete Reinhardt domain if, whenever.z1; ;zn/ 2 Dthen.1z₁; ; nzn/ 2 Dfor all complex numbersj

withjjj61.

It is well known thatVQn D f.x;y;z/ 2 C³ W xy D z^nC1gis the quotient ofC² by a cyclic group of ordernC1, i.e.ı:.z₁;z₂/D.ız₁; ıⁿz₂/, whereıis a primitive .nC1/-th root of unit. The quotient map W C² ! QV is given by.z₁;z₂/ D .z^nC₁ ¹;z^nC₂ ¹;z₁z₂/.

Definition 3.5 An open setV in theAn-varietyVQn D f.x;y;z/2C³ W xyD z^nC1g is called a complete Reinhardt domain if¹.V/is a complete Reinhardt domain inC².

Theorem 3.7 ([10]) Let Vi, iD1; 2, be two bounded complete Reinhardt domains in An-varietyVQnD f.x;y;z/2C³WxyDz^nC1g. Let

g^{.˛; ˇ/} D k₁₀k^{˛nC1n ˇ}kn;nC1k^nC1ˇ k˛ˇkk00k^˛nnC11ˇ1 : If V₁is biholomorphic to V₂, then

^{.˛; ˇ/}WDg^{.˛; ˇ/}g^{.n˛.n1/ˇ;.nC1/˛nˇ/}; ^{.˛; ˇ/}WDg^{.˛; ˇ/}Cg^{.n˛.n1/ˇ;.nC1/˛nˇ/};

^.˛;p;q/WD.g^.˛;p/g^{.n˛.n1/p;.nC1/˛np/}/.g^.˛;q/g^{.n˛.n1/q;.nC1/˛nq/}/ and

!^{.˛1; ˛2;p1;p2/}WD.g^.˛1;p1/g^{.n˛1.n1/p1;.nC1/˛1np1/}/

.g^.˛2;p2/g^{.n˛2.n1/p2;.nC1/˛2np2/}/;

where

˛>1; ˛> n

nC1ˇ; 06p; q6 nC1

n ˛

;p¤q;

06pi6 nC1

n ˛i

; ˛i>1; ˛₁ ¤˛₂;iD1; 2;

are all invariants, i.e.

V^.˛;ˇ/₁ DV^.˛;ˇ/₂ ; V^.˛;ˇ/₁ DV^.˛;ˇ/₂ ; ^.˛;V₁^p;q/D^.˛;V₂^p;q/;

!V^.˛₁^{1; ˛2;p1;p2/}D!^.˛V₂^{1; ˛2;p1;p2/}:

The following Theorem says that these invariants in Theorem3.7 determine completely the Bergman function up to automorphisms ofAn-variety.

Theorem 3.8 ([10]) Let Vi, iD 1; 2, be two bounded complete Reinhardt strictly pseudoconvex (respectively C^!-smooth pseudoconvex) domains inVQnD f.x;y;z/2 C³: xyDz^nC1}. If

V^.˛;ˇ/₁ DV^.˛;ˇ/₂ ; V^.˛;ˇ/₁ DV^.˛;ˇ/₂ ; ^.˛;V₁^p;q/D^.˛;V₂^p;q/;

!V^.˛₁^{1; ˛2;p1;p2/}D!^.˛V₂^{1; ˛2;p1;p2/}; where

˛>1; ˛> n

nC1ˇ; 06p; q6 nC1

n ˛

;p¤q;

06pi6 nC1

n ˛i

; ˛i>1; ˛₁ ¤˛₂;iD1; 2;

then there exists an automorphism‰D. ₁; ₂; ₃/of An-varietyVQnD f.x;y;z/2 C³WxyDz^nC1ggiven by either

. 1; 2; 3/D k10kM₂

k00kM₂

k00kM₁

k10kM₁

x; kn;nC1kM₂

k00kM₂

k00kM₁

kn;nC1kM₁

y;k11kM₂

k00kM₂

k00kM₁

k11kM₁

z

; or

. 1; 2; 3/D k₁₀kM₂

k₀₀kM₂

k₀₀kM₁

kn;nC1kM₁

y; kn;nC1kM₂

k₀₀kM₂

k₀₀kM₁

k₁₀kM₁

x;k₁₁kM₂

k₀₀kM₂

k₀₀kM₁

k₁₁kM₁

z

:

such that‰sends V₁to V₂.

