A Counterexample to the 2 -jet determination Chern-Moser Theorem in higher codimension
aa r X i v : . [ m a t h . C V ] M a r A COUNTEREXAMPLE TO THE − JETDETERMINATION CHERN-MOSER THEOREM INHIGHER CODIMENSION
FRANCINE MEYLAN
Abstract.
One constructs an example of a generic quadratic subman-ifold of codimension 5 in C which admits a real analytic infinitesimalCR automorphism with homogeneous polynomial coefficients of degree3 . Introduction
Let M be a real-analytic submanifold of C N of codimension d. Considerthe set of germs of biholomorphisms F at a point p ∈ M such that F ( M ) ⊂ M . By the work of Chern and Moser[9], if the codimension d = 1 , everysuch F is uniquely determined by its derivatives at p provided that its Levimap at p is non-degenerate. See also Cartan and Tanaka ( [8], [19]). Theorem 1. [9]
Let M be a real-analytic hypersurface through a point p in C N with non-degenerate Levi form at p . Let F , G be two germs of biholo-morphic maps preserving M . Then, if F and G have the same 2-jets at p ,they coincide. Note that the result becomes false without any hypothesis on the Leviform (See for instance [6]). A generalization of this Theorem to real-analyticsubmanifolds M of higher codimension d > M is Levi generating (or equivalently of finite type with 2 theonly H¨ormander number) with non-degenerate Levi map. Unfortunately, anerror has been discovered and explained in [6]. In this short note, one con-structs an example of a generic (Levi generating with non-degenerate Levimap) quadratic submanifold that admits an element in its stability groupwhich has the same 2 − jet as the identity map but is not the identity map. Inaddition, this example is Levi non-degenerate in the sense of Tumanov. Onepoints out that if M is strictly pseudoconvex, that is Levi non-degeneratein the sense of Tumanov with a positivity condition, then by a recent resultof Tumanov[20], the 2-jet determination result holds in any codimension.One also points out that finite jet determination problems for submani-folds has attracted much attention. One refers in particular to the papersof Zaitsev [21], Baouendi, Ebenfelt and Rothschild [2], Baouendi, Mir andRothschild [7], Ebenfelt, Lamel and Zaitsev [12], Lamel and Mir [17], Juhlin [13], Juhlin and Lamel [14], Mir and Zaitsev [18] in the real analytic case,Ebenfelt [10], Ebenfelt and Lamel [11], Kim and Zaitsev [15], Kolar, the au-thor and Zaitsev [16] in the C ∞ case, Bertrand and Blanc-Centi [4], Bertrand,Blanc-Centi and the author [5], Tumanov [20] in the finitely smooth case.2. The Example
Let M ⊆ C be the real submanifold of (real) codimension 5 through 0given in the coordinates ( z, w ) = ( z , . . . , z , w , . . . , w ) ∈ C , by(1) Im w = P ( z, ¯ z ) = z z + z z Im w = P ( z, ¯ z ) = − iz z + iz z Im w = P ( z, ¯ z ) = z z + z z + z z + z z Im w = P ( z, ¯ z ) = z z Im w = P ( z, ¯ z ) = z z The matrices corresponding to the P ′ i s are A = A = − i i A = A = A = Lemma 2.
The following holds:(1) the A ′ i s are linearly independent,(2) the A ′ i s satisfy the condition of Tumanov, that is, there is c ∈ R d such that det P c j A j = 0 . Proposition 3.
The real submanifold M given by (1) is Levi generating at , that is, of finite type with the only H¨ormander number, and its Levimap is non-degenerate.Proof. This follows for instance from Proposition 8, Lemma 3 and Remark4 in [6]. (cid:3)
Remark 4.
The following identity between the P ′ i s holds:(2) P + P − P P = 0 . The following holomorphic vectors fields are in hol ( M, , the set of germsof real-analytic infinitesimal CR automorphisms at 0 . (1) X := i ( z ∂∂z + z ∂∂z ) COUNTEREXAMPLE TO THE 2 − JET DETERMINATION CHERN-MOSER THEOREM IN HIGHER CODIMENSION3 (2) Y := i ( − iz ∂∂z + iz ∂∂z )(3) Z := i ( z ∂∂z )(4) U := i ( z ∂∂z ) Lemma 5.
Let P = ( P , . . . , P ) . The following holds:(1) X ( P ) = (0 , , iP , , (2) Y ( P ) = (0 , , iP , , (3) Z ( P ) = (0 , , iP , , (4) U ( P ) = (0 , , iP , , . Lemma 6.
