Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets

17 Pages·0.23 MB·English

by  Dmitri Zaitsev

GERMS OF LOCAL AUTOMORPHISMS OF REAL-ANALYTIC
CR STRUCTURES AND ANALYTIC DEPENDENCE ON k-JETS.
8
9
9 DMITRI ZAITSEV
1

n Abstract. The topic of the paper is the study of germs of local holomor-
a phisms f between Cn andCn′ such that f(M)⊂M′ and df(TcM)=TcM′
J forM ⊂CnandM′⊂Cn′ genericreal-analyticCRsubmanifoldsofarbitrary
9 codimensions. It is proved that for M minimal and M′ finitely nondegener-
ate, such germs depend analytically on their jets. As a corollary, an analytic

] structureonthesetofallgermsofthistypeisobtained.
V
C
.
h
t
a 1. Introduction.
m Let M ⊂ Cn,M′ ⊂ Cn′ be connected locally closed real-analytic submanifolds,
[ ′ ′
x ∈ M,x ∈ M be arbitrary points. The complex tangent subspace TxM ∩iTxM
1 will be denoted by TcM. M is a CR manifold, if dimTcM is constant. In this case
x x
v dim M :=dim TcM iscalledtheCRdimensionandcodim M :=dim TM−
CR C CR IR
1 dim TcM the CR codimension. A CR submanifold M ⊂ Cn is called generic, if
4 TMIR+iTM =Cn.
0
1 SupposeforthemomentthatM,M′ ⊂Cn aregenericreal-analyticCRsubman-
0 ifoldsofthesameCRdimensionandthesameCRcodimension.Baouendi,Ebenfelt
8 andRothschildfound optimalnondegeneracyconditions onM and M′ suchthat a
9 germ at x of a local biholomorphism f (between some neighborhoods in Cn) with
h/ f(M)⊂M′,f(x)=x′,isuniquelydeterminedbyitsk-jetjkf,wherekisaninteger
x
t which depends only on M and M′ (see [2], Theorem 1, Proposition 2.3). A similar
a
′
m statement for the case of hypersurfaces M and M is implicitly contained in [8].
See also[5]for the case ofLevi-nondegeneratehigher codimensionalCR manifolds.
:
v TheseresultsshouldbecomparedwiththefollowingtheoremofH.Cartan([6,11]):
i
X
Let D ⊂Cn bea bounded domain. Then the group of all biholomorphic automor-
r
a phisms Aut(D) equipped with the compact-open topology is a Lie group. Moreover,
if x ∈ D is arbitrary, an automorphism f ∈ Aut(D) is uniquely determined by its
1-jet j1f and depends analytically on it.
x
In the above CR setting it is not clear how a map (germ) f depends on its k-jet
jkf. Is it somehow continuous, analytic etc.?
x
′
ThespecialcasewhenbothM andM areLevi-nondegeneratehypersurfaceswas
previously considered by Tanaka [14] and Chern-Moser [7], where the determinacy
ofautomorphismsbytheir2-jetswasshown.Eveninthisspecialcontextananalytic
dependenceonthejetsisofinterest.Analgebraicdependenceoftheautomorphisms
1991 Mathematics Subject Classification. 32C16,32D15.
PartiallysupportedbySFB237“UnordnungundgroßeFluktuationen”.
1
2 DMITRI ZAITSEV
of bounded domains on their 1-jets is studied in [18], which can be seen as an
algebraic version of the above theorem of Cartan.
Thepresentpaperhasadifferentpointofview.Insteadofglobalautomorphisms
′
we consider locally defined germs of holomorphic maps f sending M into M with
′
no assumptions on their domains of definition. If either M or M is locally biholo-
morphic to IRs ×Cl, the space of such germs is infinite-dimensional (see also [2],
Theorem3)andhenceitcannotbeparametrizedbyafinitedimensionalk-jetspace.
