Boundary regularity of correspondences in &#x2102

Boundary regularity of correspondences in C C C

ⁿⁿⁿ

RASUL SHAFIKOV¹and KAUSHAL VERMA²

1Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7, USA

2Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India MS received 6 October 2005

Abstract. LetM, Mbe smooth, real analytic hypersurfaces of finite type inCⁿandfˆ a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toMand mapsMtoM. It is shown thatfˆmust extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich–Pinchuk extension result for CR maps.

Keywords. Correspondences; Segre varieties.

1. Introduction and statement of results 1.1 Boundary regularity

LetU, Ube domains inCⁿand letM ⊂ U, M ⊂U be relatively closed, connected, smooth, real analytic hypersurfaces of finite type (in the sense of D’Angelo). A recent result of Diederich and Pinchuk [DP3] shows that a continuous CR mappingf:M→M is holomorphic in a neighbourhood ofM. The purpose of this note is to show that their methods can be adapted to prove the following version of their result for correspondences.

We assume additionally that M (resp. M) divides the domainU (resp.U) into two connected componentsU⁺andU⁻(resp.U^±).

Theorem 1.1. Letfˆ:U⁻→Ube a holomorphic correspondence that extends continu- ously up toMand mapsMtoM, i.e.,f (M)ˆ ⊂M. Thenfˆextends as a locally proper holomorphic correspondence acrossM.

We recall that ifD⊂C^pandD⊂C^mare bounded domains, a holomorphic correspon- dencefˆ:D→Dis a complex analytic setA⊂D×Dof pure dimensionpsuch that A∩(D×∂D)= ∅, where∂Dis the boundary ofD. In this situation, the natural projec- tionπ:A→Dis proper, surjective and a finite-to-one branched covering. If in addition the other projectionπ:A→Dis proper, the correspondence is called proper. The analytic setAcan be regarded as the graph of the multiple valued mappingfˆ:=π◦π⁻¹:D→D. We also use the notationA=Graph(f ).ˆ

The branching locusσ of the projectionπis a codimension one analytic set inD. Near each point inD\σ, there are finitely many well-defined holomorphic inverses ofπ. The symmetric functions of these inverses are globally well-defined holomorphic functions on D. To say thatfˆis continuous up to∂Dsimply means that the symmetric functions extend 59

continuously up to∂D. Thus in Theorem 1.1 the various branches offˆare continuous up toMand each branch maps points onMto those onM.

We say thatfˆin Theorem 1.1 extends as a holomorphic correspondence acrossMif there exist open neighbourhoodsU˜ ofMandU˜ofM, and an analytic setA˜ ⊂ ˜U× ˜U of pure dimensionnsuch that (i) Graph(f )ˆ intersected with(U˜ ∩U⁻)×(U˜∩U)is contained in A˜ and (ii) the projectionπ˜:A˜ → ˜U is proper. Without condition (ii),fˆ is said to extend as an analytic set. Finally, the extension offˆis a proper holomorphic correspondence if in addition to (i) and (ii),π˜:A˜→ ˜Uis also proper.

COROLLARY 1.1

LetDandDbe bounded pseudoconvex domains inCⁿwith smooth real-analytic bound- ary. Letfˆ:D→Dbe a holomorphic correspondence. Thenfˆextends as a locally proper holomorphic correspondence to a neighbourhood of the closure ofD.

The corollary follows immediately from Theorem 1.1 and [BS] where the continuity offˆ is proved. This generalizes a well-known result of [BR] and [DF] where the extension past the boundary ofDis proved for holomorphic mappings.

1.2 Preservation of strata

LetM_s⁺(resp.M_s⁻) be the set of strongly pseudoconvex (resp. pseudoconcave) points on M. The set of points where the Levi formL_ρhas eigenvalues of both signs onT^C(M)and no zero eigenvalue will be denoted byM^±and finallyM⁰will denote those points where L_ρhas at least one zero eigenvalue onT^C(M).M⁰is a closed real analytic subset ofM of real dimension at most 2n−2. Then

M=M_s⁺∪M_s⁻∪M^±∪M⁰.

