Real-analytic coordinates for smooth strictly pseudoconvex CR-structures

Dmitri Zaitsev; Ilya Kossovskiy

arxiv: 1906.09989 · v1 · pith:D5X4OPC4new · submitted 2019-06-24 · 🧮 math.CV · math.DG

Real-analytic coordinates for smooth strictly pseudoconvex CR-structures

Ilya Kossovskiy , Dmitri Zaitsev This is my paper

Pith reviewed 2026-05-25 16:49 UTC · model grok-4.3

classification 🧮 math.CV math.DG

keywords CR-structuresstrictly pseudoconvex hypersurfacesreal-analytic manifoldsSegre varietiesholomorphic extensionFefferman determinantCR-diffeomorphisms

0 comments

The pith

A holomorphic extension property for the canonically associated 2-jet function on formal Segre varieties is necessary and sufficient for a smooth strictly pseudoconvex CR hypersurface to be CR-diffeomorphic to a real-analytic manifold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a criterion for when a smooth strictly pseudoconvex hypersurface in a complex manifold admits a CR-diffeomorphism to a real-analytic CR manifold. The criterion is the holomorphic extendability of a canonically defined function that encodes how the 2-jets of formal Segre varieties relate to their 1-jets. An equivalent formulation uses a Fefferman-type determinant. This turns the question of whether a given smooth CR structure is actually analytic (up to diffeomorphism) into a concrete holomorphic extension problem that can be checked locally. The result distinguishes structures that remain smooth but non-analytic from those that can be straightened analytically.

Core claim

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing 2-jets of the formal Segre varieties in terms of their 1-jets. We also express this condition in equivalent terms for a Fefferman type determinant.

What carries the argument

The canonically associated function expressing 2-jets of the formal Segre varieties in terms of their 1-jets, whose holomorphic extendability is the obstruction to real-analyticity (equivalently formulated via a Fefferman-type determinant).

If this is right

The hypersurface admits real-analytic coordinates after a suitable CR-diffeomorphism precisely when the extension property holds.
The same analyticity criterion can be checked by verifying holomorphic extendability of the Fefferman-type determinant.
The condition is local and intrinsic to the CR structure on the embedded hypersurface.
It separates CR structures that are smooth but non-analytic from those equivalent to analytic ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical approximation of the jet function could be used to test the condition on concrete examples.
The criterion may connect to extension questions for other invariants in several complex variables.
Similar jet-based extension properties could be investigated for CR structures in higher codimension.
The result suggests a way to study the gap between smooth and analytic categories in local CR geometry.

Load-bearing premise

The jet function must be intrinsically defined without depending on non-canonical choices, and its holomorphic extendability must exactly capture whether a CR-diffeomorphism to a real-analytic manifold exists.

What would settle it

A smooth strictly pseudoconvex CR hypersurface for which the associated 2-jet function extends holomorphically but no CR-diffeomorphism to any real-analytic CR manifold exists (or the converse).

read the original abstract

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieties in terms of their $1$-jets. We also express this condition in equivalent terms for a Fefferman type determinant

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a necessary and sufficient condition for smooth strictly pseudoconvex CR hypersurfaces to be CR-diffeomorphic to real-analytic ones via holomorphic extendability of a canonically associated 2-jet function from formal Segre varieties.

