Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces
aa r X i v : . [ m a t h . C V ] F e b RANGE DECREASING GROUP HOMOMORPHISMS ANDHOLOMORPHIC MAPS BETWEEN GENERALIZED LOOPSPACES
NING ZHANG
Abstract.
Let G resp. M be a positive dimensional Lie group resp.connected complex manifold without boundary and V a finite dimen-sional C ∞ compact connected manifold, possibly with boundary. Fix asmoothness class F = C ∞ , H¨older C k,α or Sobolev W k,p . The space F ( V, G ) resp. F ( V, M ) of all F maps V → G resp. V → M is a Ba-nach/Fr´echet Lie group resp. complex manifold. Let F ( V, G ) resp. F ( V, M ) be the component of F ( V, G ) resp. F ( V, M ) containing theidentity resp. constants. A map f from a domain Ω ⊂ F ( V, M ) to F ( W, M ) is called range decreasing if f ( x )( W ) ⊂ x ( V ), x ∈ Ω. Weprove that if dim R G ≥
2, then any range decreasing group homomor-phism f : F ( V, G ) → F ( W, G ) is the pullback by a map φ : W → V .We also provide several sufficient conditions for a range decreasing holo-morphic map Ω → F ( W, M ) to be a pullback operator. Then we ap-ply these results to study certain decomposition of holomorphic maps F ( V, N ) ⊃ Ω → F ( W, M ). In particular, we identify some classesof holomorphic maps F ( V, P n ) → F ( W, P m ), including all automor-phisms of F ( V, P n ). Introduction
Let M be a positive dimensional C ∞ manifold without boundary and V a finite dimensional C ∞ compact connected manifold, possibly with bound-ary (all finite dimensional manifolds considered in this paper are secondcountable). We fix a smoothness class F = C ∞ , H¨older C k,α ( k = 0 , , · · · ,0 ≤ α ≤
1, where C k, = C k ) or Sobolev W k,p ( k = 1 , , · · · , 1 ≤ p < ∞ , kp > dim R V or k = dim R V , p = 1). Then the space F ( V, M ) of all F maps V → M is a C ∞ Banach/Fr´echet manifold (see [7, 12, 21]), to which werefer as the generalized loop space of M (where F ( S , M ) is the loop spaceof M ). Note that C ∞ ( V, M ) ⊂ F ( V, M ) ⊂ C ( V, M ) . Mathematics Subject Classification.
Key words and phrases.
Range decreasing, Loop space, Holomorphic map, Group ho-momorphism, Projective space.The author is grateful to L. Lempert for stimulating discussions and very helpful com-ments on the present paper. This research was partially supported by the Scientific Re-search Foundation of Ocean University of China grant 861701013110, and the NationalNatural Science Foundation of China grants 10871002 and 11371035. If M is a Lie group G , then F ( V, G ) is a Lie group under pointwise groupoperation (see [22, 12]). If M is a complex manifold, then F ( V, M ) carriesa natural complex manifold structure (see [13, 14]). The mapping spaces F ( V, M ) and operators F ( V, N ) → F ( W, M ) are of fundamental impor-tance in analysis, geometry, mathematical physics and representation theory.Let T ( F , V, F , W ) be the space of maps φ : W → V such that x ◦ φ ∈ F ( W, R ) for every x ∈ F ( V, R ). It always contains constants. If k > k , or k = k and α ≥ α , then all C ∞ maps W → V are in T (cid:0) C k ,α , V, C k ,α , W (cid:1) . We write T ( F , V ) for the space T ( F , V, F , V ).If φ ∈ T ( F , V, F , W ), then the pullback operator φ ∗ = φ ∗ M : F ( V, M ) ∋ x x ◦ φ ∈ F ( W, M )is a well defined C ∞ map. It is holomorphic when M is a complex manifold(see Proposition 2.1). If M is a Lie group, it is clear that φ ∗ is a Lie grouphomomorphism. Let Ω be an open subset of F ( V, M ). We say that a map f : Ω → F ( W, M ) is range decreasing if f ( x ) ( W ) ⊂ x ( V ) , x ∈ Ω . It is clear that φ ∗ | Ω is range decreasing. By [26, Lemma 3.2], a range de-creasing map f : F ( S , C n ) → F ( S , C n ) (where F = C k , k = 1 , , · · · , ∞ ,or F = W k,p ) is a pullback operator if and only if f is continuous complexlinear. This lemma plays a key role in the proof of [26, Theorem 1.1]: if aholomorphic self-map f of F (cid:0) S , P n (cid:1) induces an isomorphism of the secondintegral cohomology group of F (cid:0) S , P n (cid:1) , then we have the decomposition f = γ ◦ φ ∗ P n , where φ ∈ T ( F , S ) and γ ∈ F ( S , P GL ( n + 1 , C )) (which can be con-sidered as a holomorphic automorphism of F (cid:0) S , P n (cid:1) induced by a familyof holomorphic automorphisms of P n parameterized by t ∈ S ). A rangedecreasing C ∞ map is not always (the restriction of) a pullback operator(see Subsection 2.2). As we shall prove in this paper, oftentimes the rangedecreasing condition does imply that a group homomorphism resp. holo-morphic map is (the restriction of) a pullback operator.The constant maps V → M form a submanifold of F ( V, M ), which canbe identified with M . We write F ( V, G ) for the “trivial” component of theLie group F ( V, G ) containing the identity; and if M is connected, we write F ( V, M ) for the “trivial” component of F ( V, M ) containing the constants.Recall that the evaluation at v ∈ V : E v = E v, F ( V,M ) : F ( V, M ) ∋ x x ( v ) ∈ M is a C ∞ map. If M is a Lie group resp. complex manifold, then E v is a Liegroup homomorphism resp. holomorphic map.In this paper we provide (partial) answers to the following questions: Question 1.
Let f : F ( V, G ) → F ( W, G ) resp. f : F ( V, G ) → F ( W, G ) be a range decreasing group homomorphism (in the algebraic sense). Under ange decreasing group homomorphisms and holomorphic maps 3 what conditions (on V , W , G , F and F ) must f be (the restriction of ) apullback operator? Question 2.
Let Ω be a connected open subset of F ( V, M ) and f : Ω →F ( W, M ) a range decreasing holomorphic map. Under what conditions (on V , W , M , F , F and Ω ) must f be (the restriction of ) a pullback operator? Answers to Questions 1 and 2 are useful to decompose some of the grouphomomorphisms and holomorphic maps between generalized loop spaces.As an illustration, we consider the following
Question 3.
Let Ω be a connected open subset of F ( V, N ) and f : Ω →F ( W, M ) a holomorphic map. How can we tell whether f has a decompo-sition of the form (1) f = γ ◦ φ ∗ N | Ω , where φ ∈ T ( F , V, F , W ) and γ is a holomorphic map from a connectedopen neighborhood ˜Ω ⊂ F ( W, N ) of φ ∗ N (Ω) to F ( W, M ) induced by afamily of holomorphic maps γ w : N ⊃ E w ( ˜Ω) → M with γ ( y )( w ) = γ w ( y ( w )) , w ∈ W, y ∈ ˜Ω ? Theorem 1.1.
Let f be a range decreasing group homomorphism from F ( V, G ) resp. F ( V, G ) to F ( W, G ) . Suppose that dim R G ≥ . Thenthere exists φ ∈ T ( F , V, F , W ) such that f = φ ∗ (cid:0) = φ ∗G (cid:1) on any compo-nent of F ( V, G ) containing an element whose image is nowhere dense in G .In particular, f = φ ∗ on F ( V, G ) ; and if dim R V < dim R G , then f = φ ∗ on F ( V, G ) . Theorem 1.1 does not hold if dim R G = 1 (see Section 5). The automor-phism group Aut ( G ) of G is a finite dimensional Lie group if π ( G ) is finitelygenerated (see [10, Theorem 2]). In this case, any γ ∈ F ( W, Aut ( G ))induces a Lie group automorphism of F ( W, G ) resp. F ( W, G ) (see Sub-section 2.1), which is still denoted by γ . Corollary 1.2.
Let f : F ( V, G ) → F ( W, G ) be a group homomorphism.Assume that dim R G ≥ and G is connected. Then (2) f ( F ( V, G \ { } )) ⊂ F ( W, G \ { } ) if and only if there exist γ f ∈ F ( W, Aut ( G )) and φ ∈ T ( F , V, F , W ) such that (3) f = γ f ◦ φ ∗G . It is clear that γ f ( w ) = E w ◦ f | G ∈ Aut ( G ), and the decomposition in (3)is unique.Recall that a C immersion x from V to a complex manifold M is totallyreal if for each v ∈ V , the real subspace d v x ( T v V ) ⊂ T x ( v ) M contains nocomplex subspace of positive dimension. If there exists a totally real immer-sion V → M , then dim R V ≤ dim C M . Any immersion x ∈ F (cid:0) S , M (cid:1) or N. Zhang x ∈ F ([0 , , M ) (the space of open strings) is totally real. For more infor-mation about totally real immersions and related references see [4, Section9.1]. Theorem 1.3.
Let Ω be a connected open subset of F ( V, M ) and f : Ω →F ( W, M ) a range decreasing holomorphic map. Assume that for some k =1 , , · · · and ≤ α ≤ , there is a continuous embedding (4) C k,α ( V, M ) resp. C ∞ ( V, M ) ⊂ F ( V, M ) with a dense image. Suppose that there exist a connected open subset V ⊂ V and a C k,α /C ∞ element y ∈ Ω such that y | V is a totally real embedding, y ( V \ V ) ∩ y ( V ) = ∅ and f ( y )( W ) ∩ y ( V ) = ∅ . Then f = φ ∗ | Ω for some φ ∈ T ( F , V, F , W ) . For all regularities F considered in this paper except F = C ,α , 0 < α ≤
1, there is a continuous embedding with a dense image as in (4). Note that[26, Lemma 3.2] is a special case of Theorem 1.1 and of Theorem 1.3. Theproofs of Theorems 1.1 and 1.3 are different from that of [26, Lemma 3.2].
Corollary 1.4.
