# On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

Authors: Joël Merker
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

A connected real analytic hypersurfaceM⊂Cn+1 whose Levi form is nondegenerate in at least one point—hence at every point of some Zariski-dense open subset—is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k,n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having a different signature (l,n-l). Up to signature, pseudosphericity then jumps across the Levi degenerate locus and in particular across the nonminimal locus, if there exists any.


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## Body

## 1. Introduction

The goal of this paper is to provide the complete details of an alternative direct proof of a recent theorem due to Kossovskiy and Shafikov ([1]) which relies on the explicit zero-curvature equations obtained in [2, 3], following the lines of a clever suggestion of Beloshapka. In fact, the proof we give here freely brings a more general statement.LetM⊂Cn+1 be a connected real analytic hypersurface with n⩾1. One says that M is (k,n-k)pseudospherical at one of its points p if it is locally near p biholomorphic to some Heisenberg (k,n-k)-pseudosphere having, in coordinates (z1,…,zn,w)∈Cn+1, the model quadric equation: (1)w=w¯+2i-z1z¯1-⋯-zkz¯k+zk+1z¯k+1+⋯+znz¯n,for some integer k with 0⩽k⩽n-k; when n=1, one simply says that M isspherical. One calls (k,n-k) thesignature.It is known (see [1] and the references therein) that a connected real analytic hypersurface of Cn+1 which is Levi nondegenerate at every point is (k,n-k)-pseudospherical at one point if and only if it is (k,n-k)-pseudospherical at every point. More generally, we establish that propagation of (k,n-k) pseudosphericality also holds in presence of Levi degenerate points of arbitrary kind.Theorem 1.
LetM⊂Cn+1 be a connected real analytic geometrically smooth hypersurface which is Levi nondegenerate in at least one point (hence in some nonempty open subset). Then one has the following:(a)
The set of Levi nondegenerate points ofM is a Zariski open subset of M in the sense that there exists a certainproper—that is, having dimension ⩽dim⁡M-1=2n—locally closed global real analytic subset ΣLD⊂M locating exactly the Levi degenerate points of M: (2)p∈M∖ΣLD⟺LeviformofMatpisnondegenerate.(b)
IfM is locally biholomorphic, in a neighborhood of one of its points p∈M, to some Heisenberg (k,n-k)-pseudosphere having equation, (3)w=w¯+2i-z1z¯1-⋯-zkz¯k+zk+1z¯k+1+⋯+znz¯n,for some integer k with 0⩽k⩽n-k (so that p∈M∖ΣLD necessarily is a Levi nondegenerate point of M too), then locally at every other Levi nondegenerate point q∈M∖ΣLD, the hypersurface M is also locally biholomorphic to some Heisenberg (l,n-l)-pseudosphere with possibly l≠k.Surprisingly, Example  6.2 in [1] shows that l≠k may occur, in the case of anonminimal hypersurface of Cn+1 with n⩾2, a local example for which the Levi degenerate locus ΣLD consists precisely of a complex n-dimensional hypersurface contained in M.In [1], a similar theorem was proved, assuming that ΣLD=S consists of a complex hypersurface S⊂M (nonminimal case), while M∖S consists only of Levi nondegenerate points.In the previous theorem,ΣLD can be arbitrary.