Further, letM⁺(resp.M⁻) be the pseudoconvex (resp. pseudoconcave) part ofM, which equals the relative interior of Ms⁺(resp.Ms⁻). For non-negative integersi, j such that i+j = n−1, letMi,j denote those points at whichLρ has exactlyi positive and j negative eigenvalues onT^C(M). Each (non-empty)M_i,j is relatively open inMand semi- analytic whose relative boundary is contained inM⁰. With this notation,M0,n−1 =M_s⁻ andMn−1,0 =M_s⁺. Moreover,M^±is the union of all (non-empty)M_i,j where bothi, j are at least 1 and i+j = n−1. Note that points inM_s⁻, M^±are in the envelope of holomorphy ofU⁻. Following [B], there is a semi-analytic stratification forM⁰given by

M⁰=1∪2∪3∪4, (1.1)

where4 is a closed, real analytic set of dimension at most 2n−4 and2∪3∪4

is also a closed, real analytic set of dimension at most 2n−3. Further,1, 2, 3 are either empty or smooth, real analytic manifolds;2, 3have dimension 2n−3, and1

has dimension 2n−2. Finally,2and3are CR manifolds of complex dimensionn−2 andn−3 respectively. The set of points, denoted by¹_hin1where the complex tangent space to1has dimensionn−1 is semi-analytic and has real dimension at most 2n−3, as otherwise there would exist a germ of a complex manifold inMcontradicting the finite type hypothesis. Then1\_h¹is a real analytic manifold of dimension 2n−2 and has CR dimensionn−2. Using the same letters to denote the various strata ofM⁰, there exists a

refinement of (1.1), so that1, 2, 3are all smooth, real analytic manifolds of dimensions 2n−2,2n−3,2n−3 respectively, while the corresponding CR dimensions aren−2, n−2, andn−3. Finally,4is a closed, real analytic set of dimension at most 2n−4.

Theorem 1.2. With the hypothesis of Theorem 1.1, the extended correspondencefˆ:M→ Msatisfies the additional properties:f (Mˆ ⁺)⊂M⁺,f (Mˆ ⁺∩M⁰)⊂M⁺∩M⁰and f (Mˆ ⁻)⊂M⁻,f (Mˆ ⁻∩M⁰)⊂M⁻∩M⁰. Moreover,f (Mˆ ⁺∩_j)⊂M⁺∩_jand f (Mˆ ⁻∩_j)⊂M⁻∩_jforj =1,3,4. Finally,fˆmaps the relative interior ofM^±to the relative interior ofM^±.

Preservation of2is not always possible even for holomorphic mappings as the following example shows: the domain = {(z1, z2): |z1|²+ |z2|⁴ < 1}is mapped to the unit ball inC²by the proper holomorphic mappingf (z1, z2)= (z1, z₂²). Points of the form {(e^iθ,0)} ⊂∂are weakly pseudoconvex and in fact form2⊂∂, andf maps them to strongly pseudoconvex points.

2. Segre varieties

We will writez = (z, z_n) ∈ Cⁿ⁻¹×Cfor a point z ∈ Cⁿ. The word ‘analytic’ will always mean complex analytic unless stated otherwise. The techniques of Segre varieties will be used and here is a synopsis of the main properties that will be needed. The proofs of these can be found in [DF] and [DW]. As described above, letM be a smooth, real analytic hypersurface of finite type inCⁿthat contains the origin. IfUis small enough, the complexificationρ(z, w)ofρis well-defined by means of a convergent power series in U×U. Note thatρ(z, w)is holomorphic inzand anti-holomorphic inw. For anyw∈U, the associated Segre variety is defined as

Q_w = {z∈U:ρ(z, w)=0}.

By the implicit function theorem, it is possible to choose neighbourhoodsU1⊂⊂U2of the origin such that for anyw∈U1,Q_wis a closed, complex hypersurface inU2and

Q_w = {z=(z, z_n)∈U2:z_n=h(z, w)},

whereh(z, w) is holomorphic inz and anti-holomorphic inw. Such neighbourhoods will be called a standard pair of neighbourhoods and they can be chosen to be polydiscs centered at the origin. It can be shown thatQ_w is independent of the choice ofρ. For ζ ∈ Q_w, the germ Q_w at ζ will be denoted by _ζQ_w. LetS := {Q_w: w ∈ U1} be the set of all Segre varieties, and let λ:w → Q_w be the so-called Segre map. ThenS admits the structure of a finite dimensional analytic set. It can be shown that the analytic set

I_w:=λ⁻¹(λ(w))= {z:Q_z=Q_w}

is contained inMifw∈M. Consequently, the finite type assumption onMforcesI_wto be a discrete set of points. Thusλis proper in a small neighbourhood of each point ofM.

Forw∈U₁⁺, the symmetric point^swis defined to be the unique point of intersection of the complex normal toMthroughwandQ_w. The component ofQ_w∩U₂⁻that contains the symmetric point is denoted byQ^c_w.

Finally, for all objects and notions considered above, we simply add a prime to define their corresponding analogs in the target space.