′
Thereforeboth M andM haveto satisfy certainnondegeneracyconditions.We
start with some stronger conditions formulated in terms of the Levi form. Recall
that in the case of arbitrary CR codimension the Levi form is a sesqui-linear map
(see §2):
L: T1,0M ×T1,0M →(T M/TcM)⊗ C. (1)
x x x x IR
Definition 1.1. We say that the Levi form at x is nondegenerate if for X ∈
0
T1,0M,
x
(cid:0)L(X0,Y)=0 ∀Y ∈Tx1,0M(cid:1) =⇒ X0 =0. (2)
The Levi form is said to be surjective at x, if the map (1) is surjective. The Levi
form is said to be nondegenerate (resp. surjective) if it is nondegenerate (resp.
surjective) for all x∈M.
Nowweintroducethe notionofanalytic dependency on k-jets.LetS beasubset
ofthe set ofgermsatx∈Cn ofalllocalholomorphicmaps f betweenCn andCn′.
ByJk(Cn,Cn′)denotethe k-jetspaceatx ofsuchmaps andbyjkf ∈Jk(Cn,Cn′)
x x x
the k-jet of f at x. We write also Jk(Cn,Cn′) for the (trivial) bundle of all k-jets
at all x∈Cn.
Definition 1.2. We say that the germsinS depend analyticallyontheir k-jets at
x if the following conditions are satisfied:
1. f ∈ S is uniquely determined within S by jkf, i.e. for f ,f ∈ S, jkf =
x 1 2 x 1
jkf =⇒ f =f ;
x 2 1 2
2. Foreveryf ∈S,thereexistneighborhoodsU(x)⊂Cn,U(jkf )⊂Jk(Cn,Cn′)
0 x 0 x
such that every f ∈S with jkf ∈U(jkf ) extends holomorphically to U(x);
x x 0
3. In addition there exists a holomorphic map F: U(jkf )×U(x) → Cn′ such
x 0
that for all f ∈S with jkf ∈U(jkf ),
x x 0
f(z)=F(jkf,z), z ∈U(x). (3)
x
For x∈Cn,we equipS withthe inductive limittopology,i.e. asequence (f )in
n
S converges to f ∈ S if and only if all f extend to some neighborhood of x and
0 n
converge there to f uniformly.
0
Theorem 1.1. Suppose that the Levi form of M is nondegenerate and surjective.
Then there exists an integer k >0 such that for all x∈M, the germs at x of local
biholomorphisms f with f(M)⊂M depend analytically on their k-jets. One can
take k =2(1+codim M).
CR
As an application we obtain a Lie group structure on the set of germs fixing a
point x∈M (see the end of the section for the proof).
GERMS OF LOCAL AUTOMORPHISMS 3
Corollary 1.1. Let x∈M be arbitrary and the Levi form of M be nondegenerate
and surjective at x. Then the group of all germs of local biholomorphisms f with
f(M)⊂M, f(x)=x, is a Lie group.
Noticethatthedomainofdefinitionofagermf canvaryasjkf changes.Itisnot
x
even clear a priori whether a 1-parameter family f ,t∈IR, of such automorphisms
t
yields a germ of a vector field. The analytic dependence on the k-jets guarantees,
in particular, the following extension result for the germs f with jkf sufficiently
x
close to jkf for f given.
x 0 0
Corollary 1.2. Under the assumptions of Theorem 1.1 suppose that f is a germ
0
of a local biholomorphism at x with f (M)⊂M. Then there exists a neighborhood
0
U(x)∈Cn such that all germs of local biholomorphisms f with f(M)⊂M and jkf
x
sufficiently close to jkf extend biholomorphically to U(x).
x 0
′
FollowingTanaka[14]wecallamapϕ: M →M pseudo-conformal,ifitextends
′
toaholomorphicmapbetweensomeneighborhoodsofM andM respectively.One
obtains the following global version of Corollary 1.1 (see the end of the section for
the proof).