The following identities hold:(1) P ( − Y ( P )) + P ( X ( P ) = 0 (2) P X ( P ) + P ( Y ( P ) + P ( − Z ( P )) + P ( − U ( P )) = 0 (3) P ( − Z ( P )) + P (2 Y ( P ) = 0 (4) P ( − U ( P )) + P (2 Y ( P ) = 0With the help of the Lemmata, one obtains Theorem 7.
Let Y = − Y, Y = 2 Y, Z = − Z, U = − U. The holomorphic vector field T defined by (3) T = 12 w Y + 12 w Y + w w X + w w Z + w w U + w w Y is in hol ( M, . Hence − jet determination does not hold for germs of biholomorphismssending M to M. Remark 8.
Notice that the bound for the number k of jets needed todetermine uniquely any germ of biholomorphism sending M to M is k = (1 + codim M ) ,M beeing a generic (Levi generating with non-degenerate Levi map) real-analytic submanifold: see Theorem 12.3.11, page 361 in [1]. One points outthat Zaitsev obtained the bound k = 2(1 + codim M ) in [21]. References [1] M.S. Baouendi, P. Ebenfelt, L.P. Rothschild:
Real Submanifolds in Complex Spaceand their Mappings . Princeton University Press, 1999.[2] M.S. Baoudendi, P. Ebenfelt, L.P. Rothschild,
CR automorphisms of real analytic CRmanifolds in complex space , Comm. Anal. Geom. (1998), 291-315.[3] V.K. Beloshapka: A uniqueness theorem for automorphisms of a nondegenerate sur-face in a complex space. (Russian) Mat. Zametki (3) 17-22, 1990; transl. Math.Notes (3) 239-243, 1990.[4] F. Bertrand, L. Blanc-Centi, Stationary holomorphic discs and finite jet determina-tion problems , Math. Ann. (2014), 477-509.
FRANCINE MEYLAN [5] F. Bertrand, L. Blanc-Centi, F. Meylan,
Stationary discs and finite jet determinationfor non-degenerate generic real submanifolds
Adv. Math. (2019), 910-934.[6] L. Blanc-Centi, F. Meylan,
On nondegeneracy conditions for the Levi map in highercodimension: a Survey , preprint, arXiv:1711.11481.[7] M.S. Baouendi, N. Mir, L.P. Rothschild,
Reflection Ideals and mappings betweengeneric submanifolds in complex space , J. Geom. Anal. (2002), 543-580.[8] Cartan, E: Sur la g´eom´etrie pseudo-conforme des hypersurfaces de deux variablescomplexes
Ann. Math. Pura Appl. 11 (1932), 17-90.[9] S.S. Chern, J.K. Moser: Real hypersurfaces in complex manifolds,
Acta Math. (3-4) 219-271 (1974).[10] P. Ebenfelt,
Finite jet determination of holomorphic mappings at the boundary , AsianJ. Math. (2001), 637-662.[11] P. Ebenfelt, B. Lamel, Finite jet determination of CR embeddings , J. Geom. Anal. (2004), 241-265.[12] P. Ebenfelt, B. Lamel, D. Zaitsev, Finite jet determination of local analytic CR auto-morphisms and their parametrization by 2-jets in the finite type case , Geom. Funct.Anal. (2003), 546-573.[13] R. Juhlin, Determination of formal CR mappings by a finite jet , Adv. Math. (2009), 1611-1648.[14] R. Juhlin, B. Lamel,
Automorphism groups of minimal real-analytic CR manifolds ,J. Eur. Math. Soc. (JEMS) (2013), 509-537.[15] S.-Y. Kim, D. Zaitsev, Equivalence and embedding problems for CR-structures of anycodimension , Topology (2005), 557-584.[16] M. Kol´aˇr, F. Meylan, D. Zaitsev, Chern-Moser operators and polynomial models inCR geometry , Adv. Math. (2014), 321-356.[17] B. Lamel, N. Mir,
Finite jet determination of CR mappings , Adv. Math. (2007),153-177.[18] N. Mir, D. Zaitsev,
Unique jet determination and extension of germs of CR mapsinto spheres , preprint.[19] Tanaka, N.
On the pseudo-conformal geometry of hupersurfaces of the space of ncomplex variables . J. Math. Soc. Japan 14 (1962), 397-429.[20] A. Tumanov,
Stationary Discs and finite jet determination for CR mappings in highercodimension , preprint, arXiv:1912.03782v1.[21] D. Zaitsev: Germs of local automorphisms of real analytic CR structures and analyticdependence on the k -jets. Math. Res. Lett.4