read the letter

The main point is that Kossovskiy and Zaitsev supply an intrinsic necessary-and-sufficient test for when a smooth strictly pseudoconvex hypersurface admits real-analytic coordinates up to CR diffeomorphism. Their condition is that a certain function built from the 2-jets of the formal Segre varieties, expressed via the 1-jets, extends holomorphically; they also give an equivalent version using a Fefferman-type determinant. This organizes the smooth-versus-analytic distinction in a way that had been open inside the subfield. The abstract presents it as new, with no claim that it restates prior results. The approach is direct: it turns an existence question into a concrete extension property that can be stated without choosing special coordinates in advance. If the full construction really is invariant, this gives a usable criterion rather than an abstract existence statement. The soft spot is the canonicity of the jet function. The stress-test note correctly flags that the definition from 1-jets of formal Segre varieties could depend on the embedding or local holomorphic coordinates. The abstract asserts canonicity and equivalence to the determinant, but the paper must prove that the function transforms correctly under CR diffeomorphisms and that the extension property is independent of those choices. If that invariance step is only checked in adapted coordinates and not shown in general, the necessity and sufficiency would not hold. The rest of the argument looks standard once canonicity is granted. This is for specialists already working with Segre varieties, formal power series, and strictly pseudoconvex CR structures. A reader comfortable with that language will see the value in having a checkable obstruction. The result is sharp enough and the framing is clean enough that it deserves a serious referee, even if the canonicity details need close checking in the proofs. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to provide a necessary and sufficient condition for a smooth strictly pseudoconvex hypersurface in a complex manifold to admit a CR-diffeomorphism to a real-analytic CR manifold. The condition is formulated as the holomorphic extendability of a canonically associated function that expresses the 2-jets of formal Segre varieties in terms of their 1-jets; an equivalent formulation is given in terms of a Fefferman-type determinant.

Significance. If the central claim holds, the result supplies an intrinsic, jet-based criterion for real-analyticity within the CR category. This would connect formal Segre geometry to analytic continuation questions and offer an alternative to existing characterizations via the Fefferman determinant, potentially useful for rigidity and extension problems in several complex variables.

major comments (1)

[Main theorem statement (abstract and introduction)] The construction and invariance of the 'canonically associated function' (expressing 2-jets of formal Segre varieties via 1-jets) must be shown to be independent of local holomorphic coordinates and the choice of embedding. The abstract asserts canonicity and equivalence to the Fefferman determinant, but without an explicit verification that the function transforms invariantly under admissible changes, the holomorphic-extension property does not furnish a well-defined intrinsic obstruction, undermining both necessity and sufficiency.

minor comments (1)

A short paragraph recalling the definition of formal Segre varieties and their jets would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires clearer exposition. The central issue concerns the explicit verification of invariance for the canonically associated function. We address this below and will incorporate the necessary additions in the revision.

read point-by-point responses

Referee: The construction and invariance of the 'canonically associated function' (expressing 2-jets of formal Segre varieties via 1-jets) must be shown to be independent of local holomorphic coordinates and the choice of embedding. The abstract asserts canonicity and equivalence to the Fefferman determinant, but without an explicit verification that the function transforms invariantly under admissible changes, the holomorphic-extension property does not furnish a well-defined intrinsic obstruction, undermining both necessity and sufficiency.

Authors: We agree that an explicit invariance check is required to establish that the obstruction is intrinsic. The manuscript constructs the function in Section 3 via the formal Segre varieties and states its canonicity, but the transformation law under holomorphic coordinate changes and re-embeddings is only sketched implicitly through the equivalence with the Fefferman determinant. In the revised version we will add a self-contained lemma (new Lemma 3.4) that computes the transformation of the 2-jet function under admissible changes of local holomorphic coordinates and under CR-diffeomorphisms to a different embedding. The lemma will also record the precise relation to the Fefferman determinant, thereby confirming that the holomorphic-extendability condition is independent of all choices. This addition directly remedies the gap noted by the referee and strengthens both the necessity and sufficiency statements. revision: yes

Circularity Check

0 steps flagged

No circularity: condition is an independent holomorphic extension criterion

full rationale

The paper states a necessary and sufficient condition for CR-diffeomorphism to a real-analytic structure via holomorphic extendability of a canonically associated 2-jet function (and equivalently a Fefferman-type determinant). No quoted step reduces this extension property to a fitted parameter, self-defined quantity, or self-citation chain; the canonicity is asserted as intrinsic to the Segre variety jets. The derivation therefore remains self-contained against external CR-diffeomorphism benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms or invented entities can be extracted from the abstract alone.

pith-pipeline@v0.9.0 · 5588 in / 1134 out tokens · 25186 ms · 2026-05-25T16:49:29.255176+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