Corollary 1.3. Let M be a compact CR submanifold of a complex manifold X
which in addition satisfies the assumptions of Theorem 1.1. Then the group of all
pseudo-conformal automorphisms of M is a Lie group.
In fact, all of these corollaries are proved here in the more general situation,
where M ⊂ Cn and M′ ⊂ Cn′ are generic real-analytic CR manifolds of arbitrary
CR dimension and codimension. In particular, the unique determinacy by k-jets,
i.e. the injectivity of the k-jet evaluation f 7→jkf, is also shown.
x
In this paper f always denotes a germ of a holomorphic map between open
subsetsofCn andCn′ respectively.The condition“f is biholomorphic”(see [1],[2])
is relaxed to
dfx(TxcM)=Txc′M′, (4)
where f(x)=x′ and TcM :=T M ∩iT M ⊂T Cn is the complex tangent space.
x x x x
The following simple example shows that even if both CR dimensions and
′
codimensions of M and M are equal, the case when (4) is satisfied but f is not
biholomorphic is also of interest.
Example 1.1. Let ϕ be strongly plurisubharmonic function in a neighborhood of
0∈C3 satisfying
∂ϕ
ϕ(0)=0, (0)6=0. (5)
∂z
3
Define M and M′ in C4:
M :={(z ,z ,z ,z ):ϕ(z ,z ,z )=ϕ(2z ,z ,z )=0}, (6)
1 2 3 4 1 2 3 1 2 4
′
M :={(z ,z ,z ,z ):ϕ(z ,z ,z )=Rez =0}. (7)
1 2 3 4 1 2 3 4
One has
′ ′
dim M =dim M =2, codim M =codim M =2. (8)
CR CR CR CR
The (C2-valued) Levi forms are given by the second order derivatives ∂2ϕ/∂z ∂z¯ .
i j
′
Simple calculation shows that the Levi form of M (resp. M ) is surjective (resp.
4 DMITRI ZAITSEV
′
nondegenerate). By Proposition 2.1 and Lemma 2.1, M and M satisfy conditions
of Theorem 1.2 below. The map f given by
f(z ,z ,z ,z ):=(z ,z ,z ,0), (9)
1 2 3 4 1 2 3
′
satisfies f(M)⊂M and condition (4) but is not a diffeomorphism between M and
′
M .
On the other hand, the condition (4) cannot be removed as the following ele-
mentary example shows.
Example 1.2. Define
M :={|z |2+|z |2−|z |2 =1}⊂C3. (10)
1 2 3
Then M contains the complex line
L={z =1,z =z }. (11)
1 2 3
Therefore every holomorphic map f: C3 → L satisfies f(M) ⊂ M. Although the
Levi form of M is nondegenerate and surjective, the k-jet evaluation f 7→ jkf is
x
not injective for no integer k.
As noted above, the nondegeneracy and surjectivity of the Levi form are only
sufficientconditions.The optimalconditionswhicharenecessaryinmanycasesare
given in [2] for the case f is biholomorphic. Here we reformulate them in a form
suitable for our purposes.
We first recall the notion of the Segre varieties associated to a generic real-
analytic CR submanifold M ⊂ Cn. Let x ∈ M be arbitrary and let M be defined
near x by the real-analytic equations
ρ (z,z¯)=···=ρ (z,z¯)=0, ∂ρ ∧···∧∂ρ 6=0. (12)
1 d 1 d
The complexification M⊂Cn×Cn is defined by
M:={(z,w¯)∈U(x)×U(x¯):ρ (z,w¯)=···=ρ (z,w¯)=0}, (13)
1 d
whereU(x)⊂CnandU(x¯)⊂Cnaresufficientlysmallneighborhoods.Asagermat
(x,x¯) of a complex-analytic subset, M is independent of the choice of the defining
equations (12). By fixing the coordinate w¯, we obtain the Segre variety
Q :={z ∈U(x):(z,w¯)∈M}. (14)
w
Segre varieties were introduced by Segre [13] and play an important role in the
reflection principle (see e.g. [12, 10, 9, 16, 17]). For z,w close to x, it follows that
Q is acomplexmanifoldofdimensionn−d=dim M.The followingsymmetry
w CR
property is a direct corollaryof the invariance of M under the involution (z,w¯)7→
(w,z¯):
z ∈Q ⇐⇒ w∈Q . (15)
w z
For arbitrary x ∈ Cn, denote by Jk,d(Cn) the space of k-jets at x of d-
x
codimensionalcomplexsubmanifoldsV ⊂Cn withx∈V andbyjk(V)∈Jk,d(Cn)
x x
the k-jet ofV.The following is anotherformofthe definitionof ak-nondegenerate
manifold given in [1].
