MAXIMAL COMPLEXIFICATIONS OF CERTAIN HOMOGENEOUS RIEMANNIAN MANIFOLDS

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 11, Pages S )03263-X Artile eletronially published on July 8, 2003 MAXIMAL COMPLEXIFICATIONS OF CERTAIN HOMOGENEOUS RIEMANNIAN MANIFOLDS S. HALVERSCHEID AND A. IANNUZZI Abstrat. Let M = G/K be a homogeneous Riemannian manifold with dim C G C =dim R G, where G C denotes the universal omplexifiation of G. Under ertain extensibility assumptions on the geodesi flow of M, wegivea haraterization of the maximal domain of definition in TM for the adapted omplex struture and show that it is unique. For instane, this an be done for generalized Heisenberg groups and naturally redutive homogeneous Riemannian spaes. As an appliation it is shown that the ase of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted omplex struture whih are neither holomorphially separable nor holomorphially onvex. 1. Introdution It is well known that omplexifiations of a real-analyti manifold M exist and are equivalent near M, but differ, usually very muh, in nature. If a omplete real-analyti metri on M is given, one an onstrut anonial omplexifiations of M ompatible with the metri by defining an adapted omplex struture on a domain Ω of the tangent bundle TM see [GS] and [LS]). This struture an be haraterized by the ondition that the omplexifiation x + iy) yγ x) Ω of any geodesi x γx) of M be a omplex submanifold near the zero setion. By the results of Guillemin-Stenzel and Lempert-Szőke ited above, the adapted omplex struture exists and is unique on a suffiiently small neighborhood of M. Here M is identified with its zero setion in TM. In partiular, it is natural to ask for maximal domains around M with an adapted omplex struture. By funtoriality of the definition these may be regarded as invariants of the metri, i.e., isometri manifolds have biholomorphi maximal domains. For instane, examples are known for symmetri spaes of non-ompat type [BHH]), ompat normal homogeneous Riemannian spaes [Sz2]), ompat symmetri spaes [Sz1]) and spaes obtained by Kählerian redution of these [A]). Note that in the mentioned ases maximal domains turn out to be Stein. The aim of this work is to haraterize maximal domains for the adapted omplex struture for a lass of homogeneous Riemannian spaes with big isometry group. Reeived by the editors Deember 31, 2001 and, in revised form, November 18, Mathematis Subjet Classifiation. Primary 32C09, 53C30, 32M05; Seondary 32Q28. Key words and phrases. Adapted omplex struture, homogeneous Riemannian spaes, Stein manifold. The first author was partially supported by SFB 237, Unordnung und große Fluktuationen. The seond author was partially supported by University of Bologna, funds for seleted researh topis Amerian Mathematial Soiety

2 4582 S. HALVERSCHEID AND A. IANNUZZI Let M = G/K,with G a Lie group of isometries and K ompat, and assume that dim C G C =dim R G,where G C is the universal omplexifiation of G.Then K C ats on G C and the left ation on M indues a natural G-ation on TM. Our main result is that, under ertain extensibility assumptions on the geodesi flow of M, one obtains a real-analyti and G-equivariant map P : TM G C /K C suh that see Theorem 3.2 for the preise statement) The onneted omponent of the non-singular lous of DP ontaining M is the unique maximal domain on whih the adapted omplex struture exists. This applies to the ase of naturally redutive homogeneous Riemannian spaes Corollary 3.3) and of generalized Heisenberg groups see Setions 4 and 5). As an appliation, it is shown that for all generalized Heisenberg groups suh a maximal domain is neither holomorphially separable, nor holomorphially onvex Proposition 5.1). We are not aware of previous non-stein examples. In the ase of the 3-dimensional Heisenberg group, we determine its envelope of holomorphy as well as a ertain maximal Stein subdomain Proposition 4.3). 