Let f be as in Theorem 1.3. Suppose that the embedding(4) (with a dense image) and one of the following conditions hold: (a) there is a totally real C k,α /C ∞ embedding y ∈ Ω ; (b) there is a totally real C k +sgn( α ) /C ∞ immersion y ∈ Ω ; (c) Ω contains a C k +sgn( α ) /C ∞ element and dim C M ≥ (cid:20)
32 dim R V (cid:21) (theinteger part of
32 dim R V ).Then f = φ ∗ | Ω for some φ ∈ T ( F , V, F , W ) . The map f in Question 3 is completely determined by the family of maps E w ◦ f : Ω → M , w ∈ W . If f can be decomposed as in (1), then E w ◦ f = γ w ◦ E φ ( w ) | Ω . Suppose N ⊂ Ω. Then we have γ w = E w ◦ f | N . Furthermore, if E w ◦ f | N is not constant for every w ∈ W , then the decomposition (1) is unique.For any x ∈ F ( V, M ), the pullback x ∗ T M of the tangent bundle of M by x is an F bundle (i.e. the transition functions of the bundle are F maps), andthe tangent space T x F ( V, M ) is F ( x ∗ T M ), the Banach/Fr´echet space of F sections of x ∗ T M (see [13, 14]). If x ∗ T M is trivial, then T x F ( V, M ) can beconsidered as F ( V, R m ) or F ( V, C m ) (where m = dim R M or dim C M ). Asan application of Corollary 1.2, we obtain the following Theorem 1.5.
Let Ω be a connected open subset of F ( V, N ) and f :Ω → M a holomorphic map. Assume that N, M are connected, dim C N =dim C M = n ≥ , x ∗ T N is trivial for any x ∈ Ω and (5) f ( x ) = f ( x ) for x , x ∈ Ω with x ( v ) = x ( v ) , v ∈ V. Then there is a fixed v ∈ V such that ker d x f = ker E v , F ( V, C n ) , x ∈ Ω ange decreasing group homomorphisms and holomorphic maps 5 (where ker d x f is the kernel of d x f ). In particular, f is constant on anycomponent of the complex submanifolds Ω ∩ E − v , F ( V,N ) ( ζ ) , ζ ∈ N . Submanifolds E − v , F ( V,N ) ( ζ ) ⊂ F ( V, N ) are of complex codimensiondim C N (see [13, p. 40]). Recall the local charts of F ( V, N ) as in [13, 14].It is clear that the map f in Theorem 1.5 has the following property: Forany x ∈ Ω, there exists a neighborhood Ω x ⊂ Ω of x such that f | Ω x is thecomposition of E v = E v , F ( V,N ) and an injective map E v (Ω x ) → M . Fora holomorphic map h : N → M , we write N j,h ⊂ N for the subset of points x such that the complex rank of d x h is j . Corollary 1.6.
Let O ⊂ F ( V, N ) be a connected open neighborhood of N and f : O → M a holomorphic map with dim C N = n ≤ dim C M . Then f = f | N ◦ E v | O for some v ∈ V with N n,f | N = ∅ if and only if there existsan open subset Ω ⊂ O such that Ω ∩ N = ∅ , f (Ω) is contained in an n dimensional complex submanifold of M and (5) holds on Ω . By [24, 15, 25], there does not exist a
P GL (2 , C )-equivariant holomorphicembedding of F (cid:0) S , P (cid:1) into a projectivized Banach/Fr´echet space P (wherethe action of P GL (2 , C ) on P is given by a monomorphism from P GL (2 , C )to the group of holomorphic automorphisms of P ). On the other hand, if e : M → P n is a holomorphic embedding, then e ∗ : F ( V, M ) ∋ x e ◦ x ∈F ( V, P n ) is a holomorphic embedding with e ∗ (cid:0) F ( V, M ) (cid:1) ⊂ F ( V, P n ). If e is G -equivariant for a group G , then so is e ∗ . Thus F ( V, P n ) is expectedto play a special role in the theory of generalized loop spaces of complexprojective manifolds.For integers m ≥ n ≥ d ≥
0, let H n,md be the space of holo-morphic maps P n → P m which induce multiplication by d in the secondintegral cohomology. The space H n,m consists of constants. The space H n,md with d ≥ F (cid:0) V, H n,md (cid:1) canbe considered as a holomorphic map F ( V, P n ) → F ( V, P m ). In particu-lar, H n,n = P GL ( n + 1 , C ) is the group of holomorphic automorphisms of P n , and F ( V, P GL ( n + 1 , C )) acts on F ( V, P n ) holomorphically (see Sub-section 2.1). Let f be a holomorphic map from an open neighborhood of P n ⊂ F ( V, P n ) to F ( W, P m ). We say that the degree of f is d and writedeg f = d if E w ◦ f | P n ∈ H n,md , w ∈ W . As a consequence of Corollary 1.6, we obtainthe following Theorem 1.7.
Let f : F ( V, P n ) → F ( W, P m ) be a holomorphic map ofdegree d . Suppose that one of the following conditions holds: (a) d = 1 ; (b) d = 2 and n = m = 1 . N. Zhang
Then there exist γ f ∈ F (cid:0) W, H n,md (cid:1) and φ ∈ T ( F , V, F , W ) such that (6) f = γ f ◦ φ ∗ P n . Thus any range decreasing holomorphic map f : F ( V, P n ) → F ( W, P n )must be a pullback operator ( f | P n = id, so deg f = 1 and γ f = id). On theother hand, for any nontrivial component Ω j of C ∞ ( S , P ), there exist rangedecreasing holomorphic maps Ω j → C ∞ ( S , P ) which are not pullbackoperators (see Section 4, cf. Corollary 1.4(c)).Let f be as in (6) and w ∈ W . Then ( E w ◦ f )( x ) is completely determinedby the 0-jet of x at φ ( w ). Given φ , · · · , φ p ∈ T (cid:0) C ∞ , S , F , W (cid:1) and pos-itive integers r , · · · , r p , we shall construct in Section 7 holomorphic maps g : C ∞ (cid:0) S , P (cid:1) → F ( W, P m ) for sufficiently large m such that E w ◦ g ( x ) iscompletely determined by r j -jets of x at φ j ( w ), j = 1 , · · · , p , and E w ◦ g ( x )can not be completely determined by 0-jets of x at φ j ( w ), j = 1 , · · · , p .In particular, Theorem 1.7(b) does not hold if n = 1 and m ≥
2. Theseconstructions are based on the results in [24, Section 4] about holomor-phic sections of line bundles over C ∞ (cid:0) S , P (cid:1) . For more information aboutholomorphic sections of line bundles over F (cid:0) S , P (cid:1) see [24, 25, 20].We denote by D ( F , V ) the space of bijections φ : V → V such that φ, φ − ∈ T ( F , V ). Then for any connected complex manifold M , we mayconsider D ( F , V ) as a subgroup of the group Aut (cid:0) F ( V, M ) (cid:1) of holomor-phic automorphisms of F ( V, M ). For more information about D ( F , V )see Subsection 2.2. It may happen that an element of F ( V, P GL ( n + 1 , C ))sends F ( V, P n ) to a different component of F ( V, P n ) (see Section 6). Let F I ( V, P GL ( n + 1 , C )) be the open subgroup of F ( V, P GL ( n + 1 , C )) con-sisting of elements γ with γ (cid:0) F ( V, P n ) (cid:1) = F ( V, P n ). Corollary 1.8.
The group
Aut (cid:0) F ( V, P n ) (cid:1) is the semidirect product F I ( V, P GL ( n + 1 , C )) ⋊ D ( F , V ) . The space D ( C ∞ , V ) can be endowed with a Lie group structure suchthat the action of D ( C ∞ , V ) on C ∞ ( V, P n ) is C ∞ (see [17, Theorem 11.11and Remark 11.5]). Note that there does not exist a complex structureon D (cid:0) C ∞ , S (cid:1) which is compatible with its Lie group structure (see [22,Proposition 3.3.2]).This paper is organized as follows. Section 2 contains basic facts about thetwo classes of maps in the decomposition (1) and about the space F ( V, P n ).In Sections 3 and 4, we give partial answers to Question 2. For a rangedecreasing holomorphic map f : F ( V, M ) ⊃ Ω → F ( W, M ), define(7) W f = { w ∈ W : there is a v ∈ V with f ( x )( w ) = x ( v ) for all x ∈ Ω } . It is clear that W f is a closed subset of W . With the notation in Theorem1.3, it turns out that if f ( y )( w ) ∈ y ( V ), then w is in the interior of W f (Theorem 3.1); and if dim R V ≤ dim C M , W f = ∅ , then W f = W (Lemma4.2). Theorem 1.3 is a consequence of Theorem 3.1 and Lemma 4.2. To ange decreasing group homomorphisms and holomorphic maps 7 show Corollary 1.4, we only need to find a totally real embedding (whendim R V < dim C M ) or a totally real immersion with normal crossings (whendim R V = dim C M ) in Ω (Proposition 4.3).In Section 5, we try to answer Question 1. We prove that any grouphomomorphism in Question 1 is a Lie group homomorphism (Proposition5.3). Let f be as in Theorem 1.1. To show that f = φ ∗ on F ( V, G ), wepass to the Lie algebra homomorphism ˆ f induced by f and consider themaps E w ◦ ˆ f , w ∈ W , so the general case can be reduced to the specialcase when G = R n , dim W = 0 and f is real linear. If a real linear map f = ( ˜ f , · · · , ˜ f n ) : F ( V, R n ) → R n , where n ≥
2, is range decreasing, then˜ f , · · · , ˜ f n must satisfy certain compatibility conditions, which imply that f is a pullback operator (Proposition 5.2).In Section 6, we study the decomposition of holomorphic maps as inQuestion 3. For the map f in Theorem 1.5, we show that its differential d x f at any x ∈ Ω satisfies (2) (where dim W = 0 and G = C n ). By Corollary 1.2, df can be considered as a family of maps holomorphically parameterized by x ∈ Ω, and each of them has the decomposition (3). It turns out that themap φ ∗ in the decomposition is independent of x (Lemma 6.1). Corollary1.6 is a consequence of Theorem 1.5. Some special cases of Corollary 1.6could also be proved by Theorem 1.3 or by Corollary 1.4.In the final Section 7, we focus on holomorphic maps f : F ( V, P n ) →F ( W, P m ). Such a map f can be decomposed as in Question 3 if and onlyif E w ◦ f , w ∈ W , have the same kind of decompositions ((25) and Lemma7.1). Therefore to prove Theorem 1.7, we only need to deal with the specialcase when dim W = 0. By examining how rational curves in F ( V, P n )are transformed by f , we show that f has the properties in Corollary 1.6(Proposition 7.2). Hence we have the decomposition (6). At the end of thispaper we construct holomorphic maps C ∞ (cid:0) S , P (cid:1) → F ( W, P m ) which arenot of the form (6). Still pullback operators play an important role in theseconstructions.We refer to [2, 8] for the fundamentals of infinite dimensional holomorphy.2. Background
Smooth families of holomorphic maps .