As an immediate corollary of Theorem 3.8 above, we have the following theorem.

Theorem 3.9 ([10]) The moduli space of bounded complete Reinhardt strictly pseudoconvex (respectively C^!-smooth pseudoconvex) domains in An-varietyVQn D f.x;y;z/2C³ WxyDz^nC1gis given by the image of the mapˆW fV WV a bounded complete Reinhardt strictly pseudoconvex (respectively C^!-smooth pseudoconvex) domain in VQng ! R¹, where the component function of ˆ are the invariant functions

^.˛;ˇ/; ^.˛;ˇ/; ^.˛;p;q/; !^{.˛1; ˛2;p1;p2/};

˛>1; ˛> n

nC1ˇ; 06p; q6 nC1

n ˛

;p¤q;

06pi6 nC1

n ˛i

; ˛i>1; ˛1 ¤˛2;iD1; 2:

defined in Theorem3.7.

The following theorem says that the biholomorphic equivalence problem for bounded complete Reinhardt domains inAn-varietyVQn is the same as the biholo- morphic equivalence problem for the corresponding bounded complete Reinhardt domains inC².

Theorem 3.10 ([10]) Let W C² ! QVn D f.x;y;z/ 2 C³ W xy D z^nC1gbe the quotient map given by.z₁;z₂/ D .z^nC₁ ¹;z^nC₂ ¹;z₁z₂/. Let Vi, i D 1; 2, be bounded complete Reinhardt domains inVQnsuch that WiWD ¹.Vi/;iD1; 2;are bounded complete Reinhardt domain inC². Then V₁is biholomorphic to V₂if and only if W₁ is biholomorphic to W₂. In particular, V₁is biholomorphic to V₂if and only if there exists a biholomorphismˆWV₁ !V₂given byˆ.x;y;z/D.a^nC1x;b^nC1y;abz/or ˆ.x;y;z/D.a^nC1y;b^nC1x;abz/where a;b> 0.

As a corollary of Theorems3.10and3.9, we have the following theorem.

Theorem 3.11 ([10])

(1) LetWD fWWWD¹.V/where V is a bounded complete Reinhardt domain in An-variety} be the space of bounded complete Reinhardt domains inC²which are invariant under the action of the cyclic group of order nC1onC². Then

^.˛;ˇ/; ^.˛;ˇ/; ^.˛;p;q/; !^{.˛1; ˛2;p1;p2/};

˛>1; ˛> n

nC1ˇ; 06p;q6 nC1

n ˛

;p¤q;

06pi6 nC1

n ˛i

; ˛i>1; ˛₁¤˛₂;iD1; 2;

defined in Theorem3.7are invariants ofW.

(2) LetWP D fW WW D¹.V/where V is a complete Reinhardt pseudoconvex C^!-smooth domain in An-variety} andWSP D fW W W D ¹.V/where V is a complete Reinhardt strictly pseudoconvex domain in An-variety}. Then the moduli space ofWP (respectivelyWSP) is given by the image of the mapˆQP W WP !R¹(respectivelyˆQSP WWSP!R¹), where the component functions ofˆQP(respectivelyˆQSP) are the invariant functions

^.˛;ˇ/; ^.˛;ˇ/; ^.˛;p;q/; !^{.˛1; ˛2;p1;p2/};

˛>1; ˛> n

nC1ˇ; 06p;q6 nC1

n ˛

;p¤q;

06pi6 nC1

n ˛i

; ˛i>1; ˛1¤˛2;iD1; 2;

defined in Theorem 3.7. In particular, the moduli space of WP (respectively WSP) is the same as the moduli space of bounded complete Reinhardt pseudo- convex C^!-smooth domains (respectively bounded complete Reinhardt strictly pseudoconvex domains) in An-varietyVQnD f.x;y;z/2C³WxyDz^nC1g.