2. Preliminaries Here we introdue some notation and briefly reall basi results that we will need in the present paper. Let M be a omplete real-analyti Riemannian manifold, whih will often be identified with the zero setion in its tangent bundle TM. Following [LS] see also [GS] for an equivalent haraterization), we say that a realanalyti omplex struture defined on a domain Ω of TM is adapted if all omplex leaves of the Riemannian foliation are submanifolds with their natural omplex struture, i.e., for any geodesi γ : R C, the indued map f : C TM defined by x + iy) yγ x) is holomorphi on f 1 Ω) with respet to the adapted omplex struture. Here yγ t) T γt) M is the salar multipliation in the vetor spae T γt) M. The adapted omplex struture exists and is unique on a suffiiently small neighborhood of M. If Ω is a domain around M in TM on whih it is defined, then we refer to it as an adapted omplexifiation. Sometimes these are alled Grauert tubes. For later use we need the following fat. Lemma 2.1. Let F : TM C be a real-analyti map that is holomorphi on any omplex leaf of TM in a neighbourhood of M. ThenF is holomorphi on every adapted omplexifiation. Proof. Following the proof of [Sz1, Prop. 3.2, p. 416], one heks that the restrition of F to M extends to a holomorphi map ˆF in a neighbourhood U of M TM where the adapted omplex struture J 0 exists, and, in order to have onneted leaves, U may be hosen to be starshaped. We an also assume that for any geodesi γ : R M the map x + iy F yγ x)) is holomorphi for all x + iy suh that yγ x) U. Now F = ˆF on γr) M TM; therefore, F = ˆF on every omplex leaf, i.e., on U.Inpartiular,DF J 0 = idf on U, and sine all these maps are real-analyti, the statement follows from the identity priniple. A real Lie group G ats on a omplex manifold X, i.e., X is a G-manifold, if there exists a real-analyti surjetive map G X X given by g, x) g x suh that for fixed g G the map x g x is holomorphi and gh) x = g h x) for all h, g G and x X. Furthermore, if dim R G =dim C G C,where ι : G G C is the universal omplexifiation of G see, e.g., [Ho]), then LieG C )=g C,andone

3 MAXIMAL COMPLEXIFICATIONS 4583 obtains an indued loal holomorphi G C -ation by integrating the holomorphi vetor fields given by the G-ation. Here g denotes the Lie algebra of G. Let M = G/K be a homogeneous Riemannian manifold with G a onneted Lie group of isometries and K ompat, and onsider the indued G-ation on TM defined by g w := g w for all g G and w TM. Then if Ω is a G- invariant adapted omplexifiation, as an easy onsequene of the definitions g is a biholomorphi extension of the isometry g, i.e., G AutΩ). If one assumes that dim R G =dim C G C, then the natural map ι : G G C is an immersion, and from the universality property of the universal omplexifiation K C of K it follows that the restrition ι K of ι to K extends to an immersion ι C : K C G C. Moreover, the subgroup ι C K C ) ats by right multipliation on G C, and one has a ommutative diagram G ι G/K G C /ι C K C ). Also notie the G-ation on G C /ι C K C ) defined by g hι C K C ):=ιg) hι C K C ) for all g G and h G C. Lemma 2.2. Let G be a onneted Lie group, K a ompat subgroup, and assume that dim C G C =dim R G. Then G C /ι C K C ) is a omplex G-manifold, and dim C G C /ι C K C )=dim R G/K. Proof. One needs to show that ι C K C )isloseding C.NotethatG C /ι C K C )is the orbit spae with respet to the K C -ation on G C defined by k h := hιk 1 ) for all k K C and h G C. Sine G C is Stein [He]) and K C is redutive, it follows that every fiber of the ategorial