Let
Q, N, M be finitedimensional C ∞ manifolds without boundary and β : Q × N → M a C ∞ map. Then β induces a C ∞ map β ∗ = β F ,V ∗ : F ( V, Q ) × F ( V, N ) → F ( V, M ) , where β ∗ ( γ, x )( v ) = β ( γ ( v ) , x ( v )) , v ∈ V (8)(see [12, p. 74], [7, p. 91]). Hence any γ ∈ F ( V, Q ) determines a C ∞ map F ( V, N ) → F ( V, M ), which is still denoted by γ . If N and M arecomplex manifolds and β ( q, · ) is holomorphic for every q ∈ Q , then γ : F ( V, N ) → F ( V, M ) is holomorphic (see [13, Proposition 2.3], where we setΦ to be V × N ∋ ( v, z ) β ( γ ( v ) , z ) ∈ M ; also see [13, p. 42] for the case N. Zhang when F = C k,α /W k,p ). If Q, N and M are all complex manifolds and β isholomorphic, then β ∗ is holomorphic. In particular, a C ∞ resp. holomorphicaction of a Lie group G on a manifold M induces a C ∞ resp. holomorphicaction of F ( V, G ) on F ( V, M ). Setting Q = { q } (a single point), we seethat any C ∞ resp. holomorphic map ˜ β : N → M induces a C ∞ resp.holomorphic map ˜ β ∗ : F ( V, N ) ∋ x ˜ β ◦ x ∈ F ( V, M ). Let exp : g → G bethe exponential map of a finite dimensional Lie group. Thenexp ∗ = exp F ,V ∗ : F ( V, g ) → F ( V, G )is the exponential map of F ( V, G ). If U is an open neighborhood of ∈ g such that exp | U : U → exp( U ) is a diffeomorphism, then F ( V, U ) ∋ x exp ∗ ( x ) ∈ F ( V, exp( U ))is a diffeomorphism.Let φ ∈ T ( F , V, F , W ) and γ ∈ F ( V, Q ). It is straightforward toverify that(9) φ ∗ M ◦ γ = φ ∗ Q ( γ ) ◦ φ ∗ N : F ( V, N ) → F ( W, M ) , where we consider γ on the left hand side as a map F ( V, N ) → F ( V, M ),and consider φ ∗ Q ( γ ) ∈ F ( W, Q ) as a map F ( W, N ) → F ( W, M ). Equa-tion (9) is a generalization of [26, (2)].For integers m ≥ n ≥ d ≥
1, let E = E n,md be the complexvector space of ( m + 1)-tuples of homogeneous polynomials of degree d on C n +1 . Then any h ∈ H n,md can be represented by an element of E suchthat the m + 1 polynomials have no common zeros on C n +1 \ { } , and sucha representation is unique up to an overall multiplicative constant. Thus H n,md can be considered as an open subset of the projective space P ( E ),and the evaluation H n,md × P n ∋ ( h, ζ ) h ( ζ ) ∈ P m is holomorphic. So the induced map F (cid:0) V, H n,md (cid:1) × F ( V, P n ) → F ( V, P m )as in (8) is holomorphic. The space H n,md is connected, and it is simplyconnected if n < m . For more information about the topology of H n,md see[18, 23, 3].Suppose γ : V → H n,md is a map such that γ ( · )( ζ ) : V → P m is an F mapfor any fixed ζ ∈ P n . By induction on n , one can verify that for any ζ ∈ P n and any v ∈ V , there exist a neighborhood U of ζ and a neighborhood V of v on which γ ( v )( ζ ) can be represented (under affine coordinates of P n and of P m ) by m -tuples of rational functions of ζ ∈ U whose coefficientsare F functions of v ∈ V . So γ ∈ F (cid:0) V, H n,md (cid:1) . Let x ∈ F ( V, P n ), and let(10) κ = κ γ,x : x ∗ T P n → γ ( x ) ∗ T P m be the morphism of vector bundles covering the identity map on the basespace V such that the restriction of κ to the fiber over v ∈ V is d x ( v ) γ ( v ). ange decreasing group homomorphisms and holomorphic maps 9 Note that κ is an F morphism of vector bundles (i.e. under local trivial-izations of x ∗ T P n and of γ ( x ) ∗ T P m , κ can be represented by F maps fromopen subsets of V to the space of m × n complex matrices). It is clearthat E v, F ( V, P m ) ◦ γ = γ ( v ) ◦ E v, F ( V, P n ) . So we have E v, F ( γ ( x ) ∗ T P m ) ◦ d x γ = d x ( v ) γ ( v ) ◦ E v, F ( x ∗ T P n ) , where F ( x ∗ T P n ) (resp. F ( γ ( x ) ∗ T P m )) is the tan-gent space of F ( V, P n ) (resp. F ( V, P m )) at x (resp. γ ( x )). Thus(11) d x γ (˜ x ) = κ ◦ ˜ x, ˜ x ∈ F ( x ∗ T P n ) . Basic properties of pullback operators .
Let φ ∈ T ( F , V, F , W ).For sufficiently large n , there is a C ∞ embedding x : V → R n . So φ = x − ◦ ( x ◦ φ ) is an F map. Proposition 2.1.
Suppose M is an m dimensional C ∞ manifold withoutboundary, where m = 1 , , · · · , φ : W → V is a map and Ω ⊂ F ( V, M ) isa nonempty open subset. Then φ ∈ T ( F , V, F , W ) if and only if x ◦ φ ∈F ( W, M ) for every x ∈ Ω . In this case, the map φ ∗ M is C ∞ . Furthermore,if M is a complex manifold, then φ ∗ M is holomorphic.Proof. To show the “only if” direction, let ( U, Φ) be a coordinate chart of M and x ∈ F ( V, U ) ⊂ F ( V, M ). Then x ◦ φ = (cid:0) Φ − (cid:1) F ,W ∗ ◦ φ ∗ R m ◦ Φ F ,V ∗ ( x ) ∈ F ( W, U ) ⊂ F ( W, M ) . With cut-off functions one can show that x ◦ φ ∈ F ( W, M ) for every x ∈F ( V, M ) (see the proof of [26, Proposition 2.1]).Regarding the “if” direction, recall the local chart ( U , ϕ y ) of F ( V, M ) asin [14, Subsection 1.1] (if M is a C ∞ manifold instead of a complex manifold,then similar arguments yield a C ∞ local chart rather than a holomorphicone), which maps an open neighborhood U of y ∈ F ( V, M ) to an openneighborhood of ϕ y ( y ) = ∈ F ( y ∗ T M ). Choose U ⊂ Ω. We write φ ∗ M | U for the map U ∋ x x ◦ φ ∈ F ( W, M ). It is straightforward to verify that ϕ y ◦ φ ◦ φ ∗ M | U ◦ ϕ − y is given by φ ∗ y ∗ T M | ϕ y ( U ) : ϕ y ( U ) ∋ σ σ ◦ φ ∈ F ( φ ∗ y ∗ T M ) . The linear operator φ ∗ y ∗ T M is actually well defined on all of F ( y ∗ T M ). Let˜ x ∈ F ( V, R ) and w ∈ W . Take σ ∈ F ( y ∗ T M ) with σ ◦ φ ( w ) = . Then˜ x ◦ φ = φ ∗ y ∗ T M (˜ xσ ) /φ ∗ y ∗ T M ( σ ) is an F function in a neighborhood of w .Therefore φ ∈ T ( F , V, F , W ).If a sequence { σ n } in F ( y ∗ T M ) converges to , then the sequence { σ n ◦ φ } in C ( φ ∗ y ∗ T M ) converges to . By Closed Graph Theorem, φ ∗ y ∗ T M iscontinuous. Thus φ ∗ M is C ∞ . If M is a complex manifold, then φ ∗ y ∗ T M iscomplex linear. So φ ∗ M is holomorphic. (cid:3) Let Ω be as in Proposition 2.1 and f : Ω → F ( W, M ) a map. Then f = φ ∗ M | Ω for some φ ∈ T ( F , V, F , W ) if and only if for any w ∈ W , thereexists v ( w ) ∈ V with(12) E w, F ( W,M ) ◦ f = E v ( w ) , F ( V,M ) | Ω . Suppose φ : V → V is a bijection. If φ ∈ D ( F , V ), then φ, φ − ∈ F ( V, V ).The converse is also true if F is one of the following regularities: C , C k,α with k + α ≥ W k,p with k = 2 , , · · · , p > dim R V and W k, with dim R V = 1, k = 1 , , · · · (see [1, Lemma 2.3]). If 1 < p < ∞ and dim R V = 1, then D (cid:0) W ,p , V (cid:1) is the space of bi-Lipschitz maps (see [26, p. 710]).A range decreasing C ∞ map is not always a pullback operator. For ex-ample, the map C ∞ ( S , C n ) × C ∞ ( S , S ) ∋ ( x, y ) x ◦ y ∈ C ∞ ( S , C n )is C ∞ (see [7, p. 91]). Let τ : C ∞ ( S , C n ) → C be a non-zero continuouscomplex linear functional and let T ζ ∈ C ∞ ( S , S ) be the translation P ∋ z z + ζ ∈ P , where ζ ∈ C . Then f : C ∞ ( S , C n ) ∋ x x ◦ T τ ( x ) ∈ C ∞ ( S , C n )is a range decreasing C ∞ map. It is not complex linear (the property f ( λx ) = λf ( x ), where λ ∈ C \ { } , implies that x ◦ T λτ ( x ) = x ◦ T τ ( x ) ). Soit is not a pullback operator.2.3. The space F ( V, P n ) . The sheaf of germs of complex valued F = C ∞ /C k,α /W k,p functions over V is fine. So the isomorphism class of an F complex line bundle E → V is completely determined by its first Chernclass c ( E ) ∈ H ( V, Z ). Let ξ be the universal line bundle over P n and x ∈ F ( V, P n ). Then c ( x ∗ ξ ) is constant on any component of F ( V, P n ). Let C = C ( F ( V, P n )) be the set { µ ( P ) } of curves in F ( V, P n ), where µ rangesover all holomorphic embeddings P → F ( V, P n ) such that E v ◦ µ ∈ H ,n , v ∈ V . Proposition 2.2.