It is an interesting question to study the geometry of the moduli space of bounded complete Reinhardt domains inAn-variety. As an example, we look at two families of domains inAn-variety and construct the moduli space of these families explicitly.

More specifically, consider

V_.^._a^d_;^/_b;c/D f.x;y;z/WxyDz²;ajxj^2dCbjyj^2dCcjzj^2d< "₀g:

Here we assume thata;b;care strictly greater than zero, andd is a fixed integer greater than or equal to one. This is a 3 parameters family of pseudoconvex domains inA₁-varietyVQ₁ D f.x;y;z/2 C³ Wxy Dz²g. Using our Bergman function theory, we can write down the explicit moduli space of this family as shown in the following theorem by means of the invariant

^.˛;ˇ/D .^.˛;ˇ//¹² .^{.n˛.n1/ˇ;.nC1/˛nˇ/}/¹²

.^.˛;˛//¹² ; for nD1:

Theorem 3.12 ([10]) Let

V_.^._a^d_;^/_b;c/D f.x;y;z/2C³WxyDz²;ajxj^2dCbjyj^2dCcjzj^2d< "₀g:

Let denote the biholomorphic equivalence. Then the map 'W fV_.^._a^d_;^/_b;c/g !RC;V_.^._a^d_;^/_b;c/7!^.2d1;d1/

is injective up to a biholomorphism equivalence. More precisely, the induced map 'QW fV_.^._a^d_;^/_b;c/g= !RC

is one-to-one map fromfV_.^._a^d_;^/_b;c/g= onto

0; 2

. So the moduli space offV_.^._a^d_;^/_b;c/g is an open interval

0; 2

.

The biholomorphically equivalent problem of domains inA₁-variety is not only interesting in its own right, but also has application to the classical biholomorphi- cally equivalent problem of domains inC². In fact, let

W_.^._a^d_;^/_b;c/D f.x;y/Wajxj^2dCbjyj^2dCcjxyj^d< "0g

Corollary 3.1 ([10]) The moduli space of W_.^._a^d_;^/_b;c/is the same as the moduli space of V_.^._a^d_;^/_b;c/, which is.0;²/.

As an application to the above theory, it is easy to compute explicitly the invariant ^.3;1/ for two domainsV_.1;1;1/^.1/ andV_.1;1;1/^.2/ inA₁-variety. As a consequence, we see thatV_.1;1;1/^.1/ is not biholomorphic to V_.1;1;1/^.2/ and the domain W_.1;1;1/^.1/ in C² is not biholomorphic to the domainW_.1;1;1/^.2/ inC².

One of the basic problems in complex geometry is to find a reasonable object which parametrizes all non-isomorphic complex manifolds. This is the well known moduli problem. LetD₁andD₂be two domains inCⁿ. One of the most fundamental problems in complex geometry is to find necessary and sufficient conditions which will imply that D₁ and D₂ are biholomorphically equivalent. For n D 1, the celebrated Riemann mapping theorem states that any simply connected domains inCare biholomorphically equivalent. Forn 2, there are many domains which are topologically equivalent to the ball but not biholomorphically equivalent to the ball [36]. Poincaré studied the invariance properties of theCR manifolds, which are real hypersurfaces inCⁿ, with respect to biholomorphic transformations. The systematic study of such properties for real hypersurface was made by Cartan [7]

and later by Chern and Moser [8]. A main result of the theory is the existence of a complete system of local differential invariants for CR-structures on real hypersurface. In 1974, Fefferman [13] proved that a biholomorphic mapping between two strongly pseudoconvex domains is smooth up to the boundaries and the induced boundary mapping is aCR-equivalence on the boundary. Thus, one can use Chern-Moser invariants to study the biholomorphically equivalent problem of two strongly pseudoconvex domains. Using the Chern-Moser theory, Webster [44]

gave a complete characterization when two ellipsoids inCⁿare biholomorphically equivalent. In 1978, Burns Shnider and Wells [6] showed that the number of moduli of a moduli space of a strongly pseudoconvex bounded domain has to be infinite. Thus the moduli problem of open manifolds is really a very difficult one.