quotient G C G C //K C is equivariantly biholomorphi to an affine algebrai variety on whih K C ats algebraially [Sn]). In partiular, there exists at least one losed K C -orbit, and onsequently ι C K C ) is losed in G C.ThusG C /ι C K C ) is a omplex G-manifold, and by onstrution its omplex dimension is dim R G/K. 3. A haraterization of maximal adapted omplexifiations If M = G/K is a symmetri spae of the non-ompat type, then G C /K C is a natural andidate for a omplexifiation of M,andthereexistsa G-equivariant map P : TM G C /K C embedding holomorphially a maximal adapted omplexifiation Ω M of M see [BHH], [Ha], [AG]). As a matter of fat, one may show that DP is singular on the boundary Ω M of Ω M. Here we onsider a homogeneous Riemannian manifold M = G/K endowed with the additional data of a ertain real-analyti G-equivariant map P from TM to a suitable omplex G-manifold. Proposition 3.1 haraterizes a maximal adapted omplexifiation Ω M as the onneted omponent of {DP not singular} ontaining M. UniityofΩ M follows. The existene of suh data is proved when dim C G C =dim R G and the geodesi flow extends holomorphially on G C /ι C K C ) f. Lemma 2.2). As a onsequene, the haraterization applies to the ases of naturally redutive homogeneous Riemannian spaes and of generalized Heisenberg groups. G C

4 4584 S. HALVERSCHEID AND A. IANNUZZI Proposition 3.1. Let M = G/K be an n-dimensional homogeneous Riemannian spae and X a G-omplex manifold of omplex dimension n suh that the indued loal G C -ation is loally transitive. Assume there exists a real-analyti map P : TM X that is i) G-equivariant and ii) holomorphi on every omplex leaf of TM. Then the onneted omponent Ω M of { p TM : DP p is not singular } ontaining M is the unique maximal adapted omplexifiation and P ΩM is loally biholomorphi. Proof. First we show that Ω M is well defined, i.e., DP has maximal rank along M. Sine from Lemma 2.1 it follows that P is holomorphi on M with respet to the adapted omplex struture, this is a onsequene of the following fat. Claim: Assume that P is holomorphi in p TM. Then DP p has maximal rank. Proof of the laim. Sine G C ats loally transitively on X, there exist elements ξ 1,,ξ n of g suh that the indued vetor fields ξ X,1,,ξ X,n on X span a totally real and maximal dimensional subspae V P p) of T P p) X,where ξ X,j x) := d dt exp G Ctξ j ) x 0 for j = 1,,n and all x X. By equivariane it follows that DP p V p ) = V P p),wherev p is the subspae of T p TM spanned by ξ TM,1,,ξ TM,n ;here ξ TM,j q) := d dt 0 exp G tξ) q for all q TM.Inpartiular,dim R V p = n, and sine P is holomorphi in p, V p is totally real and DP p has maximal rank, proving the laim. Now we see that the pulled-bak omplex struture J o on Ω M of the omplex struture J on X is the adapted omplex struture. For this, onsider a omplex leaf f : C TM defined by fx + iy) :=yγ x), where γ is a geodesi of M, and note that by ii) DP Dfiη) =DP J o Dfη) for all η tangent in f 1 Ω M ). Sine DP has maximal rank on Ω M, one has Dfiη) =J o Dfη), showing that J o is the adapted omplex struture. In partiular, P ΩM is loally biholomorphi. In order to prove maximality, assume that J o extends analytially in a neighborhood of a ertain p Ω M TM. By onstrution, DP J o = J DP on Ω M, and sine all maps are real-analyti, P is holomorphi in p. Then the above laim shows that DP p has maximal rank, ontraditing the definition of Ω M. Finally we want to show that any adapted omplexifiation Ω is ontained in Ω M. If this is not the ase, there exists a point p in Ω Ω M, and from Lemma 2.1 it follows that P Ω is holomorphi. In partiular, P is holomorphi in p,andone obtains a ontradition arguing as above. Thus Ω M is unique, and this onludes the proof of the statement.

