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Failure of extensions of real-analytic CR functions

In this section, we focus our attention on functions satisfying the pointwise Cauchy–Riemann conditions on a CR singular image with a nonempty CR singu-lar set S. We shall show that for each p∈S there exists a real-analytic function on a neighborhood ofpinM, satisfying all the pointwise Cauchy–Riemann condi-tions, that does not extend to a holomorphic function atp. This result generalizes Lemma4.3.

Theorem 7.1. Let M⊂Cn be a connected real-analytic CR singular subman-ifold such that there are a real-analytic generic submansubman-ifold N⊂Cn and a real-analytic CR map f:N→Cn that is a diffeomorphism ontoM=f(N). Suppose S is the CR singular set ofM andM\S is generic. Then,for anyp∈S,there exist a neighborhoodU ofpand a real-analytic functionuon U∩M such thatLu|q=0for

anyq∈U∩M and anyL∈Tq(0,1)M,butudoes not extend to a holomorphic function on any neighborhood ofpin Cn.

Proof. LetF be the unique holomorphic extension off to a neighborhood V off1(p) inCn. Letθ=JF and ϕ=θ2◦f1. Thenϕ is a real-analytic function on f(V∩N). We claim thatϕsatisfies all CR conditions onM. Indeed, for any point q∈M nearp, letL∈Tq(0,1)M. Ifq∈S, thenq:=f1(q)∈f1(S)⊂{x∈N:θ=0} and hence θ(q)=0. If q∈M\S, then Xq:=(f1)L∈Tq(0,1) N and thus Xq(θ)(q)=0.

Therefore for allq∈U∩M, we have

(Lϕ)(q) =L(θ2◦f1)(q) = ((f1)L)θ2(f1(q))

=Xqθ2(q) = 2θ(q)(Xqθ)(q) = 0.

(56)

Hence, the claim follows.

Ifϕdoes not extend to a holomorphic function in a neighborhood of pinCn, then we are done. Otherwise, suppose that ϕextends to a neighborhood. Notice thatϕ≡0 onS. Without loss of generality we can assume further thatϕis radical.

On the other hand, by Lemma 4.3, we can find a real-analytic function uon a neighborhood U of p in M such that u is CR on U\S and u does not extend holomorphically nearp. By construction,uϕandu2ϕrestricted toU are both CR functions onM∩U. We claim that at least one of the two functions and u2ϕ does not extend holomorphically pastp. Indeed, assume for a contradiction thatv1

andv2 are holomorphic functions on a neighborhood ofpin Cn whose restrictions toM areandu2ϕ, respectively. Observe that the following equalities hold onU,

(57) v12=u2ϕ2=v2ϕ.

Since M is generic at all points on M\S, and v21 and v2ϕ are holomorphic, we deduce from (57) that v21=v2ϕ in a neighborhood ofp in Cn. In other words, we have the following equality in the ringOp,

(58) v12=v2ϕ.

Note that the ringOp is a unique factorization domain andϕis radical. From (58) we obtain thatv1 dividesv2and hence v2/v1 is holomorphic nearp. Consequently, u extends to the holomorphic function v2/v1 on a neighborhood of p. We have obtained a contradiction.

8. Examples

We start this section by the following proposition, which is helpful in construct-ing examples.

Proposition 8.1. Let w=ρ(z,z)¯ define a connected CR singular manifoldM near the origin in the coordinates (z, w)∈C2×C,where ρis real-analytic and such that ρ=0 anddρ=0at the origin. If ρz¯10,thenM is Levi-flat at CR points and, furthermore,the set S of CR singularities is given by M∩{(z, w):ρz¯2(z)=0}.

Moreover,for each pointp∈M, there exists a neighborhoodU such thatU∩M is the image under a real-analytic CR diffeomorphism of an open subset of R2×C. Note that if w=ρ(z,z) and¯ ρ¯z10, we could also get that M is a complex manifold, but in this caseM is not CR singular.

Proof. If ρz¯20 then ρ is holomorphic and hence M is complex-analytic.

Otherwise,ρ¯z20 and therefore, from Proposition 5.4, we see that (59) S=M∩{(z, w) :ρz¯2(z,z) = 0¯ }.

Hence, S⊂M is a proper real subvariety and M\S is generic. To see that M\S is Levi-flat, observe that in a neighborhood of p /∈S, M\S is foliated by a family of one-dimensional complex submanifolds defined by Lt={(z, w):w=ρ(z1, t,0,¯t)} with complex parametert. Let the local mapf:R2×C→M be given by

(60) f: (x, y, ξ)→(ξ, x+iy, ρ(ξ, x+iy,0, x−iy)).

The CR structure onR2×Cis given by∂/∂ξ¯and so clearly f is a CR map. Since ρdoes not depend on ¯z1, it follows that f sendsR2×CintoM. The fact thatf is a local diffeomorphism is immediate.

Using the proposition we can easily create many examples showing that the CR singular set of a Levi-flat manifold that is an image of a CR diffeomorphism can have any possible CR structure allowed by Corollary5.2.

Example 8.2. We can obtain a 3-dimensional CR singularity by simply taking a parabolic CR singular Bishop surface in 2 dimensions and considering it in 3 dimensions. For example,

(61) w=|z2|2z22

2 .

The manifold is the image of R2×C by the construction of Proposition 8.1. The CR singular set is the set{(z, w):Rez2=0}∩M, and hence 3 real dimensional.

The submanifold is contained in the nonsingular Levi-flat hypersurface defined by Imw=−Im12z22.

Example 8.3. Next, let us consider

(62) w=z1z¯22.

The manifold is the image of R2×C by the construction of Proposition 8.1. The CR singular set is the set

({(z, w) :z1= 0}∪{(z, w) :z2= 0})∩M,

that is a union of two 2-dimensional sets, both of which are complex-analytic. Note that the set{(z, w):z1=0}∩M is a complex-analytic set that is an image of a totally real submanifold ofR2×Cunder the map of Proposition8.1. We therefore have a complex-analytic set that is a subset of M while not being an image of one of the leaves of the Levi-foliation ofR2×C.

Example 8.4. Consider

(63) w=z1z¯2¯z22

2 .

Again the manifold is the image ofR2×C. The CR singular setS is the set {(z, w) :z1= ¯z2}∩M,

which is a totally real set; to see this fact simply substitute ¯z2=z1 in the defining equation forM to find thatS is the intersection of {(z, w):z1= ¯z2}with a complex manifold.

Example 8.5. Consider

(64) w=z1z¯2−z2z¯22 2 .

The CR singular set S is the set {(z, w):z1=|z2|2}∩M, which is a CR singular submanifold.

Example 8.6. While we have mostly concerned ourselves with flat manifolds, there is nothing particularly special about flat manifolds. Even a finite-type man-ifold can map to a CR singular manman-ifold. The following example was given as Example 1.6 in [ER]. LetM⊂C3be given by

(65) M= M is taken to be the CR singular manifold

(66) {(z1, z2, w)∈C3:w= (¯z2+i|z1|2+|z1|4)2} via the finite holomorphic map

(67) (z, w1, w2)→(z, w1+iw2,(w1−iw2)2).

The map is a diffeomorphism onto its image when restricted toM.

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