Let x , x , x ∈ F ( V, P n ) be three different elements. (a) If c ( x ∗ ξ ) = c ( x ∗ ξ ) = 0 , then there is a curve in C through both x and x if and only if x ( v ) = x ( v ) for all v ∈ V . (b) If n = 1 and x i ( v ) = x j ( v ) for all v ∈ V , i, j = 1 , , , i = j , thenthere is a curve in C through x , x and x .Proof. (a) Let π : C n +1 \ { } → P n be the natural projection. Then c ( x ∗ j ξ ) = 0, j = 1 ,
2, if and only if there exist F maps ˜ x j : V → C n +1 \ { } with π ◦ ˜ x j = x j . The conclusion of (a) follows from similar arguments as inthe proof of [26, Proposition 3.3].(b) Let γ be the element of F ( V, P GL (2 , C )) such that for any v ∈ V , γ ( v ) is the unique element of P GL (2 , C ) which maps 0 , ∞ , ∈ P to x ( v ), x ( v ) and x ( v ) respectively. Consider γ as a holomorphic automorphismof F (cid:0) V, P (cid:1) . Then γ ( P ) (where P ⊂ F (cid:0) V, P (cid:1) ) is a curve in C through x , x and x . (cid:3) If dim R V ≤ n , then the map x → c ( x ∗ ξ ) induces a bijection π ( F ( V, P n )) → H ( V, Z ) ange decreasing group homomorphisms and holomorphic maps 11 (e.g. see [3, Lemma 1.1] and [21, Theorem 13.14]). In particular, F ( V, P n )consists of elements x with c ( x ∗ ξ ) = 0. If dim R V > n , then in generalthe above bijection does not hold. For example, H ( S , Z ) = 0, and itfollows from the relation between pointed and free homotopy classes that π (cid:0) F ( S , P ) (cid:1) ≃ π ( S ) /π ( S ) ≃ Z .3. Total reality and the interior of W f The following is the main result of this section.
Theorem 3.1.
Let f , F ( V, M ) , y , V be as in Theorem 1.3 and W f be asin (7). If f ( y )( w ) ∈ y ( V ) , where w ∈ W , then w ∈ W of (the interior of W f ). Proposition 3.2.
Let M be a positive dimensional topological manifold, V a compact topological manifold, possibly with boundary, Q a Hausdorfftopological space, Q ∋ λ x λ ∈ C ( V, M ) (with the compact-open topology)a continuous map, V ∗ ⊂ V an open subset and V : Q → V ∗ a map suchthat Q ∋ λ x λ ( V ( λ )) ∈ M is continuous. If for any λ ∈ Q , the map x λ is injective on the closure V ∗ of V ∗ , then V is continuous.Proof. Fix a metric on M . For any λ ∈ Q and any open neighborhood V $ V ∗ of V ( λ ), we have δ = dist (cid:0) x λ ( V ( λ )) , x λ ( V ∗ \ V ) (cid:1) > . Since the maps Q ∋ λ x λ ∈ C ( V, M ) and Q ∋ λ x λ ( V ( λ )) ∈ M arecontinuous, there is an open neighborhood Q of λ such thatdist (cid:0) x λ ( V ( λ )) , x λ ( V ∗ \ V ) (cid:1) > δ/ (cid:0) x λ ( V ( λ )) , x λ ( V ( λ )) (cid:1) < δ/ λ ∈ Q . So we have V ( Q ) ⊂ V , and V is continuous at λ . (cid:3) Proposition 3.3.
Let ∆ r = { λ ∈ C : | λ | < r } ( r > ), V ∗ a C manifold,possibly with boundary, ˜ x, ˜ y ∈ C ( V ∗ , C m ) and V : ∆ r → V ∗ a continuousmap such that the function h : ∆ r → C m defined by h ( λ ) = (˜ y + λ ˜ x ) ( V ( λ )) is holomorphic. If ˜ y + λ ˜ x : V ∗ → C m is a C embedding for every λ ∈ ∆ r ,then V is real differentiable on ∆ r . Furthermore, if ˜ y + λ ˜ x is totally real forevery λ ∈ ∆ r , then V is constant.Proof. For any λ ∈ ∆ r , h ′ ( λ ) = lim C ∋ ζ → (cid:18) (˜ y + λ ˜ x ) ( V ( λ + ζ )) − (˜ y + λ ˜ x ) ( V ( λ )) ζ + ˜ x ( V ( λ + ζ )) (cid:19) = lim C ∋ ζ → (cid:18) ρ ( ζ ) − ρ (0) ζ (cid:19) + ˜ x ( V ( λ )) , where ρ ( ζ ) = (˜ y + λ ˜ x ) ( V ( λ + ζ )) is a map from a neighborhood of 0 ∈ C to C m . It is clear that ρ is complex differentiable at ζ = 0. So V ( λ + ζ ) = (˜ y + λ ˜ x ) − ( ρ ( ζ ))is real differentiable at ζ = 0.The image of d ρ is a J -invariant subspace of T ρ (0) (˜ y + λ ˜ x ) ( V ∗ ), where J is the almost complex structure of C m . If ˜ y + λ ˜ x is totally real, then d ρ = , which implies that d λ V = . Thus V is constant. (cid:3) Lemma 3.4.
Let M be an m dimensional complex manifold without bound-ary, where m = 1 , , · · · , y ∈ F ( V, M ) and v ∈ V . Then there exist acoordinate chart ( U, Ψ) of M with y ( v ) ∈ U and a coordinate chart ( U , ϕ y ) of F ( V, M ) with y ∈ U such that (i) ϕ y ( U ) ⊂ F ( y ∗ T M ) , ϕ y ( y ) = , ϕ y ( x )( v ) ∈ T y ( v ) M for all x ∈ U and v ∈ V ; (ii) Ψ( U ) is a convex open subset of C m ; (iii) if both x ( v ) and y ( v ) are contained in U , then (13) d Ψ ( ϕ y ( x )( v )) = (Ψ ( x ( v )) − Ψ ( y ( v )) , Ψ ( y ( v ))) ∈ C m × Ψ( U ) , where we identify the tangent bundle of Ψ( U ) with C m × Ψ( U ) .Proof. Let Ψ be a biholomorphic map from an open neighborhood of y ( v )to an open subset of C m with Ψ ( y ( v )) = . For a sufficiently small ball B r ( ) ⊂ C m , define U = Ψ − ( B r ( )) and U = Ψ − (cid:0) B r/ ( ) (cid:1) . Take anopen covering { U i } i ∈ Λ of M such that every U i is biholomorphic to a convexopen subset of C m , U ∈ { U i } i ∈ Λ and U i ∩ U = ∅ if U i = U . Define thediffeomorphism F as in the proof of [14, Lemma 1.1] with the above opencovering of M . Shrinking U if necessary, we have F ( z, w ) = ( d Ψ) − (Ψ( z ) − Ψ( w ) , Ψ( w )) , z, w ∈ U. Construct the coordinate chart ( U , ϕ y ) of F ( V, M ) as on page 486 of [14]with the above diffeomorphism F . It is clear that we have (i), (ii) and(iii). (cid:3) Proof of Theorem 3.1.
By (4), we may assume that F = C k,α /C ∞ . It isclear that there is a unique v ∈ V with f ( y )( w ) = y ( v ). Let ( U, Ψ) and( U , ϕ y ) be as in Lemma 3.4. Choose a connected open neighborhood V ∗ of v such that V ∗ ⊂ V , V ∗ is a C ∞ submanifold of V , possibly with boundary,and y (cid:0) V ∗ (cid:1) ⊂ U . We claim that there exist a connected open neighborhood U ⊂ Ω ∩ U of y and a connected open neighborhood W ∗ ⊂ W of w suchthat for every x ∈ U , we have(14) ( a ) x | V ∗ is a totally real embedding;( b ) x ( V ∗ ) ⊂ U ;( c ) f ( x )( W ∗ ) ⊂ x ( V ∗ ) . The pullback operator i ∗ : F ( V, M ) → F (cid:0) V ∗ , M (cid:1) induced by the inclusion i : V ∗ → V is holomorphic. Note that i ∗ y = y | V ∗ is a totally real embedding ange decreasing group homomorphisms and holomorphic maps 13 and the subset of totally real embeddings is open in F (cid:0) V ∗ , M (cid:1) . So there isa sufficiently small connected open neighborhood U ⊂ Ω ∩ U of y such that(a) and (b) hold for every x ∈ U . Fix a metric on M . Since f ( y )( w ) = y ( v ) y ( V \ V ∗ ), there is a connected open neighborhood W ∗ ⊂ W of w such that dist (cid:0) f ( y )( W ∗ ) , y ( V \ V ∗ ) (cid:1) >
0. At the price of shrinking U , wecan arrange that ϕ y ( U ) ⊂ F ( y ∗ T M ) is an open ball centered at anddist (cid:0) f ( x )( W ∗ ) , x ( V \ V ∗ ) (cid:1) > x ∈ U . By the range decreasingcondition, we obtain (c).Next we show that f ( x )( w ) = x ( v ) for all x ∈ U , which implies that f ( x )( w ) = x ( v ) for all x ∈ Ω, hence w ∈ W f . Note that any w ∈ W ∗ satisfies the same kind of conditions as w . So we have W ∗ ⊂ W f and w ∈ W of . For any x ∈ U \ { y } , let x λ = ϕ − y ( λϕ y ( x )) ∈ U , λ ∈ ∆ , where ∆ = { λ ∈ C : | λ | < } . Then x = y . By (a) and (c) of (14), there isa unique V : ∆ → V ∗ with(15) f ( x λ ) ( w ) = x λ ( V ( λ )) : ∆ → M (where V (0) = v ). The above map is holomorphic on ∆ and continuouson ∆. By Proposition 3.2, V is continuous on ∆. Let P be the projectionfrom C m × Ψ( U ) to C m . In view of (b) of (14) and (13),˜ y ( v ) = Ψ ( y ( v )) : V ∗ → C m , ˜ x ( v ) = ( P ◦ d Ψ) ( ϕ y ( x )( v )) = Ψ ( x ( v )) − ˜ y ( v ) : V ∗ → C m are well defined F maps, andΨ ( x λ ( v )) − ˜ y ( v ) = ( P ◦ d Ψ) ( ϕ y ( x λ )( v )) = ( P ◦ d Ψ) ( λϕ y ( x )( v ))= λ ( P ◦ d Ψ) ( ϕ y ( x )( v )) = λ ˜ x ( v ) . By (a) of (14), ˜ y + λ ˜ x = Ψ ◦ x λ : V ∗ → C m is a totally real embedding forevery λ ∈ ∆. By (15), the map∆ ∋ λ (˜ y + λ ˜ x ) ( V ( λ )) = Ψ ( f ( x λ ) ( w )) ∈ C m is holomorphic. It follows from Proposition 3.3 that V is constant. So V ≡ v on ∆ and f ( x )( w ) = x ( v ). (cid:3) Range decreasing holomorphic maps
In this section, we provide partial answers to Question 2. In particular,we prove Theorem 1.3 and Corollary 1.4. Let M be a positive dimensional C ∞ resp. complex manifold without boundary, v ∈ V and J k ( v , M ), k = 1 , , · · · , the submanifold of the k -jet space J k ( V, M ) consisting of k -jets at v . It is straightforward to verify that J k ( v , M ) is a C ∞ resp.complex manifold, the target map τ k : J k ( v , M ) ∋ j kv x x ( v ) ∈ M gives rise to a C ∞ resp. holomorphic fiber bundle and the map j kv : C k,α ( V, M ) resp. C ∞ ( V, M ) ∋ x j kv x ∈ J k ( v , M )is C ∞ resp. holomorphic. Proposition 4.1.