## 2. Proof inC2

LetM⊂C2 be a connected real analytic hypersurface. Pick a point (4)p∈M,and choose some affine coordinates centered at p: (5)z,w=x+iy,u+iv,satisfying (6)T0M=Imw=0,so that the implicit function theorem represents M as (7)u=φx,y,v,in terms of somegraphing functionφ which is expandable in convergent Taylor series in some (possibly small) open bidisc: (8)□ρ02≔z,w∈C2:z<ρ0,w<ρ0,with of course ρ0>0.Classically, writing(9)w+w¯2=φz+z¯2,z-z¯2i,w-w¯2iand using the analytic implicit function theorem, one solves w in terms of z, z¯, and w¯ getting a representation of M as (10)w=Θz,z¯,w¯;recall that implicitly when one does this, one must consider (z,w,z¯,w¯) as 4 independent complex variables, which amounts tocomplexify them, namely, to introduce thecomplexified variables: (11)z,w,z_,w_∈C4;in what follows, we will work with (z,w,z¯,w¯)-variables, keeping in mind that they can be replaced by (z,w,z_,w_)since all objects are convergent Taylor series; so here, Θ(z,z_,w_) is a convergent Taylor series of (z,z_,w_) for |z|<ρ0, |z_|<ρ0, and |w_|<ρ0, after shrinking ρ0>0 if necessary.Moreover, since(12)0=φ0=φx0=φy0=φu0,one has (13)Θ=w¯+O2.Now, it is known ([4, 5])—or it could be taken here as a definition—that M isLevi nondegenerate at 0∈M when the local holomorphic map (14)C2⟶C2,z¯,w¯⟼Θ0,z¯,w¯,Θz0,z¯,w¯is of rank 2 at (z¯,w¯)=(0,0); of course, one would better think in terms of (z_,w_)-variables here.More generally,M isLevi nondegenerate at an arbitrary point close to the origin: (15)zp,wp∈M∩□ρ02when the map (16)C2⟶C2,z¯,w¯⟼Θzp,z¯,w¯,Θzzp,z¯,w¯is of rank 2 at (z¯p,w¯p), which precisely means the nonvanishing of the Jacobian determinant: (17)0≠det⁡Θz¯Θw¯Θzz¯Θzw¯=Θz¯Θzw¯-Θw¯Θzz¯at (18)z,z¯,w¯=zp,z¯p,w¯p.One may either show-check that such a definition regives the standard definition of Levi nondegeneracy (cf. [4, 5]) or prove directly that, as it stands, it really is independent of the choice of coordinates ([6]).Although we could then spend time to reprove it properly, the following fact—here admitted—is well known.Proposition 2.
If a connected real analytic hypersurfaceM2n+1⊂Cn+1 is Levi nondegenerate in at least one point, then the set of Levi degenerate points of M is a proper real analytic subset: (19)ΣLD⫋M.Suppose to begin with forM3⊂C2 that (20)0∉ΣLD.Since the above map is of rank 2 at (z¯,w¯)=(0,0), one can solve, following [2], the two equations: (21)wz=Θz,z¯,w¯,wzz=Θzz,z¯,w¯,for the two variables (z¯,w¯), and then insert the latter in (22)wzzz=Θzzz,z¯,w¯,to get a complex second-order ordinary differential equation: (23)wzzz=Φz,wz,wzz.One should notice that the possibility of solving (z¯,w¯) is expressed by the nonvanishing ofexactly and precisely the same Jacobian determinant as the one expressing Levi nondegeneracy.In [2], one deduces from an explicitly known condition on Φ for this second-order equation wzz=Φ(z,w,wz) to be pointwise equivalent to the free particle Newtonian equation (24)wz′z′′=0that a real analytic hypersurface M⊂C2 which is Levi nondegenerate at 0∈M as above isspherical in the sense—recall the definition—that it is locally biholomorphic to (25)w′=w¯′+2iz′z¯′,if and only if its complex graphing function Θ satisfies an explicit (not completely developed) equation which we now present.Introduce the