5 MAXIMAL COMPLEXIFICATIONS 4585 Now we determine a lass of homogeneous Riemannian spaes to whih Proposition 3.1 may be applied in order to determine the maximal adapted omplexifiation. Theorem 3.2. Let M = G/K be a homogeneous Riemannian spae with dim R G =dim C G C, and assume there exists a map ϕ : C T K M g C, real-analyti and holomophi on the first omponent, suh that ϕr T K M) g and t exp G ϕt, v)k is the unique geodesi tangent to v at 0 for all v T K M. Then the map P : TM G C /ι C K C ) defined by P g v)) := ιg) exp G Cϕi, v)) ι C K C ) for all g G and v T K M is as in Proposition 3.1. In partiular, the onneted omponent Ω M of { p TM : DP p is not singular } ontaining M is the maximal adapted omplexifiation, and P ΩM is loally biholomorphi. Proof. In order to prove that P is well defined, we need to show that if w = k v for some k K and v T K M,then P w) =P k v), i.e., 1) exp G C ϕ i, w ))ι C K C )=ιk) exp G C ϕ i, v ))ι C K C ). For this, note that t k exp G ϕt, v)k is the unique geodesi tangent to w at 0inK; thus, exp G ϕ t, w )K = k exp G ϕ t, v )K. Then the ommutativity of the diagram implies that g exp G G ι G C g C exp G C exp G C ϕt, w) ι C K C )=ιk) exp G C ϕ t, v ))ι C K C ) for all t R, and equation 1) is a onsequene of the identity priniple for holomorphi maps. In order to simplify notation we now assume that the anonial immersion ι : G G C is injetive, so that one we identify G with ιg) theurve γt) := exp G C ϕ t, v ))K is the unique geodesi tangent to v at 0 for all v T K M.In what follows it is easy to hek that all arguments apply to the ase where ι is a non-injetive immersion. Fix x R,letg:= exp G C ϕ x, v ) ), and note that y g exp G C ϕ y, g 1 γ x)))k is the unique geodesi tangent to γ x) at 0. Therefore one has g exp G C ϕ y, g 1 γ x)))k =exp G C ϕ x + y, v ))K for all y R, and from the identity priniple it follows that 2) g exp G C ϕ z, g 1 γ x)))ι C K C )=exp G C ϕ x + z, v ))ι C K C )

6 4586 S. HALVERSCHEID AND A. IANNUZZI for all v T K M and z C. Now for h G, v T K M,let γ, g be as above and onsider the unique geodesi γ := h γ tangent to h v) at 0. One has P y γ x)) = P h yγ x)) = hpg g 1 yγ x)) = hg exp G C ϕ i, y g 1 γ x)))ι C K C ) = hg exp G C ϕ iy, g 1 γ x)))ι C K C ) = h exp G C ϕ x + iy, v ))ι C K C ), where we used 2) and the fat that exp G C ϕ z, yv))ι C K C )=exp G C ϕ zy, v ))ι C K C ) for all z C, sine this holds for all z R. As a onsequene, the map x+iy) P y γ x)) is holomorphi for all geodesis γ of M, i.e., P is holomorphi on every omplex leaf of TM. Finally, the map P is G-equivariant by onstrution, and the G-ation on G C /ι C K C ) indues a holomorphi G C -ation, whih may be obtained through left multipliation on G C. Thus it is obviously transitive, and this yields the statement. Now let M be a naturally redutive homogeneous Riemannian spae, and let M = G/K be a natural realization of M, i.e., there exists a redutive deomposition g = LieK) m of the Lie algebra of G suh that every geodesi in M is the orbit of a one-parameter subgroup of G generated by an element of m see, e.g., [BTV]). Consider the natural projetion Π : G M and note that DΠ e m) =T K M,where e is the neutral element of G. DenotebyL : T K M m the inverse of the restrition of DΠ e to m.sinel is linear, it extends C-linearly from T K M) C to m C,andthemap ϕ : C T K M g C defined by ϕz,v) :=zlv) is as in the above theorem. Therefore one has Corollary 3.3. Let M = G/K be a natural realization of a naturally redutive homogeneous Riemannian spae, and assume that dim R G =dim C G C. Then the map P : TM G C /ιk C ) defined by P g v)) := ιg) exp G CiLv)) ιk C ) for all g G and v T K M meets the onditions of Proposition 3.1. In partiular, the onneted omponent Ω M of { p TM : DP p is not singular} ontaining M is the maximal adapted omplexifiation, and P ΩM is loally biholomorphi. 4. The 3-dimensional Heisenberg group Here we apply results of the previous setion in order to give a onrete desription of the unique maximal adapted omplexifiation for the 3-dimensional Heisenberg group. It turns out that suh a domain is neither holomorphially separable nor holomorphially onvex. We also determine its envelope of holomorphy and a partiular maximal Stein subdomain. We remark that in all previous examples of whih we are aware maximal adapted omplexifiations are Stein.