Let M be a C ∞ manifold without boundary with dim R M = p = 2 , , · · · , D ⊂ C k,α ( V, M ) resp. C ∞ ( V, M ) a nonempty open subset,where k = 1 , , · · · , and v ∈ V . If dim R V = l < p , then there exist y ∈ D and an open neighborhood V of v such that y | V is an embeddingand y ( V \ V ) ∩ y ( V ) = ∅ .Proof. Let X l ⊂ J ( v , M ) be the subset of 1-jets j v x such that the rankof d v x is l . For any y ∈ C k,α ( V, M ) resp. C ∞ ( V, M ), it follows fromLemma 3.4 that there is a coordinate chart ( U , ϕ y ) of C k,α ( V, M ) resp. C ∞ ( V, M ) with y ∈ U such that ϕ y ( U \ (cid:0) j v (cid:1) − ( X l )) is a real analyticsubset of ϕ y ( U ). So (cid:0) j v (cid:1) − ( X l ) is an open and dense subset of C k,α ( V, M )resp. C ∞ ( V, M ).Take y ∈ D ∩ (cid:0) j v (cid:1) − ( X l ), a C ∞ coordinate chart ( U, Φ) of M and anopen connected neighborhood V ∗ of v such that V ∗ is a C ∞ submanifoldof V , possibly with boundary, y | V ∗ is an embedding and y (cid:0) V ∗ (cid:1) ⊂ U . Let ξ ∈ S p − and χ ∈ C ∞ ( V ) be such that χ ≡ v and supp χ ⊂ V ∗ . For a sufficiently small ε > O ⊂ S p − of ξ , define y ξ,t ∈ D by y ξ,t ( v ) = (cid:26) Φ − (cid:0) Φ ( y ( v )) + tχ ( v ) ξ (cid:1) , v ∈ V ∗ ,y ( v ) , v ∈ V \ V ∗ , where ξ ∈ O and t ∈ ( − ε, ε ) ⊂ R . Shrinking ( − ε, ε ) and O if necessary,we may assume that y ξ,t | V ∗ is an embedding for every t ∈ ( − ε, ε ) and every ξ ∈ O . Note that the interior of the set (cid:8) Φ (cid:0) y ξ,t ( v ) (cid:1) ∈ R p : t ∈ ( − ε, ε ) , ξ ∈ O (cid:9) is non-empty and Φ ( U ∩ y ( V \ V ∗ )) ⊂ R p is nowhere dense. Thus thereexists y = y ξ ,t with y ( v ) y ( V \ V ∗ ). Take an open neighborhood V ⊂ V ∗ of v such that y ( V ) ∩ y ( V \ V ∗ ) = ∅ . Then y | V is an embedding and y ( V ) ∩ y ( V \ V ) = ∅ . (cid:3) Lemma 4.2.
Let f be as in Theorem 1.3. Assume that we have the embed-ding (4) (with a dense image). If dim R V = l ≤ m = dim C M and W f = ∅ ,then W f = W .Proof. Recall that W f is a closed subset of the connected manifold W . Weshall prove that any w ∈ W f is an interior point. Suppose that f ( x )( w ) = x ( v ) for all x ∈ Ω. By Theorem 3.1, we only need to show that there exista connected open neighborhood V of v and a C k,α /C ∞ element y ∈ Ω suchthat y | V is a totally real embedding and y ( V \ V ) ∩ y ( V ) = ∅ .Let X ⊂ J ( v , M ) be the subset of 1-jets j v x such that d v x is injectiveand totally real. Then J ( v , M ) \ X is a real analytic subset (a real linear ange decreasing group homomorphisms and holomorphic maps 15 map R l → C m ≃ R m represented by a 2 m × l matrix B is injective andtotally real if and only if the rank of the 2 m × l matrix ( B, J B ) is 2 l , where J is the standard almost complex structure on R m ). Therefore the subset (cid:0) j v (cid:1) − ( X ) ⊂ C k,α ( V, M ) resp. C ∞ ( V, M )is open and dense. By Proposition 4.1, there exist y ∈ (cid:0) j v (cid:1) − ( X ) ∩ Ωand an open neighborhood V ′ of v such that y | V ′ is an embedding and y ( V \ V ′ ) ∩ y ( V ′ ) = ∅ . Choose a connected open neighborhood V ⊂ V ′ of v such that y | V is totally real, then y ( V \ V ) ∩ y ( V ) = ∅ . (cid:3) Theorem 1.3 immediately follows from Theorem 3.1, Lemma 4.2 andProposition 2.1.
Proposition 4.3.
Let f be as in Theorem 1.3. Assume that we have theembedding (4) (with a dense image). If dim R V = dim C M and there existsa totally real C k,α /C ∞ immersion y ∈ Ω with normal crossings (i.e. theonly self-intersections of y are transversal double points), then f = φ ∗ | Ω forsome φ ∈ T ( F , V, F , W ) .Proof. We only need to consider the special case when F = C k,α /C ∞ .Assume for contradiction that f is not a pullback operator. It follows fromTheorem 1.3 that for any totally real immersion x ∈ Ω and any w ∈ W ,( E w ◦ f )( x ) is a self-intersection point of x . In particular, f ( y ) ∈ M ⊂F ( W, M ) is one of the double points of y . Suppose that f ( y ) = y ( v ) = y ( v ) , v , v ∈ V, v = v . Let V ⊂ V be an open connected neighborhood of v such that V is a C ∞ submanifold of V , possibly with boundary, and y | V is an embedding,˜ x ∈ C k,α ( y ∗ T M ) resp. C ∞ ( y ∗ T M ) with supp˜ x ⊂ V and(16) ˜ x ( v ) ∈ T y ( v ) M \ d v y ( T v V )(where we consider ˜ x ( v ) as an element of T y ( v ) M ) and let ( U , ϕ y ) be acoordinate chart of C k,α ( V, M ) resp. C ∞ ( V, M ) as in Lemma 3.4 with U ⊂ Ω. Choose δ > λ ∈ ∆ δ = { z ∈ C : | z | < δ } , we have λ ˜ x ∈ ϕ y ( U ), x λ = ϕ − y ( λ ˜ x ) is a totally real immersion and x λ | V is an embedding.Note that ( E w ◦ f )( x λ ) is a self-intersection point of x λ . Thus it is containedin the totally real immersed submanifold x λ ( V \ V ) = y ( V \ V ) ⊂ M , whichimplies that the holomorphic map ∆ δ ∋ λ ( E w ◦ f )( x λ ) ∈ M is constant.So f ( x λ ) = y ( v ) for every λ ∈ ∆ δ .On the other hand, it follows from (16) and (13) that the rank of the map V × ( − δ, δ ) ∋ ( v, λ ) x λ ( v ) ∈ M at the point ( v ,
0) is dim R V +1. So there exist a positive δ ≤ δ and an openneighborhood V ∗ ⊂ V of v with y ( v ) x λ ( V ∗ ) for every λ ∈ (0 , δ ). As y ( v ) y ( V \ V ∗ ), there is a positive δ ≤ δ such that y ( v ) x λ ( V \ V ∗ )for every λ ∈ (0 , δ ). Therefore y ( v ) x λ ( V ). Note that x λ ( v ) = y ( v ), v ∈ V \ V . So y ( v ) is not a self-intersection point of x λ , and f ( x λ ) = y ( v )for every λ ∈ (0 , δ ). We have a contradiction. (cid:3) Proof of Corollary 1.4. (a) It is an immediate consequence of Theorem 1.3.(b) The subset of C k,α ( V, M ) resp. C ∞ ( V, M ) consisting of totally realimmersions is open. When dim R V < dim C M , the subset of embeddingsis dense in C k +sgn( α ) ( V, M ) resp. C ∞ ( V, M ). So there is a totally real C k +sgn( α ) /C ∞ embedding in Ω. When dim R V = dim C M , the conclusion ofthe corollary follows from Proposition 4.3 and the fact that the subset of im-mersions with normal crossings is dense in C k +sgn( α ) ( V, M ) resp. C ∞ ( V, M )(see the proof of [6, Proposition 3.2], where we replace the multijet transver-sality theorem by [16, Theorem 11.2.2], so the arguments still hold if V is a C ∞ manifold with or without boundary).(c) If j = 2 , , · · · and dim C M ≥ (cid:20)
32 dim R V (cid:21) , then the subset of totallyreal immersions is dense in C j ( V, M ) (for the idea of a proof see [11, Ap-pendix], where we replace the jet transversality theorem by [16, Theorem11.1.5], so similar arguments can be applied to the general case discussedhere). The conclusion of the corollary follows from (b). (cid:3)
A range decreasing holomorphic map is not always a pullback operator.Let Ω j be the component of C ∞ ( S , P ) consisting of elements with topolog-ical degree j = 0. Note that x ( S ) = P for all x ∈ Ω j . Thus every holomor-phic map Ω j → C ∞ ( S , P ) is range decreasing. Any γ ∈ P GL (2 , C ) \ { } can be considered as a holomorphic automorphism of C ∞ ( S , P ). It isstraightforward to verify that γ | Ω j is not (the restriction of) a pullback op-erator. 5. Range decreasing group homomorphisms
In this section, we give (partial) answers to Question 1. In particular,we prove Theorem 1.1 and Corollary 1.2. Consider F ( V, R n ) as an addi-tive group. Let G be an additive group and ϕ : F ( V, R n ) → G a grouphomomorphism. We write S ϕ for the subset of V consisting of points v with the following property: for any open neighborhood V of v , there exists x ∈ F ( V, R n ) such that supp x ⊂ V and ϕ ( x ) = 0 ∈ G . Proposition 5.1. If ϕ , then S ϕ = ∅ .Proof. Suppose that S ϕ = ∅ . Then for any v ∈ V , there exists an openneighborhood V v of v such that ϕ ( x ) = 0 for any x ∈ F ( V, R n ) with supp x ⊂ V v . Choose a finite subcover { V v j } of the open cover { V v : v ∈ V } of V , and take a C ∞ partition of unity { χ j } subordinate to { V v j } . Then ϕ ( x ) = P j ϕ ( χ j x ) = 0 for all x ∈ F ( V, R n ), this gives a contradiction. (cid:3) Proposition 5.2.
Let f : F ( V, R n ) → F ( W, R n ) , where n ≥ , be a reallinear range decreasing map. Then there exists φ ∈ T ( F , V, F , W ) suchthat f = φ ∗ . ange decreasing group homomorphisms and holomorphic maps 17 Proof.