expression(26)AJ4Θ≔1Θz¯Θzw¯-Θw¯Θzz¯3Θzzz¯z¯Θw¯Θw¯Θz¯Θw¯Θzz¯Θzw¯-2Θzzz¯w¯Θz¯Θw¯Θz¯Θw¯Θzz¯Θzw¯+Θzzw¯w¯Θz¯Θz¯Θz¯Θw¯Θzz¯Θzw¯+Θzzz¯Θz¯Θz¯Θw¯Θw¯w¯Θzw¯Θzw¯w¯-2Θz¯Θw¯Θw¯Θz¯w¯Θzw¯Θzz¯w¯+Θw¯Θw¯Θw¯Θz¯z¯Θzw¯Θzz¯z¯+Θzzw¯-Θz¯Θz¯Θz¯Θw¯w¯Θzz¯Θzw¯w¯+2Θz¯Θw¯Θz¯Θz¯w¯Θzz¯Θzz¯w¯-Θw¯Θw¯Θz¯Θz¯z¯Θzz¯Θzz¯z¯,noticing that its denominator (27)Θz¯Θzw¯-Θw¯Θzz¯3does not vanish at the origin since 0∈M was assumed (temporarily) to be a Levi nondegenerate point. Introduce also the vector field derivation (28)D≔-Θw¯Θz¯Θzw¯-Θw¯Θzz¯∂∂z¯+Θz¯Θz¯Θzw¯-Θw¯Θzz¯∂∂w¯.Then the main and unique theorem of [2] states that M is spherical at 0 if and only if (29)0≡DDAJ4Θ,identically in C{z,z¯,w¯}.Unfortunately, it is essentially impossible to print in a published article what one obtains after a full expansion of these two derivations; other instances of this phenomenon appear in [7, 8].Nevertheless, by thinking a bit, one convinces oneself that, after full expansion and reduction to a common denominator, one obtains a kind of expression that we will denote in summarized form as(30)polynomialΘzjz¯kw¯l1⩽j+k+l⩽6Θz¯Θzw¯-Θw¯Θzz¯7,and hence instantly, sphericity of M is characterized by(31)0≡polynomialΘzjz¯kw¯lz,z¯,w¯1⩽j+k+l⩽6.One notices that the complex graphing function Θ is differentiated always at least once.Interpretation. Then the true thing is that, after erasing the Levi determinant lying at denominator place, if this explicit equation vanishes in some very small neighborhood of some point(32)zp,z¯p,w¯p∈□ρ03of the polydisc of convergence of Θ, then by the uniqueness principle for analytic functions, the concerned polynomial numerator (33)polynomialΘzjz¯kw¯lz,z¯,w¯1⩽j+k+l⩽6immediately vanishes identically all over □ρ03, so that sphericity at one point should freely propagate to all other Levi nondegenerate points q∈M∩□ρ02.Before providing rigorous details to explain the latter assertion, a further comment is in order.The explicit sphericity formula brings the information that denominator places are occupied by nondegeneracy conditions, so that division is allowed only at points where these conditions are satisfied, but numerators happen to be polynomial, a computational fact which hence enables one to jump across degenerate points through the “bridge-numerator” from one nondegenerate point to another nondegenerate point.Now, the local version of Theorem1 is as follows. Notice that from now on one does not assume anymore that the origin 0∈M∖ΣLD is a Levi nondegenerate point.Proposition 3.
WithM3⊂C2, □ρ02, (z,z¯,w¯), φ, and Θ as above, assuming that the real analytic subset ΣLD of Levi degenerate points is proper, if M is spherical at one Levi nondegenerate point (34)p∈M∖ΣLD∩□ρ02,then M is also spherical at every other Levi nondegenerate point: (35)q∈M∖ΣLD∩□ρ02.Proof.
Take a (possibly much) smaller bidisc(36)p+□ρ′2⊂⊂□ρ02,with 0<ρ′≪ρ0 to be chosen below, and center new coordinates at (37)p=zp,wp;that is to say, introduce the new translated coordinates: (38)z′≔z-zp,w′≔w-wp.The graphed complex equation (39)w=Θz,z¯,w¯then becomes (40)w′+wp=Θz′+zp,z¯′+z¯p,w¯′+w¯p′.Of course, the fact that p∈M reads (41)wp=Θzp,z¯p,w¯p,and hence, in the new coordinates (z′,w′) centered at p, the equation of M becomes (42)w′=Θz′+zp,z¯′+z¯p,w¯′+w¯p-Θzp,z¯p,w¯p≕Θ′z′,z¯′,w¯′,in terms of a new graphing function Θ′ that visibly satisfies (43)Θ′0′,0′,0′=0.
Observation  4. For any integers (44)j,k,l∈N3,with (45)j+k+l⩾1,one has (46)Θz′jz¯′kw¯′l′0′,0′,0′=Θzjz¯kw¯lzp,z¯p,w¯p.
Proof. Indeed, the constant -Θ(zp,z¯p,w¯p) disappears after just a single differentiation. This completes the proof.