Lempert [27] made significant progress in the subject. He was able to construct the moduli space of bounded strictly convex domains ofCⁿwith marking at the origin.

Although the theory established by Lempert is beautiful, the computation of his invariants is a hard problem.

In [10], Du and Yau studied the moduli problem of complete Reinhardt domains inC². The main tool to solve this moduli problem with geometry information is the new biholomorphic invariant Bergman function defined by Yau [50]. In fact Yau’s Bergman function theory can also solve the biholomorphic equivalence problem or moduli problem for complete Reinhardt pseudoconvex domains in Cⁿ for all n 2. In order to describe the complete biholomorphic invariants of bounded complete Reinhardt domains inCⁿ, we introduce the following notations. Let Sn

be the symmetric group of degreen. Recall that group ringRŒSnis a ring of the formRŒ 1; 2; : : : ; nŠwith i 2 Sn for1 6 i 6 nŠ. LetP

i

xi iandP

j

yj j, where xi;yjare inR, be two elements inRŒSn. Then

.X

i

xi i/.X

j

yj j/WDX

i;j

xiyj. i j/;

where i j is the product in the groupSn. We shall considerRŒSn RŒSn the product of the group ring with itself. Such a product has a naturalSn-module structure in the following manner. Let 2 Snand.P

i

xi i; ;P

i

yi i/2.RŒSn RŒSn/. Then

.X

i

xi i; ;X

i

yi i/D.X

i

xi. i/; ;X

i

yi. i//:

Definition 3.6 Two elementsf;g inRŒSn RŒSnare said to be equivalent and denoted byf gif there exists a2Snsuch that.f/Dg.

Let ˛E D .˛1; : : : ; ˛n/be ann-tuple of nonnegative integers. Denote ˛E D Qn

˛ED1z^˛_iⁱ dz₁ ^dz₂^ ^ dzn: For a domain Din Cⁿ, we shall use notation k˛Ek²_DWD R

D˛E^˛E:In [9], the authors showed that all biholomorphic invariants of a bounded complete Reinhardt domains are contained in.RŒSn RŒSn/=

where there arenŠcopies of RŒSn and is the equivalence relation defined in Definition3.6.

Theorem 3.13 ([9]) Let D be a bounded complete Reinhardt domain in Cⁿ. Let

˛ED.˛1; : : : ; ˛n/be a n-tuple of nonnegative integers and 2Sn. Denote

g_D.˛/E D kE0k^†˛_D ⁱ¹k _.˛/EkD

Qn

iD1keEik^˛_D ^.i/

where .˛/E D .˛ _.1/; : : : ; ˛ _.n//andeEi D .0; : : : ; 0; 1; 0; : : : ; 0/with 1 in the ith component. Then for all n-tuple of nonnegative integers

ˇE1; ;ˇEnŠ; D^{.ˇE1;;ˇEnŠ/}D.X

2Sn

g_D.ˇE1/ ; ;X

2Sn

g_D.ˇEnŠ/ /

as an element in.RŒSn RŒSn/= is a biholomorphic invariant. In fact, if D₁ and D₂are two such domains which are biholomorphically equivalent, then there exists a2Snsuch that

g_D.˛/E Dg_D2.˛/E 8 2Snand8 E˛n-tuple of nonnegative integers.

The invariants in Theorem3.13 are complete invariants for bounded complete Reinhardt pseudoconvex domains withC¹boundaries.

Theorem 3.14 ([9]) Let Di, i D 1; 2, be two bounded complete Reinhardt pseu- doconvex domains in Cⁿ with C¹ boundaries. If for all ˛E1; ;˛EnŠ n-tuples of non-negative integers,D^.˛E₁^1;;˛EnŠ/DD^.˛E₂^1;;˛EnŠ/in.RŒSn RŒSn/= ;where

D^{.˛E1;;˛EnŠ/}D.X

2Sn

g_D.˛E1/ ; ;X

2Sn

g_D.˛EnŠ/ /;

then there exists2Snand a biholomorphic map

‰.z₁; : : : ;zn/D.a₁z_.1/; : : : ;anz_.n//;

where aiD _k^{kE0kD1keiEkD2}

Ee.i/kD1k_E0kD2, such that‰sends D₁onto D₂.