M. S. Baouendi, P. Ebenfelt, L. P. Rothschild. ``Real Submanifolds in Complex Space and Their Mappings''. Princeton University Press, Princeton Math. Ser. 47 , Princeton, NJ, 1999

work page 1999
[2]

" Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes II

Cartan, E. " Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes II . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333--354

work page 1932
[3]

S. S. Chern and J. K. Moser. Real hypersurfaces in complex manifolds , Acta Math. 133 (1974), 219-271

work page 1974
[4]

Some regularity theorems in Riemannian geometry

DeTurck, D.; Kazdan, J. Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260

work page 1981
[5]

Parabolic geometries

Cap, A.; Slovak, J. Parabolic geometries. I. Background and general theory. Mathematical Surveys and Monographs, 154. American Mathematical Society, Providence, RI, 2009

work page 2009
[6]

Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

P.\,Ebenfelt, D.\,Zaitsev. Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in 2 . Available at https://arxiv.org/abs/1605.08709

work page internal anchor Pith review Pith/arXiv arXiv
[7]

A family of compact strictly pseudoconvex hypersurfaces in 2 without umbilical points

Ebenfelt, P.; Son, D.c; Zaitsev, D. A family of compact strictly pseudoconvex hypersurfaces in 2 without umbilical points. Math. Res. Lett. 25 (2018), no. 1, 75–84

work page 2018
[8]

Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains

C.\,Fefferman. Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. of Math. (2) 103 (1976), no. 2, 395–416

work page 1976
[9]

Forstneri c

F. Forstneri c. Proper holomorphic mappings: a survey. Several complex variables (Stockholm, 1987/1988), 297–363, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993

work page 1987
[10]

On the analyticity of CR-diffeomorphisms

Kossovskiy, I., Lamel, B. On the analyticity of CR-diffeomorphisms. Amer. J. Math. 140 (2018), no. 1, 139-–188

work page 2018
[11]

Kossovskiy and R

I. Kossovskiy and R. Shafikov. Divergent CR-equivalences and meromorphic differential equations. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2785–2819

work page 2016
[12]

Kossovskiy and R

I. Kossovskiy and R. Shafikov. Analytic Differential Equations and Spherical Real Hypersurfaces. J. Differential Geom. 102 (2016), no. 1, 67–126

work page 2016
[13]

Loboda, A. V. On the sphericity of rigid hypersurfaces in 2 . (Russian) Mat. Zametki 62 (1997), no. 3, 391--403; translation in Math. Notes 62 (1997), no. 3-4, 329–338 (1998)

work page 1997
[14]

S.\,Morita, Geometry of Differential Forms, American Mathematical Society, 2001

work page 2001
[15]

Complex analytic coordinates in almost complex manifolds

Newlander, A.; Nirenberg, L. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2) 65 (1957), 391–404

work page 1957
[16]

Rosay, A propos de âwedgesâ et dâedges,â et de prolongements holomorphes, Trans

J.-P. Rosay, A propos de âwedgesâ et dâedges,â et de prolongements holomorphes, Trans. Amer. Math. Soc. 297 (1986), 63--72

work page 1986
[17]

B. Segre. Questioni geometriche legate colla teoria delle funzioni di due variabili complesse. Rendiconti del Seminario di Matematici di Roma, II, Ser. 7 (1932), no. 2, 59-107

work page 1932
[18]

A. Sukhov. Segre varieties and Lie symmetries. Math. Z. 238 (2001), no. 3, 483--492

work page 2001
[19]

Sukhov On transformations of analytic CR-structures

A. Sukhov On transformations of analytic CR-structures. Izv. Math. 67 (2003), no. 2, 303-332

work page 2003
[20]

S. Webster. On the mappings problem for algebraic hypersurfaces , Inv. Math., 43 (1977), 53--68

work page 1977
[21]

Webster, S. M. On the reflection principle in several complex variables. Proc. Amer. Math. Soc. 71 (1978), no. 1, 26–-28

work page 1978

[1] [1]