7 MAXIMAL COMPLEXIFICATIONS 4587 Consider the 3-dimensional Heisenberg group defined as a subgroup of GL 3 R) by H := 1 α γ 0 1 β : α, β, γ R, fix the inner produt of the tangent spae T e H in the neutral element e for whih the anonial basis determined by the global natural hart α, β, γ) is orthonormal, and let a, b, ) be oordinates of T e H with respet to this basis. Endow H with the indued H-invariant metri dα) 2 +dβ) 2 +dγ α dβ) 2, let h = LieH) and define ϕ : R T e H h by ϕt, a, b, )) := a sint) b 1 ost), b sint) + a 1 ost), t + a2 + b 2 t sint) )) ), 2 2 where the oordinates of h are indued by those of T e H via the natural identifiation h = T e H. Note that all singularities are removable, and onsequently ϕ is real-analyti. Following [BTV, the theorem on p. 31], one heks that t exp H ϕt, a, b, )) is the unique geodesi tangent to a, b, ) at 0. Furthermore, by expanding the power series it is easy to verify that ϕ, a, b, )) extends holomorphially on C to T e H) C, and by onsidering the polar deomposition H h H C of H C given by g, ξ) g exp H Ciξ), one obtains real-analyti funtions a, b, ) h a,b,) H and a, b, ) ξ a,b,) h suh that exp H C ϕi, a, b, )) = h a,b,) exp H Ciξ a,b,) ). Define P : TH H C = H h by g a, b, ) g exp H C ϕi, a, b, )) = ) gh a,b,),ξ a,b,). Then Theorem 3.2 implies that the onneted omponent Ω H ontaining H of {DP not singular} is the maximal adapted omplexifiation. Note that sine P is H-equivariant, Ω H is H-invariant. Moreover, T e H is a global slie for the H- ation on TH, i.e., the map H T e H TH given by g, a, b, )) g a, b, ) is an H-equivariant real-analyti diffeomorphism; thus, Ω H is ompletely determined by its slie Ω H T e H. Furthermore, sine H ats freely on the first omponent of H h, then the H-equivariane of P implies that DP g a,b,) has maximal rank if and only if D P a,b,) has maximal rank, where P := p 2 P TeH : T e H h is given by P a, b, ) = ξ a,b,) = a sinh), b sinh), 1+ a2 + b ) ) sinh) osh)). Here p 2 : H h h is the anonial projetion. It follows that Ω H = H O 0, where O 0 is the onneted omponent of {detd P ) 0} ontaining 0 in T e H. Now a straightforward omputation shows that detd P a,b,) )= sinh) sinh) sinh) osh) +a 2 + b 2 ) 3 )),

8 4588 S. HALVERSCHEID AND A. IANNUZZI and therefore { O 0 = a, b, ) T e H : a 2 + b 2 2 } sinh) <. osh) sinh) We want to disuss injetivity of P ΩH :Ω H H C = H h, and again this is equivalent to injetivity of P O0. Note that P is equivariant with respet to rotations around the -axis as well as to the refletion σ with respet to the plane { =0}.Inpartiular,forany a, b, ) {1+ a2 + b 2 } 2 3 sinh) osh)) = 0, one has P a, b, ) = P a, b, ) = to investigate the domain O 1 := a sinh),b sinh), 0 { a, b, ) T e H : a 2 + b 2 < ). Therefore we are indued 2 3 }. sinh)osh) Lemma 4.1. The domain O 1 is the maximal σ -invariant subdomain of O 0 ontaining 0 on whih P is injetive. In partiular, P O0 is not injetive. Proof. Let f j be the real funtion defining O j = { a, b, ) T e H : a 2 + b 2 <f j ) } for j =0, 1.Firstwewanttoshowthat O 1 is a subdomain of O 0, i.e., f 1 ) f 0 ) for all R, whih is equivalent to 2osh) sinh) Expanding in power series, one obtains ) sinh2 ) 2 osh) ) ) All oeffiients are non-negative, and one easily heks that for k 2 the oeffiient of 2k in the last series on the right side is stritly greater than that in the series on the left; hene O 1 O 0.Moreover, O 1 O 0 = { a, b, 0) T e H : a 2 +b 2 =3}, and thus O 1 is a proper subdomain of O 0. Furthermore, by the previous remarks, any σ-invariant domain ontaining 0 on whih P is injetive is neessarily ontained in O 1. Assume that there exist a,b, ), a,b, ) O 1 suh that P a,b, )= P a,b, )=:A, B, C). If C =0,then = = 0, and onsequently a = a = A and b = b = B. If C 0, by eventually ating with σ and a rotation around the -axis we may assume that a, A 0, b = B =0 and >0. Now one has a sinh ) = a sinh ) = A.