Theorems3.13and3.14above give a complete characterization of two bounded complete Reinhardt domains with real analytic boundaries inCⁿto be biholomor- phically equivalent in terms of the group ring.RŒSn RŒSn/= . In case n D 2, we can actually write down the complete numerical invariants for two bounded complete Reinhardt domains with real analytic boundaries in C² to be biholomorphically equivalent.

Theorem 3.15 ([9]) Let D₁;D₂be two bounded complete Reinhardt pseudoconvex domains inC²with C¹boundaries. Then D₁is biholomorphic to D₂if and only if

.1/gD₁.˛₁; ˛₂/CgD₁.˛₂; ˛₁/DgD₂.˛₁; ˛₂/CgD₂.˛₂; ˛₁/ .2/gD₁.˛₁; ˛₂/gD₁.˛₂; ˛₁/DgD₂.˛₁; ˛₂/gD₂.˛₂; ˛₁/ .3/ .gD₁.˛1; ˛2/gD₁.˛2; ˛1//.gD₁.ˇ1; ˇ2/gD₁.ˇ2; ˇ1//

D.gD₂.˛1; ˛2/gD₂.˛2; ˛1//.gD₂.ˇ1; ˇ2/gD₂.ˇ2; ˇ1//

for all non-negative integers˛i; ˇi, where

gDi.˛1; ˛2/D kE0k^˛_D¹_i^C˛21k.˛1;˛2/kDi

Q2 jD1keEjk^˛_D^j_i

Corollary 3.2 ([9]) The moduli space of bounded complete Reinhardt domains with C¹boundaries inC²can be constructed explicitly as the image of the complete family of numerical invariants: gD.˛₁; ˛₂/CgD.˛₂; ˛₁/, gD.˛₁; ˛₂/gD.˛₂; ˛₁/and

.gD.˛₁; ˛₂/gD.˛₂; ˛₁//.gD.ˇ₁; ˇ₂/gD.ˇ₂; ˇ₁//

8˛i; ˇinon-negative integers.

In order to find the complete numerical biholomorphic invariants of bounded complete Reinhardt domain in Cⁿ for n > 3, we need to consider the finite symmetric groupSn D f₁; ₂; : : : ; nŠg of degree n acting on the affine space C^nŠnŠ DC^nŠ C^nŠ;which is the product ofnŠcopies ofC^nŠ, in the following manner. Let 2Snand

.x₁; : : : ;xnŠI Iy₁; : : : ;ynŠ/2C^nŠ C^nŠDC^nŠnŠ: Then

.x1; : : : ;xnŠI Iy₁; : : : ;ynŠ/D.x1 ; : : : ;xnŠ I Iy₁ ; : : : ;ynŠ /:

SinceSnis linearly reductive, by Hilbert Theorem, the ring of invariants CŒx1; : : : ;xnŠI Iy₁; : : : ;ynŠ^Sn

is finitely generated. Moreover, the generators can be listed explicitly by Göbel’s theorem [15]. Before we give the statement of Göbel’s theorem, we shall introduce some definitions first.

Definition 3.7 Suppose that a finite groupGacts as permutations on a finite setX.

We then refer toXtogether with theG-action as a finiteG-set. A subsetB Xis called an orbit ifGpermutes the elements ofBamong themselves and the induced permutation action ofGonBis transitive.

Definition 3.8 IfK D .k1; ;kn/is ann-tuple of non-negative integers, then K is called an exponent sequence. The associated partition of K is the ordered set consisting of thennumbersk₁; ;knrearranged in weakly decreasing order. We denote by.K/the partition associated toK, so

.K/D.1.K/2.K/ n.K//