3. Localization and extension across an open dense subset ofMMM

In the proof of Theorem 1.1 in order to show extension offˆas a holomorphic correspon- dence, it is enough to consider the problem in an arbitrarily small neighbourhood of any pointp∈M. The reason is the following. Firstly, since the projectionπ: Graph(f )ˆ →U⁻ is proper, the closure of Graph(f )ˆ has empty intersection withU⁻×∂U. Therefore, by [C] § 20.1, to prove the continuation offˆacrossMas an analytic set, it is enough to do that in a neighbourhood of any point inM. Secondly, once the extension offˆas a holomorphic correspondence in a neighbourhood of any pointp∈Mis established, then globally there exists a holomorphic correspondence defined in a neighbourhoodU˜ ofMwhich extends fˆ. To see that simply observe that ifF ⊂ ˜U× ˜Uis an analytic set extendingfˆ, then by choosing smallerU˜ we may ensure that the projection to the first component is proper, as otherwise there would exist a pointzonMsuch thatF (z)ˆ has positive dimension (hereFˆ is the map associated with the setF). This however contradicts local extension offˆnear zas a holomorphic correspondence.

Since the projection π: Graph(f )ˆ → U⁻ is proper, Graph(f )ˆ is contained in the analytic set A ⊂ U⁻×U, defined by the zero locus of holomorphic functions P1(z, z₁), P2(z, z₂), . . . , P_n(z, z_n)given by

Pj(z, z_j)=z_j^l+aj1(z)z_j^l−1+ · · · +aj l(z), (3.1) wherelis the generic number of images inf (z)ˆ , and 1≤j ≤n(for details, see [C]). The coefficientsaµν(z)are holomorphic inU⁻and extend continuously up toM. This is the definition of continuity of the correspondencefˆup toMwhich is equivalent to that given in §1.1.

The discriminant locus is{Rj(z)=0}, 1≤j ≤n, whereRj(z)is a universal polyno- mial function ofaj µ(z)(1≤µ≤l) and hence by the uniqueness theorem, it follows that {R_j(z)=0} ∩Mis nowhere dense inM, for allj. The set of pointsSonMwhich do not belong to{R_j(z)=0} ∩Mfor anyjis therefore open and dense inM. Near each pointp onS,fˆsplits into well-defined holomorphic mapsf1(z), f2(z), . . . , f_l(z)each of which is continuous up toM.

Ifp∈S∩(M⁻∪M^±), the functionsa_µν(z)extend holomorphically to a neighbourhood of p and hence fˆextends as a holomorphic correspondence across p. It is therefore sufficient to show thatfˆextends across an open dense subset ofS∩M⁺. But this follows from Lemma 3.2 and Corollary 3.3 in [DP3]. We denote by⊂Mthe non-empty open dense subset ofMacross whichfˆextends as a holomorphic correspondence.

4. Extension as an analytic set

Fix 0∈Mand letp₁, p₂, . . . , p_k ∈ ˆf (0)∩M. The continuity offˆallows us to choose neighbourhoods 0 ∈ U1andp_i ∈ U_i and local correspondencesfˆ_i:U₁⁻ → U_i that are irreducible and extend continuously up toM. Moreover,fˆ_i(0)=p_ifor all 1≤i ≤k. It will suffice to focus on one of thefˆ_i’s, sayfˆ₁and to show that it extends holomorphically across the origin. Abusing notation, we will writefˆ₁= ˆf,U₁ =Uandp₁=0. Thus fˆ:U₁⁻→Uis an irreducible holomorphic correspondence andf (0)ˆ =0. Define

V⁺= {(w, w)∈U₁⁺×U:f (Qˆ ^c_w)⊂Q_w}.

ThenV⁺is non-empty. Indeed,fˆextends across an open dense set near the origin and [V] shows that the invariance property of Segre varieties then holds. Moreover, a similar argument as in [S2] shows that V⁺ ⊂ U₁⁺×Uis an analytic set of dimensionnand exactly the same arguments as in Lemmas 4.2 – 4.4 of [DP3] show that: first, the projection π:V⁺→U⁺:=π(V⁺)⊂U₁⁺is proper (and hence thatU⁺⊂U₁⁺is open) and second, the projectionπ:V⁺→Uis locally proper. Thus, toV⁺is associated a correspondence F⁺:U⁺→Uwhose branches areFˆ⁺=π◦π⁻¹.