9 MAXIMAL COMPLEXIFICATIONS 4589 Therefore a, 0, )anda, 0, ) lie on the same level urve ρ A : R T e H given by ρ A t) := A t ) sinht), 0,t. One has the following assertion. Claim: Let A 0andt 0 R 0 be suh that ρ A t 0 ) O 0.Thenρ A t) O 0 for all t>t 0. Proof of the laim. One needs to show that A 2 t2 is, 3) A 2 < sinh2 t) f 0 t). t 2 sinh 2 t) <f 0t) for all t>t 0 ;that By expanding in power series as above one has the estimate 2t osh 2 t) 3osht) sinht)+t>0 for all t>0, whih by a straightforward omputation implies that the derivative of the funtion on the right-hand side of 3) is positive for all t>0, proving the laim. Now let t 0 := min, ) and note that sine O 1 O 0, as a onsequene of the above laim there exists ɛ>0 suh that ρ A t) O 0 for t > t 0 ɛ. In partiular, a, 0, )anda, 0, ) lie in the same onneted real one-dimensional submanifold N := ρ A t 0 ɛ, ) of O 0,and P N : N {A, 0, ) T e H} = R is loally diffeomorphi. Then a lassial argument implies that P N is injetive; thus a, 0, )=a, 0, ), as wished. We also want to determine the image of P ΩH in H C.NotethatPΩ H )ishinvariant, and the polar deomposition implies that exp H Cig) is a global slie for the H-ation on H C. Then this an be ahieved by desribing exp H Ci P O 0 )) = P Ω H ) exp H Cig). Lemma 4.2. P O0 )=h \{A, B, C) h : A 2 + B 2 =3, C =0}. Proof. Let a, 0,) {a 2 = f 1 )} O 1 with >0. From the proof of Lemma 4.1 it follows that a, 0,) O 0.Sine P a, 0,)=a sinh), 0, 0) and f1 0) = 3 and lim f1 ) sinh) =, it follows that A, 0, 0) P O 0 ) for all A> 3. For A> 3leta, 0,) O 0 be suh that P a, 0,)=A, 0, 0). Bythelaim in Lemma 4.1 one has ρ A t) O 0 for all t. Moreover, one sees that 4) P ρa t)) = A, 0,C A t)) with lim C A t) =. t Then by σ-invariane of O 0 and σ-equivariane of P it follows that A, 0,C) P O 0 ) for all C R and A> 3. Now note that P a, 0, 0) = a, 0, 0) and f 0 0) = 3 ; thus A, 0, 0) P O 0 ) for all A< 3, and, arguing as above, it follows that A, 0,C) P O 0 ) for all C R and A< 3.

10 4590 S. HALVERSCHEID AND A. IANNUZZI Figure 1. Finally, ρ 3 0) O 0. Thus ρ 3 t) O 0 for all t > 0. It follows that t P ρ 3 t)) is injetive for t>0, and sine lim C t t) = 0 and lim t C 3 t) =, one has 3, 0,C) P O 0 ) if and only if C 0. The statement follows from the invariane of O 0 and the equivariane of P with respet to the group of rotations around the -axis. In Figure 1 one sees the boundary of O 0 and O 1 defined by f 0 and f 1 respetively as well as the level urve ρ 2 in the upper half-plane { b =0,a 0} of T e H. Sine O 0 and O 1 are invariant with respet to rotations around the -axis, this ompletely determines their shape and, by H-equivariane, that of Ω H. Proposition 4.3. The maximal domain Ω H is neither holomorphially separable, nor holomorphially onvex. Its envelope of holomorphy is biholomorphi to C 3. The maximal σ-invariant Stein subdomain of Ω H is biholomorphi to { z 1,z 2,z 3 ) C 3 : Imz 1 ) 2 +Imz 2 ) 2 < 3 }. Proof. Note that the elements of H with integer