For any w ∈ W , E w ◦ f : F ( V, R n ) → R n is a real linear rangedecreasing map. It is enough to assume that dim W = 0 (see (12)).Let e , · · · , e n be the standard basis of R n ⊂ F ( V, R n ). For any x ∈F ( V, R n ), we write x = P nj =1 x j e j , where x j ∈ F ( V, R ). In view of therange decreasing condition, we have f ( x ) = n X j =1 f j ( x j ) e j , where f j : F ( V, R ) → R is real linear and range decreasing, j = 1 , · · · , n .Note that f j | R = id. Take ¯ v ∈ S f (see Proposition 5.1). Next we show that(17) S f = · · · = S f n = { ¯ v } . Let v j ∈ S f j , where j = 1. Assume that v j = ¯ v . Then we can find anopen neighborhood V of ¯ v , an open neighborhood V j of v j with V ∩ V j = ∅ and x , x j ∈ F ( V, R ) such that supp x ⊂ V , supp x j ⊂ V j , f ( x ) = 0and f j ( x j ) = 0. By the range decreasing condition, for any t ∈ (0 , v t ∈ V such that f ( tx e + (1 − t ) x j e j ) = tf ( x ) e + (1 − t ) f j ( x j ) e j (18) = tx ( v t ) e + (1 − t ) x j ( v t ) e j . (19)Since supp x ∩ supp x j = ∅ , the value of (19) is either in { R e } or in { R e j } .On the other hand, the value of (18) is not contained in { R e } ∪ { R e j } . Wehave a contradiction. Therefore v j = ¯ v and S f j = { ¯ v } . Similarly S f = { ¯ v } .For any x ∈ F ( V, R ) with x (¯ v ) = 0, we claim that(20) f j ( x ) = 0 , j = 1 , · · · , n. Otherwise f j ( x ) = 0 for some 1 ≤ j ≤ n . Take an open neighborhood V of ¯ v such that x ( v ) = 0 for any v ∈ V . If j = j , by (17), we can find x ∈ F ( V, R ) with supp x ⊂ V and f j ( x ) = 0. Note that f ( x e j + x e j ) = f j ( x ) e j ∈ { R e j } \ { } . As f is range decreasing, there exists v ∈ V such that f ( x e j + x e j ) isequal to ( x e j + x e j )( v ) = (cid:26) x ( v ) e j , v V ,x ( v ) e j + x ( v ) e j , v ∈ V , which is not in { R e j } \ { } . This gives a contradiction.Note that f | R n = E ¯ v | R n = id. So both ker f and ker E ¯ v are of codimension n . It follows from (20) that ker f ⊂ ker E ¯ v . Thus ker f = ker E ¯ v and f = E ¯ v . (cid:3) Proposition 5.3.
Any range decreasing group homomorphism f : F ( V, G ) → F ( W, G ) resp. f : F ( V, G ) → F ( W, G ) , where dim R G = 1 , , · · · , is aLie group homomorphism. Proof.
The inclusion I : F ( W, G ) → C ( W, G ) is an injective Lie group ho-momorphism. It follows from the range decreasing condition that the grouphomomorphism I ◦ f is continuous at 1 ∈ F ( V, G ). So it is C ∞ (see [19,Theorem IV.1.18]). Let exp : g → G be the exponential map of G , B a ballcentered at ∈ g such that exp | B : B → exp( B ) is a diffeomorphism and ˆ I the Lie algebra homomorphism induced by I . Recall that the exponentialmap of F ( V, G ) is simply exp F ,V ∗ (see Subsection 2.1). Defineˆ f (ˆ x ) = (cid:0) exp F ,W ∗ (cid:1) − ◦ f ◦ exp F ,V ∗ (ˆ x ) ∈ F ( W, B ) , ˆ x ∈ F ( V, B ) . Thenˆ I ◦ ˆ f (ˆ x ) = (cid:0) exp C,W ∗ (cid:1) − ◦ I ◦ f ◦ exp F ,V ∗ (ˆ x ) ∈ C ( W, B ) , ˆ x ∈ F ( V, B ) . Thus ˆ I ◦ ˆ f is (the restriction of) the Lie algebra homomorphism inducedby I ◦ f . Now ˆ f = ( ˆ I ) − ◦ ( ˆ I ◦ ˆ f ) is real linear. Hence it is well definedon all of F ( V, g ). Since f is range decreasing, we have(21) ˆ f (ˆ x )( W ) ⊂ ˆ x ( V ) , ˆ x ∈ F ( V, g ) . It follows from Closed Graph Theorem that ˆ f is continuous. So f is C ∞ . (cid:3) Proof of Theorem 1.1. (i) By (21), Propositions 5.2 and 5.3, f is a Liegroup homomorphism which induces the same Lie algebra homomorphismas a pullback operator φ ∗ = φ ∗G . So f = φ ∗ on F ( V, G ).(ii) By (i), the closed subset { x ∈ F ( V, G ) : f ( x ) = φ ∗ ( x ) } is also open.So f ( x ) = φ ∗ ( x ) implies that f = φ ∗ on the component of F ( V, G )containing x .Let x ∈ F ( V, G ) be such that x ( V ) is nowhere dense in G . We claimthat f ( x ) = φ ∗ ( x ). Otherwise there exists w ∈ W with f ( x )( w ) = x ( φ ( w )). Note that f ( x )( w ), x ( φ ( w )) ∈ x ( V ), hence they are in thesame component of G . Fix a left invariant metric on G and set δ = dist ( f ( x )( w ) , x ( φ ( w ))) > . Choose an open neighborhood ˜ U of x ( φ ( w )) ∈ G with diameter < δ/ U of 1 ∈ G and an openneighborhood V of φ ( w ) ∈ V such thatdist ( f ( x )( w ) a, x ( φ ( w ))) > δ/ , a ∈ U, and x ( v ) a ∈ ˜ U , v ∈ V , a ∈ U. The triangle inequality gives us(22) dist ( f ( x )( w ) x ( φ ( w )) , x ( v ) x ( v )) > δ/ , v ∈ V , x ∈ F ( V, U ) . Now take x ∈ F ( V, U ) such that x ( v ) = 1 for any v ∈ V \ V . It followsfrom (i), (22) and the range decreasing condition of f that(23) f ( x )( w ) x ( φ ( w )) = f ( x x )( w ) ∈ x x ( V \ V ) = x ( V \ V ) . ange decreasing group homomorphisms and holomorphic maps 19 With different choices of x , the left hand side of (23) cannot always becontained in the nowhere dense subset x ( V \ V ), we have a contradiction.So f ( x ) = φ ∗ ( x ).If dim R V < dim R G , then the image of any C map V → G is nowheredense. As any component of F ( V, G ) contains a C element, we have f = φ ∗ on F ( V, G ). (cid:3) Theorem 1.1 and Proposition 5.2 do not hold if dim R G = 1. Let v , v ∈ V be two different points and g : W → (0 ,
1) an F map. Define a real linearmap f : F ( V, R ) → F ( W, R ) by f ( x )( w ) = g ( w ) x ( v ) + (1 − g ( w )) x ( v ) , w ∈ W, x ∈ F ( V, R ) . It is clear that f is range decreasing. We claim that f is not a pullbackoperator. Assume that f = φ ∗ for some φ : W → V . Fix a point w ∈ W .Choose x ∈ F ( V, R ) with x ( v ) = 0 and x ( v ) = 0. Then f ( x )( w ) = x ( v ), hence φ ( w ) = v . Similarly φ ( w ) = v . Take x ∈ F ( V, R ) with x ( v ) = x ( v ) = 0 and x ( φ ( w )) = 0. Then f ( x )( w ) = φ ∗ ( x )( w ),which is a contradiction. Proof of Corollary 1.2.
One direction being trivial, we shall only verify thenecessity part of the claim.(i) First we consider the special case when dim W = 0 (i.e. F ( W, G ) = G ). It follows from (2) that f | G : G → G is injective, which implies that f | G ∈ Aut ( G ). Let h = ( f | G ) − ◦ f . Then h | G = id and h still satisfies (2).For any x ∈ F ( V, G ), we have h (cid:0) x ( h ( x )) − (cid:1) = 1, where ( h ( x )) − ∈ G ⊂F ( V, G ). By (2), 1 ∈ x ( h ( x )) − ( V ). Hence h ( x ) ∈ x ( V ) (i.e. h is rangedecreasing). By Theorem 1.1, h = E v for some v ∈ V . So f = f | G ◦ E v .(ii) For the general case, it follows from (i) that there exists a map φ : W → V such that E w ◦ f = ( E w ◦ f ) | G ◦ E φ ( w ) . Define γ f : W ∋ w ( E w ◦ f ) | G ∈ Aut ( G ) . For any a ∈ G , the map W ∋ w γ f ( w )( a ) ∈ G is just f ( a ) ∈ F ( W, G ).Thus W ∋ w d γ f ( w ) ∈ Aut ( g ) is an F map, where Aut ( g ) is the auto-morphism group of the Lie algebra g of G . Recall that the map Aut ( G ) ∋ γ d γ ∈ Aut ( g ) is an injective Lie group homomorphism onto a closed sub-group of Aut ( g ) (e.g. see [9, Subsection 11.3.1]). So γ f ∈ F ( W, Aut ( G )).Now φ ∗ = ( γ f ) − ◦ f is a well defined map from F ( V, G ) to F ( W, G ). ByProposition 2.1, we have φ ∈ T ( F , V, F , W ). (cid:3) Decomposition of holomorphic maps
In this section, we answer Question 3 partially. In particular, we proveTheorem 1.5 and Corollary 1.6.
Lemma 6.1.
Let M be a connected complex locally convex manifold, G apositive dimensional connected complex Lie group and g : M × F ( V, G ) → F ( W, G ) a holomorphic map such that for any ˆ x ∈ M , g (ˆ x, · ) is a group ho-momorphism with g (cid:0) ˆ x, F ( V, G \ { } ) (cid:1) ⊂ F ( W, G \ { } ) . Then there exist φ ∈ T ( F , V, F , W ) independent of ˆ x and γ g (ˆ x, · ) ∈ F ( W, Aut ( G )) with g (ˆ x, · ) = γ g (ˆ x, · ) ◦ φ ∗G , ˆ x ∈ M (cf. (3)).Proof. The group
Aut ( G ) of complex Lie group automorphisms of G is a finitedimensional complex Lie group (e.g. see [9, Section 15.4]). For any fixed w ∈ W , the map M ∋ ˆ x E w ◦ g (ˆ x, · ) | G ∈ Aut ( G ) is holomorphic. It followsfrom Corollary 1.2 that for any ˆ x ∈ M , there exists φ ˆ x ∈ T ( F , V, F , W )such that( E w ◦ g (ˆ x, · ) | G ) − ◦ ( E w ◦ g (ˆ x, · )) = E φ ˆ x ( w ) : F ( V, G ) → G . Therefore for any ˜ x ∈ F ( V, G ), the map M ∋ ˆ x E φ ˆ x ( w ) (˜ x ) ∈ G isholomorphic. Let ρ : R → G be a C ∞ embedding and µ ∈ F ( V, R ). Then ρ ◦ µ ∈ F ( V, G ), and M ∋ ˆ x E φ ˆ x ( w ) ( ρ ◦ µ ) ∈ G is a holomorphic mapwhose image is contained in ρ ( R ). Thus E φ ˆ x ( w ) ( ρ ◦ µ ) is independent of ˆ x for every µ ∈ F ( V, R ), which implies that φ ˆ x ( w ) is independent of ˆ x . (cid:3) Proof of Theorem 1.5.