Now, the Levi nondegeneracy ofM at p which reads according to what precedes as (47)0≠Θz¯Θzw¯-Θw¯Θzz¯zp,z¯p,w¯preads in the new coordinates as (remember that inequation (17) is independent of coordinates) (48)0≠Θz¯′′Θz′w¯′′-Θw¯′′Θz′z¯′′0′,0′,0′,which means as we know Levi nondegeneracy at (z′,z¯′,w¯′)=(0′,0′,0′).
Precisely because in [2] one needs only this condition to hold in order to associate as was explained above a second-order complex ordinary differential equation (49)wz′z′′=Φ′z′,w′z′,wz′′z′for some possibly very small (50)z′<ρ′,w′<ρ′,this is where one has to choose ρ′ with 0<ρ′≪ρ0, the possible presence of rather close Levi degenerate points being a constraint in the needed application(s) of the implicit function theorem, one has the impression that one can in principle only determine whether M is spherical restrictively in such a very narrow neighborhood □ρ′2 of p in C2, when one applies the main result of [2].
But looking just at the numerator of the equation which expresses thatM is spherical at p in the coordinates (z′,w′)(51)0≡polynomial︷sameuniversalexpressionΘz′jz¯′kw¯′l′1⩽j+k+l⩽6Θz¯′′Θz′w¯′′-Θw¯′Θz′z¯′′7︸nonvanishingat0′,0′,0′,if one takes into account the above observation, one readily realizes that (52)polynomialΘz′jz_′kw_′l′z′,z_′,w_′1⩽j+k+l⩽6=polynomialΘzjz_kw_lz,z_,w_1⩽j+k+l⩽6,so that the identical vanishing of the left-hand side for (53)z′<ρ′≪ρ0,z_′<ρ′≪ρ0,w_′<ρ′≪ρ0means the identical vanishing of the right-hand side for (54)z-zp<ρ′≪ρ0,z_-z¯p<ρ′≪ρ0,w_-w¯p<ρ′≪ρ0,which lastly yieldsthanks to the uniqueness principle enjoyed by analytic functions the identical vanishing of the original numeratorin the whole initial domain of convergence: (55)0≡polynomialΘzjz_kw_lz,z_,w_1⩽j+k+l⩽6z<ρ0,z_<ρ0,w_<ρ0.
Take now any other Levi nondegenerate point:(56)q∈M∩□ρ02.The goal is to prove that M is also spherical at q. Center similarly new coordinates at q=(zq,wq): (57)z′′≔z-zq,w′′≔w-wq.Introduce the new graphed equations: (58)w′′=Θz′′+zq,z¯′′+z¯q,w¯′′+w¯q-Θzq,z¯q,w¯q≕Θ′′z′′,z¯′′,w¯′′.At such a point, since the Levi determinant is nonvanishing, one can for completeness construct the associated second-order complex ordinary differential equations (59)wz′′z′′′′z′′=Φ′′z′′,w′′z′′,wz′′′′z′′or question directly whether local sphericity holds near q by plainly applying the main theorem of [2]; namely, question whether the following equation holds: (60)0≡?polynomial︷againsameuniversalexpressionΘz′′jz¯′′kw¯′′l′′1⩽j+k+l⩽6Θz¯′′′′Θz′′w¯′′′′-Θw¯′′Θz′z¯′′′′7︸nonvanishingat0′′,0′′,0′′.But then by exactly the same application of the above observation, we know that this last numerator satisfies (61)polynomialΘz′′jz_′′kw_′′l′′z′′,z_′′,w_′′1⩽j+k+l⩽6=polynomialΘzjz_kw_lz,z_,w_1⩽j+k+l⩽6,when (62)z′′<ρ′′≪ρ0,z_′′<ρ′′≪ρ0,w_′<ρ′′≪ρ0,z-zq<ρ′′≪ρ0,z_-z¯q<ρ′′≪ρ0,w_-w¯q<ρ′′≪ρ0,and since we already know that the latter right-hand side vanishes, according to the last boxed equation, we conclude that M is indeed spherical