Leta ∈ M be a point close to the origin, across whichfˆextends as a holomorphic correspondence. Iffˆis well-defined in the ballB(a, r),r >0 andw∈B(a, r)⁻, it follows from Theorem 4.1 in [V] that all points inf (w)ˆ have the same Segre variety. By analytic continuation, the same holds for allw∈U₁⁻. Using this observation, it is possible to define another correspondenceF⁻:U₁⁻→Uwhose branches areFˆ⁻(w)=(λ)⁻¹◦λ◦ ˆf (w). LetU:=U₁⁻∪U⁺∪(∩U1). The invariance property of Segre varieties shows that the correspondencesFˆ⁺,Fˆ⁻can be glued together near points on∩U1. Hence, there is a well-defined correspondenceFˆ:U→Uwhose values overU⁺andU₁⁻areFˆ⁺andFˆ⁻ respectively. Note that

F :=Graph(F )ˆ = {(w, w)∈U×U:w∈ ˆF (w)}

is an analytic set inU×Uof pure dimensionn, with proper projectionπ:F →U. Once again, the invariance property shows that all points inF (w),ˆ w∈U, have the same Segre variety.

Lemma4.1. The correspondenceFˆ satisfies the following properties:

(i) Forw0∈∂U∩U₁⁺, cl_Fˆ(w0)⊂∂U. (ii) cl_Fˆ(0)⊂Q₀.

(iii) If cl_Fˆ(0)= {0}, then 0∈.

(iv) F ⊂(U1\(M\))×Uis a closed analytic set.

Proof.

(i) Choose(w_j, w_j)∈F converging to(w0, w₀)∈(∂U∩U₁⁺)×U. Thenf (Qˆ ^c_wj)⊂ Q_w

j

for allj. Ifw₀ ∈U, then passing to the limit, we getf (Qˆ ^c_w0)⊂Q_w 0 which shows that(w0, w₀) ∈ F and hencew0 ∈ U, which is a contradiction. This also proves (iv).

(ii) Choosew_j ∈ U converging to 0. There are two cases to consider. First, if w_j ∈ U₁⁻∪(∩U1)for allj, it follows thatf (wˆ j)→0. Moreover, for anyw_j ∈ ˆF (wj), Q_w

j =Q_ˆ

f (wj). IfUis small enough, the equalityQ_w =Q₀ implies thatw=0 and thus we conclude thatw_j → 0 ∈ Q₀. Second, ifw_j ∈ U⁺ for allj, then f (Qˆ ^c_wj) ⊂ Q_w

j

for any w_j ∈ ˆF (w_j). Let w_j → w₀ ∈ U. If ζ ∈ Q^c_wj, then f (ζ )ˆ ∈ Q_w

j → Q_w

0. But w_j → 0 implies that dist(Q^c_wj,0) → 0 and hence f (ζ )ˆ →0. Thus 0∈Q_w

0which shows thatw₀ ∈Q₀.

(iii) If cl_Fˆ(0)= {0}, then (i) shows that 0∈/∂U∩U₁⁺. LetB(0, r)be a small ball around the origin such thatB(0, r)∩∂U = ∅. The correspondenceFˆ overB(0, r)⁺is the union of some components of the zero locus of a system of monic pseudo-polynomials

whose coefficients are bounded holomorphic functions onB(0, r)⁺. By Trepreau’s theorem, all these coefficients extend holomorphically toB(0, r), and the extended zero locus contains the graph offˆnear the origin since is dense. It follows that

0∈. 2

Following [S1], for anyw0∈U, it is possible to find a neighbourhoodofw0, relatively compact inUand a neighbourhoodV ⊂U1ofQ_w0∩U1such that forz∈V,Q_z∩is non-empty and connected. Associated with the pair(, V )is

F˜ := ˜F (w0, , V )= {(z, z)∈V ×U:F (Qˆ _z∩)⊂Q_z} (4.1) which (see [DP4]) is an analytic set of dimension at mostn. Ifw0∈, then Corollary 5.3 of [DP3], shows thatF∩(V×U)is the union of irreducible components ofF˜of dimension n. As in [DP3] we call(w0, z0)∈U×Q_w0a pair of reflection if there exist neighbourhoods (w0) w0and(z0) z0such that for allw ∈ (w0),F (Qˆ _w ∩(z⁰)) ⊂Q_ˆ

F (w). It follows from the invariance property of Segre varieties that the definition of the pair of reflection is symmetric. As an example we note that if the setF˜ defined in (4.1) contains F∩(V×U), then(w0, z)is a point of reflection for any pointzin a connected component ofQ_w0∩Ucontainingw0.