GERMS OF LOCAL AUTOMORPHISMS 5
Definition 1.3. A generic real-analytic CR manifold M ⊂ Cn is called k-
nondegenerate at x∈M (k ≥1), if the (antiholomorphic) map
ϕk: Q →Jk,d(Cn), w 7→jk(Q ), (16)
x x x w
is of the rank n −d = dimQ at x. M is called finitely nondegenerate, if it is
x
k-nondegenerate for some k.
A manifold M is called essentially finite at x if the germ of Q at x is different
x
from germs of Q at x for all w ∈ Q \{x} sufficiently close to x. This notion
w x
was introduced in [4] and earlier implicitly discussed in [8]. By Proposition 1.3.1
in [1], if M is essentially finite at x, there exists the so-called Levi number l(M).
ThenM isl(M)-nondegenerateatallpoints p∈M\W,whereW ⊂M is aproper
′
real-analyticsubset.Clearlythe k-nondegeneracyguaranteesthe k -nondegeneracy
′
for all k ≥k.
ThesecondimportantnotionistheminimalityconditionintroducedbyTumanov
in [15]:
Definition 1.4. A CR manifold M is called minimal at x∈M, if there does not
existaproper CRsubmanifold N ⊂M with x∈N suchthat dim N =dim M.
CR CR
Tumanov [15] shows that, if M is minimal at x, all CR functions on M extend
holomorphically to a wedge with the edge M.
Definition 1.5. Wecallagermf admissibleifitsatisfiesthefollowing conditions:
f(M)⊂M′, df(TcM)=Tc M′. (17)
x f(x)
Theorem 1.2. Suppose that M′ is r-nondegenerate, M is minimal at x and k :=
2r(1+codim M). Let S be the set of all admissible germs f at x. Then the germs
CR
in S depend analytically on their k-jets at x.
A different proof of Theorem 1.2 in the case of hypersurfaces (codim M =
CR
′
codim M = 1) with a sharper estimate for the jet order (2r-jets instead of
CR
4r-jets) has been obtained recently by Baouendi, Ebenfelt and Rothschild [3].
Remark.Thecase,whereoneofthemanifoldsM,M′isnotgeneric,canbereduced
to the generic case. Every real-analytic CR submanifold M ⊂ Cn is generic in the
so-calledintrinsic complexification V ⊂Cn defined to be the minimal (in the sense
ofgerms)complex-analyticsubvarietywhichcontainsM.IfM isCR,V is smooth.
After a change of local coordinates V becomes open in Cm ⊂ Cn near x ∈ M.
′ ′ ′
The condition f(M) ⊂ M automatically implies f(V) ⊂ V . If M is not generic,
we can therefore replace Cn′ with V′. If M is not generic, Theorem 1.2 yields
a parametrization (3) for the restricted admissible germs f: V → Cn′, whereas
outside V the germs can be chosen arbitrary and cannot be determined by their
s-jets even for s arbitrary large.
As a corollary,we obtainthe followingdescriptionof the space ofadmissible germs
as a (locally closed) real-analytic subset of Jk(Cn,Cn′).
x
Corollary 1.4. Let k,r,x,M,M′,S satisfy the conditions of Theorem 1.2.