entries determine a disrete oompat subgroup Γ of H, and from [GH] it follows that H C /Γ is Stein in fat it is easy to hek that H C /Γ is biholomorphi to C ) 3 ). Then the proposition on p. 543 of [CIT] applies to show that any holomorphially separable Riemann H-domain over H C is univalent. Moreover, Lemma 4.1 implies that P ΩH :Ω H H C is not injetive. Therefore Ω H is not holomorphially separable. By a result of Loeb [L, the theorem on p. 186]), a Stein H-invariant domain U of H C is geodesially onvex, i.e., it is onvex with respet to all urves of the form t g exp H Citξ) with g U and ξ h. Sine H C admits polar deomposition and U is H-invariant, it is enough to onsider urves of the form exp H Ciη) exp H Citξ) withexp H Ciη) U and ξ h. Furthermore, for a two-step

11 MAXIMAL COMPLEXIFICATIONS 4591 nilpotent Lie group, one has exp H Ciη) exp H Citξ) =exp H Ciη + itξ t [η, ξ]) 2 =exp H C t 2 [η, ξ]) exp HCiη + tξ)), and, using H-invariane one more time, we onlude that if U = H exp H CiD), with D a domain in h,isstein,then D is onvex in the usual affine sense. Sine P Ω H )=H exp H Ci P O 0 )) and, as a onsequene of Lemma 4.2, the domain P O 0 ) is not onvex, it follows that P Ω H ) is not Stein. Now H C is Stein and P ΩH is loally biholomorphi. Therefore by [R] there exists a ommutative diagram Ω H j Ω H P P H C where ˆΩ H is the envelope of holomorphy of Ω H. Moreover, H ats on ˆΩ H and all maps are H-equivariant. Furthermore, ˆP is injetive by the proposition on p. 543 of [CIT], and if Ω H is holomorphially onvex, then j is surjetive, and onsequently ˆΩ H is biholomorphi to P Ω H ), giving a ontradition. Hene Ω H is not holomorphially onvex. Notie that ˆΩ H = ˆP ˆΩH )ontainspω H )=H exp H Ci P O 0 )) and the onvex envelope of P O0 )ish; thus, by the above arguments, the envelope of holomorphy ˆΩ is biholomorphi to H C = C 3. Finally, the maximal onvex σ-invariant subdomain of P O0 ) is { A, B, C) h : A 2 + B 2 < 3 } = P O 2 ), where O 2 := { a, b, ) h : a 2 + b 2 < 3 2 sinh 2 ) }. One heks that O 2 O 1 ;thus P O2 is injetive and H O 2 is biholomorphi to H exp H Ci P O 2 )). Moreover, one has exp H CA,B,C )+ia, B, C)) =exp H A,B,C )exp H C ia, B, C 1 2 A B AB ))). It follows that exp 1 H exp H C H Ci P O 2 )) = { Im z 1 ) 2 +Imz 2 ) 2 < 3 }, where z 1,z 2,z 3 )=A + ia, B + ib, C + ic) are natural omplex oordinates of h C = C 3, and this yields the statement. Remark. Sine P is injetive on O 1,the H-invariant domain defined by O 1 is holomorphially separable. As a matter of fat, one may show that P O1 ) = h \{A, B, C) h : A 2 + B 2 3, C =0}, and analogous arguments as above show that suh an H-invariant domain is not holomorphially onvex. 5. Generalized Heisenberg groups Here we apply results of the previous setion to generalized Heisenberg groups, exhibiting additional examples of non-stein maximal domains of existene for the adapted omplex struture. We refer to [BTV] for the basi properties of generalized Heisenberg groups.