Letw0∈U,z0∈Q_w0∩be a pair of reflection. FixB(z0, r), a small ball aroundz0

wherefˆis well-defined and letS(w0, z0)⊂ ˜F∩((Q_w0∩U1)×U)be the union of those irreducible components that contain Graph(f )ˆ overQ_w0 ∩B(z0, r). Note thatS(w0, z0) is an analytic set of dimensionn−1 and is contained in(Q_w0∩U1)×Uand moreover, the invariance property shows that

S(w0, z0)⊂((Qw0∩U1)×(Q_ˆ

F (w0)∩U)).

Furthermore, from the above considerations it follows that for anyz∈π(S(w0, z0))the point(w0, z)is a pair of reflection. Finally, let the cluster set of a sequence of closed sets {Cj} ⊂D, whereDis some domain, be the set of all possible accumulation points inD of all possible sequences{c_j}wherec_j ∈C_j.

PROPOSITION 4.1

Let{z_ν} ∈converge to0. Suppose that the cluster set of the sequence{S(z_ν, z_ν)}contains a point(ζ0, ζ₀)∈U×U. Thenfˆextends as an analytic set across the origin.

Proof. First, the pair(z_ν, z_ν)is an example of a pair of reflection and henceS(z_ν, z_ν) is well-defined. Also, note that(z_ν,f (zˆ _ν)) → (0,0). Choose(ζ_ν, ζ_ν) ∈ S(z_ν, z_ν)that converges to(ζ0, ζ₀)∈ U×U. It follows that(ζ_ν, z_ν)is a pair of reflection. Let, V be neighbourhoods ofζ0 andQζ0 as in the definition ofF (ζ˜ 0, , V ). Sinceζ0 ∈ U, it follows thatF (ζ˜ 0, , V )is a non-empty, analytic set inV ×U. ShrinkingU1if needed, Q_ζν∩U1⊂V andζ_ν ∈for all largeν. This shows thatF (ζ˜ _ν, , V )= ˜F (ζ0, , V )for all largeν. Lemma 5.2 of [DP3] shows thatF (ζ˜ _ν, , V )contains the graph of all branches of fˆnear z_ν and hence F (ζ˜ 0, , V )contains the graph of fˆnear (0,0). Therefore, F (ζ˜ 0, , V )extends the graph offˆacross the origin. 2 Remarks. First, as in [DP3] this proposition will be valid if the pair(z_ν, z_ν)were replaced by a pair of reflection(w_ν, z_ν)∈ U×that converges to(0,0)andF (wˆ _ν)clusters at some point in U. Second, this proposition shows the relevance of studying the cluster

set of a sequence of analytic sets (see [SV] also). In general, the hypothesis that the cluster set of {S(z_ν, z_ν)} (or S(w_ν, z_ν) in case (w_ν, z_ν) is a pair of reflection) con- tains a point in U ×U cannot be guaranteed since the projection π: S(z_ν, z_ν) → U is not known to be proper. However, the following version of Lemma 5.9 in [DP3]

holds.

Lemma4.2. There are sequences(w_ν, z_ν) ∈ U ×, w_ν ∈ ˆF (w_ν)and analytic sets σ_ν ⊂Uof pure dimensionp≥1 (pindependent ofν) such that:

(i) (w_ν, z_ν)→(0,0)and(w_ν, z_ν)is a pair of reflection for allν. (ii) w_ν →w₀∈Uandz_ν ∈σ_ν ⊂π(S(w_ν, z_ν)).

Proof. Choose a sequence zν ∈ that converges to the origin. If the projections π:S(z_ν, z_ν)→U were proper for allν, then for some fixedr >0 andνlarge enough, let σ_ν := Q_zν ∩B(z_ν, r),w_ν = z_ν and w_ν ∈ ˆf (z_ν). It can be seen that the lemma holds with these choices. On the other hand, ifπis not known to be proper onS(z_ν, z_ν), no fixed value of r, as described above, exists. Hence, for arbitrarily small values of r >0, there exist(w_ν, w_ν)∈S(z_ν, z_ν)∩(U⁺×U)such thatw_ν →0 andw_ν →w₀ with |w₀| = r. SinceM is of finite type, we may assume thatQ_w

0 = Q₀. More- over, note thatw₀ ∈ Q₀ ∩U (which shows that 0 ∈ Q_w

0) and(wν, zν)is a pair of reflection for all ν. By making a small holomorphic perturbation of coordinates in the target space, if needed, it follows that 0 ∈ Q_w

0 ∩ {z ∈ U:z₂ = · · · = z_n =0}is an isolated point. Therefore, there exists an > 0 such that after shrinkingU, if needed, q₀ :=Q_w

0∩ {z∈U:z₂= · · · =z_n−1=0,|z_n|< }(which is an analytic set of dimen- sion 1 inU∩ {|z_n|< }containing the origin) has no limit points on∂U∩ {|z_n|< }.