at q.To finish the proof of Theorem1 in the case n=1, it remains only toglobalize this local propagation of sphericity.Assume therefore thatM⊂C2 is a connected real analytic smooth hypersurface, that is ΣLD≠M, so that the set M∖ΣLD of Levi nondegenerate points is Zariski open, everywhere dense in M. Assume that there exists aspherical pointp∈M∖ΣLD (abbreviation for “an open neighborhood of p in M is locally biholomorphic to the unit 3-sphere S3⊂C2”). Take any other point q∈M∖ΣLD.By connectedness ofM, there exists a finite (possibly large) number ν⩾1 of points (63)p=p0,p1,…,pν-1,pν=q,and there exist small radii ρ0,…,ρν>0 such that, in the open polydiscs, (64)Δκ≔pκ+□ρκ20⩽κ⩽ν,the local hypersurface M∩Δκ is connected and graphed either as (65)w-wpκ=Θκz-zpκ,z-zpκ¯,w-wpκ¯or as (66)z-zpκ=Ξκw-wpκ,w-wpκ¯,z-zpκ¯,and such that [M∩Δκ-1]∩[M∩Δκ] isopen nonempty in M.SinceΣLD⫋M is locally stratified by a finite union of real submanifolds of dimension <3=dimRM, this insures that (67)∅≠M∖ΣLD∩Δκ∀0⩽κ⩽ν,∅≠M∖ΣLD∩Δκ-1∩M∖ΣLD∩Δκ∀1⩽κ⩽ν.Remember that the main Proposition 3 that precedes has shown that, for every 0⩽κ⩽ν, (68)∃r∈M∖ΣLD∩Δκspherical⟹allr∈M∖ΣLD∩Δκarespherical.Then an immediate induction yields propagation from (69)p=p0∈M∖ΣLD∩Δ0isspherical⟹allr∈M∖ΣLD∩Δ0∩Δ1arespherical⟹allr∈M∖ΣLD∩Δ1arespherical⋮⟹allr∈M∖ΣLD∩Δν-1∩Δνarespherical⟹allr∈M∖ΣLD∩Δνarespherical,up to q∈(M∖ΣLD)∩Δν thereby shown to be spherical too.


## 3. Proof inCn+1(n⩾2)

We briefly summarize the quite similar arguments, relying upon [3], which is unpublished and several times rejected, although useful here.The Levi determinant becomes(70)Δ≔Θz¯1⋯Θz¯nΘw¯Θz1z¯1⋯Θz1z¯nΘz1w¯⋯⋯⋯⋯Θznz¯1⋯Θznz¯nΘznw¯.It is nonzero at one point (71)p=z1p,…,znp,wp∈Mif and only if M is Levi nondegenerate at p and also if and only if one can associate with M a completely integrable system of second-order partial differential equations: (72)wzk1zk2z=Φk1,k2z,wz,wz1z,…,wznz1⩽k1,k2⩽n.Hachtroudi ([9]) established that such a system is pointwise equivalent to (73)wzk1′zk2′′z′=01⩽k1,k2⩽nif and only if (74)0≡∂2Φk1,k2∂wzl1wzl2-1n+2∑l3=1nδk1,l1∂2Φl3,k2∂wzl3∂wzl2+δk1,l2∂2Φl3,k2∂wzl1∂wzl3+δk2,l1∂2Φk1,l3∂wzl3∂wzl2+δk2,l2∂2Φk1,l3∂wzl1∂wzl3+1n+1n+2δk1,l1δk2,l2+δk2,l1δk1,l2∑l3=1n∑l4=1n∂2Φl3,l4∂wzl3∂wzl41⩽k1,k2⩽n1⩽l1,l2⩽n.When one does apply Hachtroudi’s results to CR geometry (instead of Chern-Moser’s results, which is up to now not sufficiently explicit to be applied), the signature of the Levi forms disappears for the following reason.The infinite-dimensional local Lie (pseudo)group of biholomorphic transformations(75)z1,…,zn,w⟼z1′,…,zn′,w′=z1′z•,w,…,zn′z•,w,w′z•,wacts simultaneously on (z•,w)-variables and on (z¯•,w¯)-variables as (76)z¯1,…,z¯n,w¯⟼z¯1′,…,z¯n′,w¯′=z¯1′z¯•,w¯,…,z¯n′z¯•,w¯,w¯′z¯•,w¯.But when one passes to the extrinsic complexification, one replaces(z¯,…,z¯n,w¯)-variables by new independent variables: (77)z_1,…,z_n,w_,considered as theconstants of integration for the system of