12 4592 S. HALVERSCHEID AND A. IANNUZZI Let G be a generalized Heisenberg group with Lie algebra g. Consider its abelian subalgebra z := [g, g] and the subspae v orthogonal to z with respet to the G-invariant metri, ) of G. Then for all V + Y v z = g = T e G, the unique geodesi tangent to V + Y at 0 an be expliitly given by t exp G ϕ G t, V + Y ) for a ertain real-analyti map ϕ G : R T e G g see [BTV, the theorem on p. 31]). A straightforward omputation shows that ϕ G extends holomorphially on C T e G to g C, and arguments analogous to the previous setion imply that Ω G := G O G TG is the maximal adapted omplexifiation, where O G is the onneted omponent of { V + Y T e G : detd P G ) V +Y 0} ontaining 0, and P G : T e G g is given by V + Y l Y )V + 1+ V 2 m Y ))Y. Here denotes the norm indued by, ), and the real-analyti funtions l, m : R R are defined by lt) := sinht) t sinht) osht), mt) := t 2t 3. Now for V + Y v z = T e G with Y 0 and U + X T V +Y T e G = T e G,one has D P G ) V +Y U + X) = t P G V + tu)+y + tx)) t=0 = l Y ) Y,X) V + l Y )U Y + 2V,U)m Y )+ V 2 m Y ) Y,X) Y +1+ V 2 m Y ))X. Note that the equation is written aording to the splitting v z, and sine l Y ) never vanishes, the v -part vanishes if and only if U = Y,X) l Y ) V. Y l Y ) It follows that the entral z-part also vanishes if and only if 1 + V 2 m Y ))X = V 2 X, Y ) ) 2 l Y ) Y l Y ) m Y ) m Y ) Y. In partiular, Y and X have to be proportional. Sine both sides are homogeneous of degree 1 in X, then Dϕ) V +Y ) is singular if and only if ) 5) 1 + V 2 m Y ) = V 2 Y 2 l Y ) l Y ) m Y ) m Y ). An analogous omputation shows that D P G ) V +Y has maximal rank if Y =0; thus equation 5) desribes the singular lous of D P G. It is remarkable that this identity is independent of the fine struture of the generalized Heisenberg group, e.g., of its dimension or the dimension of its enter. In partiular, if H is the 3-dimensional Heisenberg group onsidered in the previous setion, equation 5) determines the boundary of O H =Ω H T e H. Using this fat, we are now going ) Y

13 MAXIMAL COMPLEXIFICATIONS 4593 to show that for a generalized Heisenberg group G there exist many opies of Ω H embedded as losed submanifolds in Ω G. Let G be a generalized Heisenberg group and hoose non-zero elements V 1 v and Ȳ z. Then there exists an element V 2 v suh that the losed subgroup exp G span{ V 1, V 2, Ȳ }) is a totally geodesially embedded 3-dimensional Heisenberg group see [BTV, p. 30]). Denote by I : H G suh an embedding, and note that sine exp G C : g C G C is a biholomorphism, I extends to a holomorphi embedding I C : H C G C of the universal omplexifiation of H into the universal omplexifiation of G suh that the diagram h C DIC g C exp H C H C I C G C exp G C ommutes. Now I : H G is totally geodesi; thus, t I exp H ϕ H t, v) the unique geodesi of G tangent to DIv) at 0 for all v h.then I exp H ϕ H t, v) =exp G ϕ G t, DIv)), and by the identity priniple, 6) I C exp H C ϕ H z,v) =exp G C ϕ G z,div)) for all z C, sine this holds for all z R. The ommutativity of the diagram 7) TH DI TG is P H P G H C I C G C follows. For this, note that, sine I is a group homomorphism, then DI : TH TG is H-equivariant, i.e., DIg w)=ig) DIw) for all g H and w TH. In partiular, P G DIg v)=p G Ig) DIv))=Ig) exp G C ϕ G i, DIv)) for all g H and v T e H. On the other hand, using equation 6), one obtains I C P H g v)=i C g exp H C ϕ H i, v)) = I C g) I C exp H C ϕ H i, v)) = Ig) exp G C ϕ G i, DIv)), showing that the above diagram is ommutative. From the equivariane of DI it follows that DIΩ H )= DIH O H )=IH) DIO H ). Sine DI is isometri and the boundary of O H is defined by equation 5), whih also desribes the singular lous of PG, one has DIO H ) O G span{ V 1, V 2, Ȳ }, DI O H) O G. Thus DIΩ H ) = IH) DIO H ) is losed in Ω G = G OG. Furthermore, DI is injetive and P H, P G are loally biholomorphi where the adapted omplex struture is defined f. Theorem 3.2). Thus diagram 7) shows that DIΩ H ) = Ω H is a losed omplex submanifold of Ω G. Finally, by Proposition 4.3, the domain Ω H is neither holomorphially separable, nor holomorphially onvex. Thus one has

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