Letl be the multiplicity offˆ:U₁⁻ → U. Letf (zˆ _ν) = {ζ_ν^j},1 ≤ j ≤ l counted with multiplicity. For largeν, thelsets

q_ν,j =Q_w

ν ∩ {z∈U:z_k=(ζ_ν^j)_k, 2≤k≤n−1, |z_n|< }

are analytic, of dimension 1, inU∩ {|z_n|< }without limit points on∂U∩ {|z_n|< } and clearly contain(z_ν, ζ_ν^j). Sinceπ(S(w_ν, z_ν))⊂Q_w

ν,

s_ν,j :=S(w_ν, z_ν)∩ {(z, z):z_k =(ζ_ν^j)_k, 2≤k≤n−1}

are analytic sets of dimension at least 1 inU1×(U∩ {|z_n| < })for all 1 ≤ j ≤ l.

By construction, the analytic sets q_ν,j do not have limit points on ∂U ∩ {|z_n| < } and hence sν,j do not have limit points on U1×(∂U∩ {|z_n| < }). By Lemma 4.1, cl_Fˆ(0)⊂Q₀ = {z_n=0}and by shrinkingU1if needed, this shows thats_ν,jhave no limit points onU1×(U∩ {|z_n| =}). Thus for largeνand allj, the projectionsπ:s_ν,j →U1

are proper and their imagesσ_ν,j :=π(s_ν,j)are analytic sets inU1of dimension at least 1 andz_ν ∈ σ_ν,j for allν, j. It remains to pass to subsequences if necessary to chooseσ_ν,j

with constant dimension. 2

One conclusion that follows now is: iffˆdoes not extend as an analytic set across the origin, then cl(σ_ν) ⊂ M\. Indeed, if there existsζ0 ∈ cl(σ_ν)∩(U1\(M\)), let (ζ_ν, ζ_ν) ∈ S(w_ν, z_ν)converge to(ζ0, ζ₀) ∈ U1×U. Proposition 4.1 now shows that ζ0∈∂U∩U1. But sinceζ0∈/M\, it follows from Lemma 4.1 thatζ₀∈∂Uwhich is a contradiction.

The goal will now be to show thatfˆextends as an analytic set across the origin. For this, choose{z_ν} ∈converging to the origin and consider the analytic setsS(z_ν, z_ν). By Proposition 4.1, it suffices to show thatπ(cl(S(z_ν, z_ν))∩U = ∅. Let

S:=π(cl(S(z_ν, z_ν))∩({0} ×U))⊂Q₀

and letmbe the dimension ofSˆ– the smallest closed analytic set containingS(the so- called Segre completion of [DP3]). Ifm=0, then 0 is an isolated point inS and after shrinkingU1, Usuitably, it follows that cl(S(z_ν, z_ν))has no limit points onU1×∂U. Thusπ:S(z_ν, z_ν)→U1are proper projections and thereforeπ(S(z_ν, z_ν))=Q_zν ∩U1

for all largeν. Henceπ(cl(S(z_ν, z_ν)))=Q0∩U1. Iffˆdid not extend as an analytic set across the origin, the aforementioned remark shows that withσ :=Q_zν∩U1,Q0∩U1= cl(σν)⊂M\⊂M. This cannot happen asMis of finite type. Hencefˆextends as an analytic set across the origin in casem=0. We may therefore suppose thatm > 0. We recall the following lemma proved by Diederich and Pinchuk:

Lemma4.3 ([DP3], Lemma 9.8). LetSbe a subset ofQ₀, 0∈Sandm=dimSˆ. Then after possibly shrinkingU1, there are pointsw¹, . . . , w^k∈S(k≤n−1)such that one of the following holds:

(1) k=manddim(Sˆ∩Q_w1∩ · · · ∩Q_wk)=0;

(2) k≥2m−n+1 and dim(Sˆ∩Q_w1 ∩ · · · ∩Q_wk)=m−k. Thus there are two cases to consider.