1. Then the set A ⊂ Jk(Cn,Cn′) of the k-jets of all f ∈ S is a locally closed
x
real-analytic subset.
6 DMITRI ZAITSEV
2. There exists a neighborhood U of A×{x} ⊂ A×Cn and a (unique) real-
analytic map
G: U →Cn′, (18)
which is holomorphic in the Cn-factor and is such that every f ∈S satisfies
f(w)=G(jkf,w). (19)
x
Proof. By Theorem 1.2, the germs in S depend analytically on their k-jets. Let
′
M,M be closed in some open subsets
U(M)⊂Cn, U(M′)⊂Cn′ (20)
respectively. For every f ∈ S, Theorem 1.2 yields the local parametrization (3).
0
Without loss of generality, U(x)⊂U(M) and
F(U(jkf )×U(x))⊂U(M′). (21)
x 0
Then the above set A⊂Jk(Cn,Cn′) is locally defined by
x
A∩U(jkf ):={j ∈U(jkf ):
x 0 x 0
F(j,w)⊂M′ for all w ∈M ∩U(x), j =jkF(j,·)}. (22)
x
′
Itfollowsfromthereal-analyticityofM andM thatAislocallydefinedbyfinitely
manyreal-analyticequations,i.e.itisareal-analyticsubset.TherestrictionsofF’s
to A×U(x) for different f ’s glue together to a well-defined real-analytic map G
0
satisfying the required properties.
Proof of Corollary 1.1. By Corollary 1.4, the set A ⊂ Jk(Cn,Cn′) of the k-jets
x
(k := 2(d+1)r) of all admissible germs is real-analytic. Denote by A ⊂ A the
0
subset of the k-jets fixing x. Clearly A is also real-analytic and locally closed.
0
SinceA isalsoclosedunderthecompositionofthejets,itisaclosedLiesubgroup
0
of the Lie group of all k-jets fixing x.
Proof of Corollary 1.3. By Theorem 1.2 and the definition of analytic dependence
on k-jets, there exists a finite open covering of M with coordinate neighborhoods
U(x ) with parametrizations
i
F : U(jk f )×U(x )→Cn, (23)
i xi 0 i
where f := id ∈ Aut(M). An automorphism f ∈ Aut(M) in a neighborhood of id
0
is therefore uniquely determined by finitely many parameters
p =jk f, q =jk f−1. (24)
i xi i xi
Set f := F (p ,·), g := F (q ,·). A tuple (p ,... ,p ,q ,... ,q ) determines an
i i i i i i 1 m 1 m
automorphism of M if and only if the following is satisfied:
1. f =f , g =g on U(x )∩U(x ) for all i,j;
i j i j i j
2. f (M)⊂M, g (M)⊂M for all i;
i i
3. The gluedmapsf andg satisfy f ◦g =g ◦f =id,jk f =p ,jk g =q .
p q p q q p xi p i xi q i
Conditions(1.),(2.),(3.)defineananalyticsubsetoftheparameterspaceofp ’sand
i
q ’s. In these coordinates the group operation is given by
i
((p,q),(p′,q′))7→(jxki(fp′ ◦fp),jxki(gq◦gq′))
and is therefore real-analytic. Hence Aut(M) is a Lie group.
GERMS OF LOCAL AUTOMORPHISMS 7
2. The Levi form and nondegeneracy conditions.
Let M ⊂ Cn be a CR submanifold and x ∈ M an arbitrary point. Recall that
a (1,0)-vector field on M is a vector field X in TcM ⊗ C such that JX = iX,
IR
where J: TcM ⊗ C → TcM ⊗ C is the complexification of the CR structure
IR IR
J: TcM → TcM and iX is the multiplication by i in the component C of the
tensor product.