It is enough to show that for any y ∈ Ω, there exista connected open neighborhood U y ⊂ Ω of y and v = v ( y ) ∈ V independentof x ∈ U y such that ker d x f = ker E v, F ( V, C n ) for every x ∈ U y . As Ω isconnected, v is also independent of y .Let Φ : U → C n be a local chart of M with f ( y ) ∈ U and Φ ( f ( y )) = , ϕ y : U y → F ( V, C n ) a local chart of F ( V, N ) as in [14, Subsection 1.1]such that U y is connected, y ∈ U y , ϕ y ( y ) = and f ( U y ) ⊂ U , h = Φ ◦ f ◦ ϕ − y : ϕ y ( U y ) → C n and ˆ x , ˆ x ∈ ϕ y ( U y ) with ˆ x ( v ) = ˆ x ( v ), v ∈ V . By [14, Lemma 1.1],(24) ϕ − y (ˆ x )( v ) = ϕ − y (ˆ x )( v ) , v ∈ V. It follows from (5) that h (ˆ x ) = h (ˆ x ). In particular, h | ϕ y ( U y ) ∩ C n is injective,so it is an embedding.Next we show that d h ( F ( V, C n \ { } )) ⊂ C n \ { } . Let λ ∈ C \ { } ,˜ x ∈ F ( V, C n \ { } ), A = d (cid:0) h | ϕ y ( U y ) ∩ C n (cid:1) ∈ GL ( C n ), m (˜ x ) = min v ∈ V | ˜ x ( v ) | > , | A | = min ζ ∈ C n , | ζ | =1 | A ( ζ ) | > B ⊂ ϕ y ( U y ) ∩ C n a small ball centered at such that | h ( ζ ) | > | A || ζ | / , ζ ∈ B \ { } . Note that h ( λ ˜ x ) ∈ h ( B ) when | λ | is small enough. In this case, there is aunique ζ ( λ ) ∈ B with h ( λ ˜ x ) = h ( ζ ( λ )). Since h has the property as in (5),we must have ζ ( λ ) ∈ λ ˜ x ( V ). Therefore | h ( λ ˜ x ) | > | A || ζ ( λ ) | / ≥ | A || λ | m (˜ x ) / , ange decreasing group homomorphisms and holomorphic maps 21 which implies that d h (˜ x ) ∈ C n \ { } . Similarly d ˆ x h ( F ( V, C n \ { } )) ⊂ C n \ { } for every ˆ x ∈ ϕ y ( U y ). Applying Lemma 6.1 (where we set dim W =0) to the holomorphic map ϕ y ( U y ) × F ( V, C n ) ∋ (ˆ x, ˜ x ) d ˆ x h (˜ x ) ∈ C n , we obtain that ker d ˆ x h = ker E v, F ( V, C n ) = ker d x f , where x = ϕ − y (ˆ x ) ∈ U y and v ∈ V is independent of ˆ x ∈ ϕ y ( U y ). (cid:3) Let x ∈ F (cid:0) V, P (cid:1) be such that the bundle x ∗ T P is trivial. By (24), wecan find x , x in a neighborhood of x with x i ( v ) = x j ( v ) for all v ∈ V , i, j =1 , , i = j . It follows from the proof of Proposition 2.2(b) that there existsan element of F ( V, P GL (2 , C )) which maps 0 ∈ P to x . In particular, theaction of F ( V, P GL (2 , C )) on F (cid:0) V, P (cid:1) is transitive if H ( V, Z ) = 0. Recallthat F ( S , P ) has infinitely many components. Therefore F ( S , P ) is notan invariant subset of F ( S , P ) under the action of F (cid:0) S , P GL (2 , C ) (cid:1) . Proof of Corollary 1.6.
Suppose that f = f | N ◦ E v | O with N n,f | N = ∅ .Take an open subset U ⊂ N such that f | U is an embedding and let Ω = E − v ( U ) ∩ O . Then f (Ω) = f ( U ) and (5) holds on Ω. For the other direction,it follows from Theorem 1.5 that f = f | N ◦ E v on an open subset of Ω, thusalso on the connected manifold O . (cid:3) Some special cases of the sufficiency part of Corollary 1.6 could also beproved by Theorem 1.3 or by Corollary 1.4 (instead of Theorem 1.5): By (5), f | Ω ∩ N is injective. Take a component U of Ω ∩ N . Then f | U : U → f ( U )is biholomorphic. Let Ω be the component of Ω ∩ f − ( f ( U )) containing U . Then f (Ω ) = f ( U ). Define˜ f = ( f | U ) − ◦ f | Ω : Ω → U ⊂ N. We only need to show that ˜ f = E v | Ω for some v ∈ V . It follows from(5) that for any x ∈ Ω and any ζ ∈ U \ x ( V ) (if U \ x ( V ) = ∅ ), we have˜ f ( x ) = ˜ f ( ζ ) = ζ . So ˜ f ( x ) ∈ x ( V ) , x ∈ Ω (i.e. ˜ f is range decreasing with dim W = 0). If ˜ f satisfies the conditions ofTheorem 1.3 or of Corollary 1.4, then we obtain ˜ f = E v | Ω .7. Holomorphic maps F ( V, P n ) → F ( W, P m )In this final section, we study the decomposition of holomorphic maps F ( V, P n ) → F ( W, P m ) as in Question 3. Let O ⊂ F ( V, P n ) be a con-nected open neighborhood of P n and f : O → F ( W, P m ) a holomorphicmap with deg f = d ≥ m ≥ n ). Then f induces a map γ f : W ∋ w E w ◦ f | P n ∈ H n,md . For any ζ ∈ P n , the map W ∋ w γ f ( w )( ζ ) ∈ P m is exactly f ( ζ ) ∈F ( W, P m ). So(25) γ f ∈ F (cid:0) W, H n,md (cid:1) , and γ f can be considered as a holomorphic map F ( W, P n ) → F ( W, P m )with γ f | P n = f | P n (see Subsection 2.1). Lemma 7.1.
Let f, γ f be as above and let φ : W → V be a map. If (26) E w, F ( W, P m ) ◦ f = γ f ( w ) ◦ E φ ( w ) , F ( V, P n ) | O , w ∈ W, then φ ∈ T ( F , V, F , W ) .Proof. It is enough to show that for any w ∈ W , there is an open neigh-borhood W of w such that its closure W is a contractible C ∞ subman-ifold of W , possibly with boundary, and φ | W ∈ T (cid:0) F , V, F , W (cid:1) . Let I : W → W be the inclusion and I ∗ : F ( W, P m ) → F (cid:0) W , P m (cid:1) thepullback operator induced by I . Replacing f by I ∗ ◦ f , we may assumethat W = W . Then F ( W, P m ) is connected. We identify the tangentspace of F ( W, P m ) (resp. F ( V, P n )) at any point with F ( W, C m ) (resp. F ( V, C n )).First we consider the case when m = n . Shrinking W = W if necessary,we can find ζ ∈ P n ⊂ F ( W, P n ) such that the tangent map d ζ γ f ( w ) is ofrank n for any w ∈ W . Then the map κ γ f ,ζ : ζ ∗ T P n → γ f ( ζ ) ∗ T P n as in(10) is an F isomorphism of trivial vector bundles. By (11), the tangent map d ζ γ f is an automorphism of the complex Banach/Fr´echet space F ( W, C n ).Note that γ f ( ζ ) = f ( ζ ). It follows from (26) that E φ ( w ) , F ( V, C n ) = ( d ζ γ f ( w )) − ◦ E w, F ( W, C n ) ◦ d ζ f, w ∈ W. So the pullback by φ from F ( V, C n ) to the space of maps W → C n is exactly( d ζ γ f ) − ◦ d ζ f : F ( V, C n ) → F ( W, C n ). Thus φ ∈ T ( F , V, F , W ).When m > n , take a hyperplane P m − ⊂ P m and a point p ∈ P m \ P m − such that γ f ( w )( P n ) ⊂ P m \ { p } for any w ∈ W (which can be done forsufficiently small W = W ). In view of (26), we have f ( O ) ⊂ F ( W, P m \{ p } ) . Let π : P m \ { p } → P m − be the projection from p to P m − , π ∗ : F ( W, P m \ { p } ) ∋ y π ◦ y ∈ F ( W, P m − )the induced holomorphic map and f = π ∗ ◦ f . By (26), we have E w, F ( W, P m − ) ◦ f = γ f ( w ) ◦ E φ ( w ) , F ( V, P n ) | O , w ∈ W ;and the conclusion of the lemma follows from an induction on m . (cid:3) Consider P n as C n ∪ P n − , where we identify C n with { [ Z , Z , · · · , Z n ] ∈ P n : Z = 0 } . Let x ∈ F ( V, C n ) and y = ( y , · · · , y n ) ∈ F ( V, C n \ { } ) . Recall the class of rational curves C in Subsection 2.3. Define C x, y = { x + λy ∈ F ( V, P n ) : λ ∈ P } ∈ C , where λ = ∞ ∈ P corresponds to the point(27) P ( y ) = [0 , y , · · · , y n ] ∈ F ( V, P n − ) ∩ F ( V, P n ) . If n = 1, then P ( y ) = ∞ ∈ P ⊂ F (cid:0) V, P (cid:1) . ange decreasing group homomorphisms and holomorphic maps 23 Proposition 7.2.