partial differential equations. Hence the local Lie (pseudo)group considered by Hachtroudi becomes enlarged as the group of transformations: (78)z•,w,z_•,w_⟼holomorphicmapz•,w,otherholomorphicmapz_•,w_in which the transformations on the “constant-of-integration” variables (z_1,…,z_n,w_) become completely discoupled from the group of transformations on the true variables (z1,…,zn,w). By definition (cf. the explanations in [2]), transformations on differential equations, when viewed in the space of solutions, are always of this general form.It is then clear that allcomplexified Heisenberg (k,n-k) pseudospheres (79)w=w_+2i-z1z_1-⋯-zkz_k+zk+1z_k+1+⋯+znz_nbecome all pairwise equivalent through such transformations, because one is allowed to replace z_1,…,z_k by -z_1,…,-z_k without touching z1,…,zk; even the factor 2i can be erased: (80)w=w_+z1z_1+⋯+zkz_k+zk+1z_k+1+⋯+znz_n.Therefore, when passing to systems of partial differential equations associated with CR manifolds,Levi form signatures drop.Consequently, when one applies the main theorem of [3], according to which a Levi nondegenerate M⊂Cn+1having given Levi form signature(k,n-k) is pseudospherical if and only if (notation same as in [3]) its Hachtroudi system is equivalent to (81)wzk1′zk2′′z′=01⩽k1,k2⩽nand moreover if and only if—after translating back to the graphing function Θ the explicit condition of Hachtroudi—the identical vanishing property holds, (82)0≡1Δ3∑μ=1n+1∑ν=1n+1·Δ·∂4Θ∂zk1∂zl3∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zk1∂zl3∂t¯τΔ01+l1μ≡·Δ01+l2νΔ·∂4Θ∂zk1∂zk2∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zk1∂zk2∂t¯τ0-δk1,l1n+2∑l3=1nΔ01+l3μ0·Δ01+l2νΔ·∂4Θ∂zl3∂zk2∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zl3∂zk2∂t¯τ0-δk1,l2n+2∑l3=1nΔ01+l1μ0·Δ01+l3νΔ·∂4Θ∂zl3∂zk2∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zl3∂zk2∂t¯τ0-δk2,l1n+2∑l3=1nΔ01+l3μ0·Δ01+l2νΔ·∂4Θ∂zk1∂zl3∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zk1∂zl3∂t¯τ0-δk2,l2n+2∑l3=1nΔ01+l1μ0·Δ01+l3νΔ·∂4Θ∂zk1∂zl3∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ·∂3Θ∂zk1∂zl3∂t¯τ0+1n+1n+2·δk1,l1δk2,l2+δk2,l1δk1,l20·∑l3=1n∑l4=1nΔ01+l3μ·Δ01+l4νΔ·∂4Θ∂zl3∂zl4∂t¯μ∂t¯ν-∑τ=1n+1Δt¯μt¯ντ0·∂3Θ∂zl3∂zl4∂t¯τ1⩽k1,k2⩽n;1⩽l1,l2⩽n,one can reason exactly as in the preceding section for M3⊂C2—noticing that the denominator is the same as 1/Δ3, noticing that the numerator is similarly polynomial in the partial derivatives of the graphing function Θ—but when one jumps from a Levi nondegenerate point p∈M∩□ρ0n+1 to another Levi nondegenerate point q∈M∩□ρ0n+1, from the property of local equivalence at q to (83)wzk1′′zk2′′′′z′′=01⩽k1,k2⩽n,one can only conclude that the complexification of M near q is equivalent near q to (84)w′′=w_′′+z1′′z_1′′+⋯+zk′′z_k′′+zk+1′′z_k+1′′+⋯+zn′′z_n′′so that the Levi form signature can in principle change—and it really does in Example  6.3 of [1]—through Levi degenerate points.



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