The Levi form of M at x is the hermitian (vector-valued) form
L: T1,0M ×T1,0M →(T M/TcM)⊗ C, (25)
x x x x IR
given by
1
L(X,Y):= π([X,Y¯]), (26)
2i
where X,Y are (1,0)-vector fields on M and
π: T M ⊗ C→(T M/TcM)⊗ C (27)
x IR x x IR
is the canonical projection. Notice that L(X,Y)(x) depends only on X(x),Y(x).
Remark. If M ⊂ Cn is a hypersurface defined by M ={ϕ=0} with dϕ6= 0, the
standard Levi form of ϕ coincides with the evaluation of ∂ϕ on L(X,Y).
TheLeviformisafirstorderholomorphicinvariantofM whereastheconditions
in Theorem 1.2 are of possibly higher order. However,we obtain the following
Proposition 2.1. Agenericreal-analyticCRmanifoldM ⊂Cn is1-nondegenerate
at x if and only if the Levi form at x is nondegenerate.
Proof. Let M be locally defined near x by the real-analytic equations
ρ (z,z¯)=···=ρ (z,z¯)=0, (28)
1 d
where ∂ρ ∧···∧∂ρ 6=0.
1 d
The differentials ∂ρ vanish on TcM ⊗ C. Hence, for a (1,0)-vector field X ⊂
j IR
T1,0M,
∂ρ (X)=∂ρ (X¯)=0. (29)
j j
Denote by Xψ the derivative of ψ along X.
Applying the formula
dω(X ,X )=X ω(X )−X ω(X )−ω([X ,X ]) (30)
1 2 1 2 2 1 1 2
for ω =∂ρ , X =X,X =Y¯, we obtain
j 1 2
∂¯∂ρ (X,Y¯)=−∂ρ ([X,Y¯]). (31)
j j
The Segre varieties Q are given by
w
ρ (z,w¯)=···=ρ (z,w¯)=0 (32)
1 d
and their 1-jets j1Q by the linear equations
x w
∂ρ (z,w¯)=···=∂ρ (z,w¯)=0. (33)
1 s
Let e (z,w¯),... ,e (z,w¯) be a collection of (1,0)-vector fields on the complex-
1 d
ification M which is a basis of T Q at every point (z,w¯) ∈ M close to (x,x¯).
z w
′ ′
Let e ,... ,e be a similar collection of (1,0)-vector fields in the w-direction, i.e.
1 d
8 DMITRI ZAITSEV
pointwise a basis of T Q . The rankof the map ϕ1 in Definition 1.3 is the same as
w¯ z¯
the rank of the matrix with the rows
∂¯∂ρ (e ,e¯′) with l fixed . (34)
j k l
By (31), this matrix has the rows
′
−∂ρ ([e ,e¯]) with l fixed . (35)
j k l
Since∂ρ ∧···∧∂ρ 6=0,thetuple(∂ρ ,... ,∂ρ )definesanisomorphismbetween
1 d 1 d
(TM/TcM)⊗ C andCd. Hence the matrix with the rows(35)has the same rank
IR
as the matrix with the rows
L(e ,e ) with l fixed . (36)
k l
This rank equals to n−d if and only if the Levi form of M is nondegenerate.
The minimality condition (Definition 1.4) involves high order commutators and
therefore cannot be formulated in terms of the Levi form. However, one obviously
has the following sufficient condition.
Lemma 2.1. SupposethattheLeviformofM atxissurjectiveonto(T M/TcM)⊗
x x IR
C. Then M is minimal at x.
NowwecanformulatespecialcasesofTheorem1.2andCorollary1.4underLevi
form conditions.
Theorem 2.1. Suppose that the Levi form of M′ is everywhere nondegenerate, the
Levi form of M is surjective at x and d = codim M. Then the conclusions of
CR
Theorem 1.2 and Corollary 1.4 hold.