Let f be as in Theorem 1.7 and assume that dim W = 0 (i.e. F ( W, P m ) = P m ). Then there exist ζ ∈ C n and an open neighborhood Ω ⊂ F ( V, C n ) of ζ such that f (Ω) ⊂ f ( P n ) and (5) holds on Ω .Proof. If d = 1, then f ( P n ) is an n -plane in P m . For any curve C ∈ C in F ( V, P n ), f | C is injective and f ( C ) is a projective line in P m . Next weshow that(28) f ( F ( V, C n )) ⊂ f ( P n )by induction on n . Suppose n = 1. For any x ∈ F ( V, C ), take ζ , ζ ∈ P \ x ( V ) with ζ = ζ . By Proposition 2.2(b), there is a curve C ∈ C through ζ , ζ and x . Then f ( C ) is the projective line through f ( ζ ) and f ( ζ ). So f ( x ) ∈ f ( P ). Assume that (28) holds for n = k −
1, where k ≥ m ≥ n . When n = k , consider P k as C k ∪ C k − ∪ P k − . Let O ⊂F ( V, C k \ { } ) be the open subset consisting of y with P ( y ) ∈ F ( V, C k − )(thus f ( P ( y )) ∈ f ( P k − )). For any y ∈ O , f ( C , y ) is the projective linethrough f (0) and f ( P ( y )). Thus f ( O ) ⊂ f ( P k ). Note that O ∩ ( C k \ { } )is dense in C k \ { } . For any x ∈ F (cid:0) V, C k (cid:1) , take ζ ∈ O ∩ ( C k \ { } ) \ x ( V ).By Proposition 2.2(a), there is a curve C ∈ C through x and ζ . Theopenness of O implies that C contains infinitely many points of O . Hence f ( C ) ⊂ f ( P k ) and (28) holds for n = k . Let Ω = F ( V, C n ) and x , x ∈ Ωwith x ( v ) = x ( v ) for all v ∈ V . Then there is a curve C ∈ C through x and x . So f ( x ) = f ( x ).For the map f in Theorem 1.7(b), let ζ , ζ ∈ C be regular points of f | P with ζ = ζ and f ( ζ ) = f ( ζ ), D ⊂ P an open neighborhood of f ( ζ ) and U j ⊂ C an open neighborhood of ζ j , j = 1 ,
2, such that U ∩ U = ∅ and f maps U j biholomorphically onto D . Take a connected open neighborhoodΩ ⊂ F ( V, U ) of ζ with f (Ω) ⊂ D . Let x , x ∈ Ω with x ( v ) = x ( v )for every v ∈ V . Then we can find ζ = ζ ( x ) ∈ U with f ( ζ ) = f ( x ).By Proposition 2.2(b), there is a curve C ∈ C through ζ , x and x . As f | C : C → P is of topological degree two, we have f ( x ) = f ( x ). (cid:3) Proof of Theorem 1.7.
By Proposition 7.2 and Corollary 1.6, there is a map φ : W → V such that E w ◦ f = ( E w ◦ f ) | P n ◦ E φ ( w ) : F ( V, P n ) → P m , w ∈ W. The conclusion of the theorem follows from (25) and Lemma 7.1. (cid:3)
Corollary 1.8 follows from Theorem 1.7 and (9).Next we construct holomorphic maps C ∞ (cid:0) S , P (cid:1) → F ( W, P m ) whichare not of the form (6). The constructions are closely related to the C ∞ maps J k : S × C ∞ (cid:0) S , P (cid:1) ∋ ( t, x ) j kt x ∈ J k ( S , P ) , the target maps τ k : J k ( S , P ) → P of the jet spaces, k = 0 , , · · · , and theevaluation E = τ k ◦ J k : S × C ∞ (cid:0) S , P (cid:1) ∋ ( t, x ) x ( t ) ∈ P . Recall that for any fixed t ∈ S , the map j kt = J k ( t , · ) : C ∞ (cid:0) S , P (cid:1) → J k ( t , P ) and the restriction of τ k to J k ( t , P ) are holomorphic (see Section4). If h : N → M is a holomorphic map between complex manifolds and L is a holomorphic line bundle over M , then we write H ( M, L ) for the spaceof holomorphic sections of L , and denote by h ∗ H ( M, L ) the subspace of H ( N, h ∗ L ) consisting of pullback sections. For any positive integer n , thepullback of the line bundle O ( n ) → P by E t = τ k ◦ j kt : C ∞ (cid:0) S , P (cid:1) → P is actually the bundle Λ ϕ , where ϕ = E n t , in [24, Section 4]. By [24,Theorem 1.2], n + 1 ≤ dim H (cid:0) C ∞ ( S , P ) , Λ ϕ (cid:1) < ∞ . For any ν ∈ H (cid:0) P , O ( n ) (cid:1) , E ∗ ν = ( J k ) ∗ τ ∗ k ν is a C ∞ section of thebundle E ∗ O ( n ) → S × C ∞ (cid:0) S , P (cid:1) , where τ ∗ k ν is a C ∞ section of the bundle τ ∗ k O ( n ) → J k ( S , P ) which isholomorphic on each of the submanifolds J k ( t , P ) ⊂ J k ( S , P ), t ∈ S ,and we may consider E ∗ t ν as the restriction of E ∗ ν to { t }× C ∞ (cid:0) S , P (cid:1) . Let { σ a ∈ H (cid:0) C ∞ (cid:0) S , P \ { a } (cid:1) , Λ ϕ (cid:1) : a ∈ P } be the family of non-vanishingsections in [24, Proposition 2.7]. Then there are sections s a ∈ H (cid:0) P , O (1) (cid:1) such that(29) σ a = E ∗ t s n a | C ∞ ( S , P \{ a } ) = ( j kt ) ∗ τ ∗ k s n a | C ∞ ( S , P \{ a } ) , a ∈ P (see the proof of [24, Proposition 2.7]). By Proposition 4.2 and Theorem4.7 of [24], for any σ ∈ H (cid:0) C ∞ ( S , P ) , Λ ϕ (cid:1) , σ/σ ∞ : C ∞ (cid:0) S , C (cid:1) → C is a polynomial in x ( t ), x ′ ( t ), · · · , x ( n − ( t ), where x ∈ C ∞ (cid:0) S , C (cid:1) .Hence there is a holomorphic function ς ∞ on τ − n − ( C ) ∩ J n − ( t , P ) with σ/σ ∞ = ( j n − t ) ∗ ς ∞ . For any γ ∈ P GL (2 , C ), we have γ ∗ Λ ϕ ≃ Λ ϕ (see[24, Section 2]). Similar to ς ∞ , we can find holomorphic functions ς a on τ − n − ( P \ { a } ) ∩ J n − ( t , P ) with(30) σ/σ a = ( j n − t ) ∗ ς a , a ∈ P . Combination of (29) and (30) gives(31) H (cid:0) C ∞ ( S , P ) , Λ ϕ (cid:1) = ( j n − t ) ∗ H (cid:0) J n − ( t , P ) , τ ∗ n − O ( n ) (cid:1) . If n ≥
2, then( j k − t ) ∗ H (cid:16) J k − ( t , P ) , τ ∗ k − O ( n ) (cid:17) $ ( j kt ) ∗ H (cid:16) J k ( t , P ) , τ ∗ k O ( n ) (cid:17) , where k = 1 , · · · , n −
1, and (cid:0) j t (cid:1) ∗ H (cid:0) J (cid:0) t , P (cid:1) , τ ∗ O ( n ) (cid:1) = E ∗ t H (cid:0) P , O ( n ) (cid:1) . If we consider t in the expression of σ/σ ∞ as a variable in S , then weobtain a C ∞ function S × C ∞ (cid:0) S , C (cid:1) → C which is the pullback of a C ∞ function ˜ ς ∞ on the open subset τ − n − ( C ) of J n − ( S , P ) by J n − . Simi-larly, each σ/σ a induces a C ∞ function ˜ ς a on the open subset τ − n − ( P \ { a } ) ange decreasing group homomorphisms and holomorphic maps 25 of J n − ( S , P ). Hence we obtain a C ∞ section ˜ σ of the bundle E ∗ O ( n ) → S × C ∞ (cid:0) S , P (cid:1) such that(˜ σ/E ∗ s n a ) | S × C ∞ ( S , P \{ a } ) = (cid:0) J n − (cid:1) ∗ ˜ ς a (and σ can be considered as the restriction of ˜ σ to { t } × C ∞ (cid:0) S , P (cid:1) ). Let E ϕ = (cid:8) ˜ σ : σ ∈ H (cid:0) C ∞ ( S , P ) , Λ ϕ (cid:1)(cid:9) . Choose sections ˜ σ , · · · , ˜ σ m ′ ∈ E ∗ H (cid:0) P , O ( n ) (cid:1) ⊂ E ϕ , where m ′ ≥ σ m ′ +1 , · · · , ˜ σ m +1 ∈ E ϕ \ E ∗ H (cid:0) P , O ( n ) (cid:1) (where E ϕ \ E ∗ H (cid:0) P , O ( n ) (cid:1) = ∅ if n ≥ E ϕ spanned by ˜ σ , · · · , ˜ σ m +1 gives rise to a C ∞ map G : S × C ∞ (cid:0) S , P (cid:1) → P m (e.g. see [5, Section 1.4]) such that G ( t , · ) : C ∞ (cid:0) S , P (cid:1) → P m is aholomorphic map of degree n for every t ∈ S . The map G induces aholomorphic map(32) ˜ g : C ∞ (cid:0) S , P (cid:1) → C ∞ (cid:0) S , P m (cid:1) , where ˜ g ( x )( t ) = G ( t , x ) . By (31), we may consider E t ◦ ˜ g = G ( t , · ) as the composition of j n − t anda holomorphic map J n − ( t , P ) → P m . If n ≥ m ≥ m ′ , then ˜ g isnot of the form (6).More generally, let Υ : P m × · · · × P m p → P m , g j : F ( V, P n ) →F ( V, P m j ) be holomorphic maps and φ j ∈ T ( F , V, F , W ), j = 1 , · · · , p .Recall that Υ induces a holomorphic mapΥ ∗ : F ( W, P m ) × · · · × F ( W, P m p ) → F ( W, P m )as in (8). Define a holomorphic map g = g Υ ,g , ··· ,g p ,φ , ··· ,φ p : F ( V, P n ) → F ( W, P m ) by g ( x ) = Υ ∗ (cid:0) ( φ ) ∗ P m ( g ( x )) , · · · , ( φ p ) ∗ P mp ( g p ( x )) (cid:1) . (33)Now choose g j to be maps C ∞ (cid:0) S , P (cid:1) → C ∞ (cid:0) S , P m j (cid:1) as in (32), φ j ∈ T (cid:0) C ∞ , S , F , W (cid:1) , where j = 1 , · · · , p , and take Υ to be injective. If p ≥ φ , · · · , φ p are the same, then the map g : C ∞ (cid:0) S , P (cid:1) →F ( W, P m ) as in (33) is not of the form (6).By similar arguments as above, we could also construct holomorphic maps f : F (cid:0) S , P (cid:1) → F ( W, P m ), where F = C k or W k,p (see [24, Section 4]). 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