Case1. Choose(w1ν, w_1ν), (w2ν, w_2ν), . . . , (w_mν, w_mν)∈ S(z_ν, z_ν)so thatw_µν → 0 andw_µν →w_µ for all 1≤µ≤m. A generic choice ofw_µν(see p. 136 in [DP3]) ensures thatq_mν :=Q_w1ν∩Q_w2ν∩ · · · ∩Q_wmν has dimensionn−m. Each(w_µν, z_ν)is a pair of reflection and hence the analytic set

S_ν^m:=

1≤µ≤m

S(wµν, zν)⊂(q^mν×q^mν)∩(U1×U)

is well-defined. Ifm=n−1, then Lemma 9.7 of [DP3] shows that the germ ofq⁽ⁿ⁻¹⁾at the origin has dimension 1. Moreover,Sˆ=Q₀and Lemma 4.3 implies thatq⁽ⁿ⁻¹⁾∩Q₀

contains 0as an isolated point. Since cl_Fˆ(0)⊂Q₀, it follows that 0is an isolated point of π(cl(S_νⁿ⁻¹)∩({0} ×U))⊂q⁽ⁿ⁻¹⁾∩Q₀ = {0}.

ShrinkingU1, the projectionπ:S_νⁿ⁻¹→U1becomes proper andπ(Sⁿ⁻_ν ¹)=q^n−1,ν∩U1. By Theorem 7.4 of [DP3], there is a subsequence of q^n−1,ν ∩U1 that converges to an analytic setA ⊂ U1 of pure dimension 1 and contains the origin.Acontains pointsζ0

that do not belong toMbecause of the finite type assumption andζ0 ∈ π(cl(S_νⁿ⁻¹)) ⊂ π(cl(S(wµν, zν))). By Proposition 4.1,fˆextends as an analytic set across the origin.

Ifm < n−1, the dimension ofS_ν^m∩S(z_ν, z_ν)is at leastn−m−1>0. Now π(cl(S_ν^m∩S(z_ν, z_ν))∩({0} ×U))⊂q^m∩ ˆS= {0},

the last equality following from Lemma 4.3. The projectionπ:S_ν^m∩S(z_ν, z_ν)→ U1is therefore proper for smallU1and thatπ(S_ν^m∩S(z_ν, z_ν))=q^mν∩Q_zν ∩U1. Again, by

Theorem 7.4 of [DP3], there is a subsequence ofq^mν ∩Qzν ∩U1that converges to an analytic setA⊂U1of positive dimension and as before this shows thatfˆextends as an analytic set across the origin.

Case2. As before, choose(w1ν, w_1ν ), (w2ν, w_2ν ), . . . , (wkν, w_kν )∈S(zν, zν)such that wµν → 0 andw_µν → w_µ for all 1≤ µ ≤ k andqkν = Qw1ν ∩Qw2ν ∩ · · · ∩Qwkν,

˜

q^kν := Q_zν ∩q^kν have dimensionn−k andn−k−1 respectively. Now note that dim(S_ν^k∩S(z_ν, z_ν))≥n−k−1>1. Indeed, the inequalities 2m−n+1≤k < mshow thatm≤n−2 and hencek < n−2. Since the dimension ofSˆ∩q^k ism−k, choose coordinates so that

Sˆ∩q^k∩ {z∈U:z₁=z₂= · · · =z_m−k =0} = {0}.

Letf (zˆ ν)= {ζν^j}, 1≤j ≤l,lbeing the multiplicity offˆ. Thelsets Tν,j = {(z, z)∈S_ν^k∩S(zν, zν):z₁=(ζ_ν^j)₁,

z₂ =(ζ_ν^j)₂, . . . , z_m−k=(ζ_ν^j)_m−k},

where 1≤j ≤lare analytic sets inU1×Uand have dimension at leastn−k−1−(m−k)= n−m−1>0. By construction,

π(cl(T_ν,j)∩({0} ×U))⊂ ˆS∩q^k

∩ {z∈U:z₁ =z₂= · · · =z_m−k=0} = {0}

and hence by shrinkingU1, U, the projectionsπ:T_ν,j →U1are proper and the images σ_ν,j :=π(T_ν,j)⊂U1are analytic and have dimensionn−m−1. Moreoverσ_ν,j ⊂ ˜q^kν, and since q˜^kν depend anti-holomorphically on the k-tuple defining it, Theorem 7.4 of [DP3] shows thatq˜^kνconverges to an analytic setA˜ ⊂U1of dimensionn−k−1, after passing to a subsequence. Working with this subsequence, we see that cl(σ_ν,j)⊂ ˜A. On the other hand, since 2m−n+1≤k, it follows, as in [DP3], that

dimA˜ =n−k−1≤2(n−m−1)=2 dimσ_ν,j.

Proposition 8.3 of [DP3] shows that cl(σ_ν,j)⊂Mand hence by Proposition 4.1, it follows thatfˆextends as an analytic set across the origin.

To complete the proof, it suffices to show that extension as an analytic set implies extension as a locally proper holomorphic correspondence. This is achieved in the next lemma.