3. Local parametrization of jets of holomorphic maps.
One of the technical tools for proving Theorem 1.2 is the following connection
between the k-jets at z and the (k +r)-jets at w¯. For simplicity, we write for a
subset U(j)⊂Jk+r(Cn,Cn′),
U (j):=U(j)∩Jk+r(Cn,Cn′). (37)
z z
Proposition 3.1. Suppose that M′ is r-nondegenerate at x′ and f is an admissi-
0
′
ble germ at x∈M, x =f (x). Then, for every integer k there exist neighborhoods
0
U(jk+rf ) ⊂ Jk+r(Cn,Cn′), U(x,x¯) ⊂ M and for every (z,w¯) ∈ U(x,x¯) a holo-
x 0
morphic map Fk : U (jk+rf )→Jk(Cn,Cn′), such that for all admissible germs
(z,w¯) z x 0 w¯
f at x with jk+rf ∈U(jk+rf ),
x x 0
jkf¯=Fk (jk+rf), (38)
w¯ (z,w¯) z
where (z,w¯)⊂M is sufficiently close to (x,x¯). Moreover, the map Fk depends
(z,w¯)
holomorphically on (z,w¯)∈U(x,x¯).
AdifferentproofofProposition3.1isgivenin[1](Assertion3.3.1)and[2](Propo-
sition 2.2).
Proof. Let (z,w¯) ∈ M be close to (x,x¯). By the construction (see (12),(13),(14)),
the SegrevarietyQ is smoothatz andd:=dim Q is constant.We chooselocal
w z w
GERMS OF LOCAL AUTOMORPHISMS 9
coordinates z = (z ,z ) ∈ Cd ×Cn−d near x such that the Segre variety Q has
1 2 w
the form of the graph:
Q ={(z ,ϕ (z )):z ∈U(x )}, x=(x ,x ), (39)
w 1 w 1 1 1 1 2
where U(x ) ⊂ Cn is an open neighborhood and ϕ : U(x ) → Cn−d is holomor-
1 w 1
phic. The map ϕ is uniquely defined and depends holomorphically on w¯ in some
w
neighborhood U(x¯)⊂Cn . Denote by jrQ the holomorphic r-jet evaluation
z
jrQ: U(x¯)→Jr (Cd,Cn−d), w¯ 7→jr ϕ . (40)
z z1 z1 w
′ ′ ′ ′ ′
A similar evaluation is obtained for (z ,w¯ )∈M sufficiently close to (x,x¯):
jzr′Q′: U(x¯′)→Jzr′(Cd′,Cn′−d′), w¯′ 7→jzr′ϕ′w′. (41)
1 1
′ ′ ′ ′
Weclaimthat(41)isanimmersion.SinceM isr-nondegenerateatx =(x ,x ),
1 2
jrQ′ restricted to the Segre variety Q′ is an immersion. By the construction, the
x x′
0-jetevaluationj0 Q′ =x′ isconstantonQ′ .Ontheotherhand,therestrictionof
x′ 2 x′
j0 Q′ to the transversaldirection, i.e. {x′}×Cn′−d′, is also an immersion. Indeed,
x′ 1
since M′∩({x′}×Cn′−d′) is totally real in {x′}×Cn′−d′, it is locally biholomor-
1 1
phically equivalent to IRn′−d′. For IRs ⊂ Cs, the immersion property of j0 Q′ can
x′
be directly verified.
Hencethetotalmap(41)splitsintoj0 Q′whichisanimmersionon{x′}×Cn′−d′
x′ 1
′ ′
and constant on Q and the remainder which is an immersion on Q . Therefore
x′ x′
(41) is an immersion for z close to x and locally there exists a left inverse map
(jr Q′)−1 which satisfies (jr Q′)−1◦(jr Q′)=id.