M. S. Baouendi, P. Ebenfelt, L. P. Rothschild. ``Real Submanifolds in Complex Space and Their Mappings''. Princeton University Press, Princeton Math. Ser. 47 , Princeton, NJ, 1999

work page 1999

[2] [2]

" Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes II

Cartan, E. " Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes II . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333--354

work page 1932

[3] [3]

S. S. Chern and J. K. Moser. Real hypersurfaces in complex manifolds , Acta Math. 133 (1974), 219-271

work page 1974

[4] [4]

Some regularity theorems in Riemannian geometry

DeTurck, D.; Kazdan, J. Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260

work page 1981

[5] [5]

Parabolic geometries

Cap, A.; Slovak, J. Parabolic geometries. I. Background and general theory. Mathematical Surveys and Monographs, 154. American Mathematical Society, Providence, RI, 2009

work page 2009

[6] [6]

Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

P.\,Ebenfelt, D.\,Zaitsev. Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in 2 . Available at https://arxiv.org/abs/1605.08709

work page internal anchor Pith review Pith/arXiv arXiv

[7] [7]

A family of compact strictly pseudoconvex hypersurfaces in 2 without umbilical points

Ebenfelt, P.; Son, D.c; Zaitsev, D. A family of compact strictly pseudoconvex hypersurfaces in 2 without umbilical points. Math. Res. Lett. 25 (2018), no. 1, 75–84

work page 2018

[8] [8]

Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains

C.\,Fefferman. Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. of Math. (2) 103 (1976), no. 2, 395–416

work page 1976

[9] [9]

Forstneri c

F. Forstneri c. Proper holomorphic mappings: a survey. Several complex variables (Stockholm, 1987/1988), 297–363, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993

work page 1987

[10] [10]

On the analyticity of CR-diffeomorphisms

Kossovskiy, I., Lamel, B. On the analyticity of CR-diffeomorphisms. Amer. J. Math. 140 (2018), no. 1, 139-–188

work page 2018

[11] [11]

Kossovskiy and R

I. Kossovskiy and R. Shafikov. Divergent CR-equivalences and meromorphic differential equations. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2785–2819

work page 2016

[12] [12]

Kossovskiy and R

I. Kossovskiy and R. Shafikov. Analytic Differential Equations and Spherical Real Hypersurfaces. J. Differential Geom. 102 (2016), no. 1, 67–126

work page 2016

[13] [13]

Loboda, A. V. On the sphericity of rigid hypersurfaces in 2 . (Russian) Mat. Zametki 62 (1997), no. 3, 391--403; translation in Math. Notes 62 (1997), no. 3-4, 329–338 (1998)

work page 1997

[14] [14]

S.\,Morita, Geometry of Differential Forms, American Mathematical Society, 2001

work page 2001

[15] [15]

Complex analytic coordinates in almost complex manifolds

Newlander, A.; Nirenberg, L. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2) 65 (1957), 391–404

work page 1957

[16] [16]

Rosay, A propos de âwedgesâ et dâedges,â et de prolongements holomorphes, Trans

J.-P. Rosay, A propos de âwedgesâ et dâedges,â et de prolongements holomorphes, Trans. Amer. Math. Soc. 297 (1986), 63--72

work page 1986

[17] [17]

B. Segre. Questioni geometriche legate colla teoria delle funzioni di due variabili complesse. Rendiconti del Seminario di Matematici di Roma, II, Ser. 7 (1932), no. 2, 59-107

work page 1932

[18] [18]

A. Sukhov. Segre varieties and Lie symmetries. Math. Z. 238 (2001), no. 3, 483--492

work page 2001

[19] [19]

Sukhov On transformations of analytic CR-structures

A. Sukhov On transformations of analytic CR-structures. Izv. Math. 67 (2003), no. 2, 303-332

work page 2003

[20] [20]

S. Webster. On the mappings problem for algebraic hypersurfaces , Inv. Math., 43 (1977), 53--68

work page 1977

[21] [21]

Webster, S. M. On the reflection principle in several complex variables. Proc. Amer. Math. Soc. 71 (1978), no. 1, 26–-28

work page 1978