Lemma4.4. There exist neighbourhoodsU of0 andUof0such thatF ⊂U×Uis a proper holomorphic correspondence which extendsfˆ.

Proof. Extension as a holomorphic correspondence essentially follows from [DP4]. All nuances in the proof of Proposition 2.4 in [DP4] work in this situation as well provided the following two modifications are made. LetU, Ube neighbourhoods of 0,0respectively and suppose thatF ⊂U×Uextendsfˆas an analytic set inU×U. Then it needs to be checked thatF ∩(U⁺×U)= ∅and that there exists a sequence{z_ν} ∈Mconverging to 0 such thatfˆextends as a correspondence across eachz_ν.

Suppose thatF ∩(U⁺×U)= ∅. In this case, the proof of Proposition 3.1 (or even Proposition 4.1 in [SV]) shows that(0,0)is in the envelope of holomorphy ofU⁻×U. The coefficientsa_µν(z)in (3.1) can be regarded as holomorphic functions onU⁻×U (i.e., independent of thezvariables) and thus eacha_µν(z)extends holomorphically across (0,0). This extension must be independent of thezvariables by the uniqueness theorem and hencea_µν(z)extends holomorphically across the origin. This shows thatfˆextends as a holomorphic correspondence across the origin. To show the existence of the sequence {zν}claimed above, letπ:F →Ube the natural projection and define

A= {(z, z)∈F: dim(π⁻¹(z))_(z,z)≥1},

where(π⁻¹(z))_(z,z)denotes the germ of the fiber overzat(z, z). ThenAis an analytic subset ofF, and sinceFcontains the graph offˆoverU⁻, it follows that the dimension of Ais at mostn−1. Since Lipschitz maps do not increase Hausdorff dimension, it follows that the Hausdorff dimension ofπ(A)is at most 2n−2. Pickp ∈M\π(A). The fiber F ∩π⁻¹(p)is discrete and this shows thatfˆextends as a holomorphic correspondence acrossp.

Finally, we show that U can be chosen so small that the projectionπ:F → U is also proper. Indeed, forz ∈ M,π⁻¹(z)is an analytic subset ofF. Sinceπ is proper, it follows by Remmert’s theorem that Fˆ⁻¹(z) = π ◦π⁻¹(z)is an analytic set. The invariance property of Segre varieties yields F (Qˆ _z∩U ) ⊂ Q_z for any z ∈ ˆF⁻¹(z).

SinceMis of finite type, the set∪_{z∈ ˆF}−1(z)Q_zhas Hausdorff dimensionn, and therefore cannot be mapped byFˆintoQ_zwhich has dimensionn−1. This shows that projectionπ has discrete fibers onM. It follows from the Cartan–Remmert theorem that there exists a neighbourhoodUofMsuch thatπhas only discrete fibers, and therefore the projection πfromF toUwill be proper.

This completes the proof of Theorem 1.1. 2

5. Preservation of strata

Fixp∈Mand letp₁, p₂, . . . , p_k ∈ ˆf (p)⊂M. Choose neighbourhoodsU, Uofp, p₁ respectively and let fˆ1: U⁻ → U be a component of fˆsuch that fˆ1(p) = p₁. Then fˆ1 extends as a holomorphic correspondenceF ⊂ U ×U and to prove Theorem 1.2, it suffices to focus on fˆ1, which will henceforth be denoted by fˆ. The following two general observations can be made in this situation. First, the branching locus σˆ of Fˆ is an analytic set inU and the finite-type assumption onMshows that the real dimension of σˆ ∩M is at most 2n−3. The branching locus of fˆdenoted byσ, is contained in

ˆ

σ ∩U⁻. Second, the invariance property of Segre varieties in [DP1], [V] shows thatFˆ, the extended correspondence, preserves the two componentsU^±. That is, after possibly re-labellingU^±, it follows thatF (Uˆ ^±) ⊂ U^±andF (M)ˆ ⊂ M. The same holds for Gˆ := ˆF⁻¹:U→U.

Proof of Theorem1.2. Letp∈M⁺and suppose that{ζ_j} ∈Mis a sequence converging top₁ with the property that the Levi formLρrestricted to the complex tangent space toM atζ_j has at least one negative eigenvalue. Fixζ_j0 ∈Ufor some largej0. By shiftingζ_j0 slightly, we may assume thatζ_j0 ∈ ˆ/σ∪ ˆF (M⁰∩U ), whereσˆis the branching locus ofGˆ, and at the same time retain the property of having at least one negative eigenvalue. Letg1

be a locally biholomorphic branch ofGˆ nearζ_j0. Theng1(ζ_j0)is clearly a pseudoconvex