z′ z′ z′
′
Let f be an admissible germ. Complexifying the condition f (M) ⊂ M we
0 0
obtain (f ,f¯)(M)⊂M′, which means
0 0
′
f (Q )⊂Q . (42)
0 w f0(w)
Since dimQ is constant, the second condition df(TcM)=Tc M′ implies
w x f(x)
dimf (Q )=dimdf (T Q )=dimdf (TcM)
0 w 0 x x 0
=dimTcM′ =dimQx′ =dimQ′f0(w). (43)
Along with (42) this yields
′ ′
f0(Qw)=Qf0(w), df0(TzQw)=Tz′Qw′. (44)
Aftergoingtothek-jetevaluationsweobtainthefollowingcommutativediagram
U(x¯) −−j−zr−Q→ Jr (Cd,U(x ))
z1 2
f¯0 f0∗ (45)
y y
U(x¯′) −−jzr−′Q−→′ Jr (Cd′,U(x′)),
z′ 2
1
where f0∗ is the corresponding map on the jet level defined as follows.
By (44), there exists a local splitting of the source space Cd =Cd′×Cd−d′ near
x =(x ,x ) such that the restriction
1 11 12
′
f0: Qw∩{z12 =x12}→Qw′ (46)
is locally biholomorphic for all (z,w¯)∈M close to (x,x¯).
10 DMITRI ZAITSEV
We write f : U(x) → Cd′, f : U(x) → Cn′−d′ for the components of f (here
01 02 0
Cd′ and Cn′−d′ are in the target space). Since (46) is locally biholomorphic, the
map
f˜ : Cd′ →Cd′, z 7→f (z ,x ,ϕ (z ,x )) (47)
0 11 01 11 12 w¯ 11 12
is also locally biholomorphic.
Let f˜−1 be the local inverse. By (39) and (44)
0
ϕ′w¯′(z1′)=f02(f˜0−1(z1′),x12,ϕw¯(f˜0−1(z1′),x12)) (48)
and hence, passing to the r-jets,
jzr1′ϕw′¯′ =(jzrf02)((jzr11f˜0)−1,x12,(jzr1ϕw¯)((jzr11f˜0)−1,x12)), (49)
where by (47),
jr f˜ =(jrf )(z ,x ,(jr ϕ )(z ,x )) (50)
z11 0 z 01 11 12 z1 w¯ 11 12
with obvious notation.
The formulae (49,50) can be written together in the form
jzr1′ϕw′¯′ =Φ(jzrf0,jzr1ϕw¯), (51)
where
Φ: Uz(jxrf0)×Uz1(jxr1ϕx¯)→Jzr1′(Cd′,U(x′2)) (52)
isaholomorphicfamilyofmapsandU(jrf )⊂Jr(Cn,Cn′),U(jr ϕ )⊂Jr(Cd,Cn−d)
x 0 x1 x¯
are sufficiently small neighborhoods.
Define f0∗ as in (45) by
f0∗ :=Φ(jzrf0,·). (53)
By (51), the diagram (45) is commutative.
The commutativity means that
(jzr′Q′)◦f¯0 =f0∗◦(jzrQ). (54)
Applying the left inverse (jr Q′)−1 to both sides we obtain
z′
f¯0 =(jzr′Q′)−1◦f0∗◦(jzrQ) (55)
and, passing to the k-jets,
jwk¯f¯0 =jgk′(jzr′Q′)−1◦jgkf0∗◦jwk¯(jzrQ), (56)
where g :=(jzrQ)(w¯) and g′ :=jgkf0∗(g).
The k-jet jgkf0∗ can be calculated from (53):
jgkf0∗ :=(jskΦ)(jzk+rf0,·), s:=(jzrf0,g). (57)
We write (56) and (57) together in the form
jkf¯ =Fk (jk+rf ), (58)
w¯ 0 z,w¯ z 0
where
Fk : U (jk+rf )→Jk(Cn,Cn′) (59)
z,w¯ z x 0 w¯
isaholomorphicfamilyofmapsandU(jk+rf )⊂Jk+r(Cn,Cn′)asufficientlysmall
x 0
open neighborhood.
Now we notice that f was an arbitrary admissible germ. Let f be another one
0
with jk+rf ∈U(jk+rf ). Then for (z,w¯)∈M sufficiently close to (x,x¯), (48)-(58)
x x 0