Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds

arXiv:1808.08034v3 [math.DG] 28 Aug 2018 Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds Hiroki Yagisita (Kyoto Sangyo University) Abstract: An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product Cnof the complex number field. On the other hand, when Ais a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of Ais holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product An. Therefore, a manifold modeled on An(an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n= 1, that is, Riemann surfaces). By the way, if Xis a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on Xis the most basic example of a commutative Banach algebra (furthermore, a commutative C∗-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family Mof compact complex manifolds (a topological deformation M of compact complex analytic structures) on Xis denoted by Γ(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on Xis introduced into Γ(M). Therefore, especially, if Xis contractible, then Γ(M) is a finite-dimensional C(X)-manifold. Japanese: https://www.researchgate.net/publication/327236804 1 Keywords: Frechet differentiable manifold, infinite-dimensional (almost) complex manifold, functional complex geometry, abstract differential geometry, sheaf of sets, Serre-Swan theorem, finitely generated projective module, normed free module, Hilbert module, K-theory, Chern class, characteristic class, Hermitian metric, Riemannian metric, Finsler metric, fiber bundle, real tangent bundle, parallel transport, symmetric affine connection, spray, geodesic line, exponential map, local normal coordinate system, tubular neighborhood 2 1 Compact continuous families of complex manifolds In this and the next sections, we introduce two basic concepts of this note. Definition 1.1 (Canonical real coordinate system) : The mapping that maps a complex vector z=x+√−1y∈Cnto its real and imaginary pair (x, y)∈R2n, i.e., (x1+√−1y1, x2+√−1y2,···, xn+√−1yn) 7→ ((x1, x2,···, xn),(y1, y2,···, yn)) is called the canonical real coordinate system. — Hereafter, when we consider an open set of Cnto be a C∞-manifold, we use the canonical real coordinate system as its local coordinate system. Definition 1.2 (Open polydisk) : For r > 0, let Dn r:= {(z1, z2,···, zn)∈Cn|max k|zk|< r } ={((x1, x2,···, xn),(y1, y2,···, yn)) ∈R2n| max kpxk2+yk2< r }. Let Dn:= Dn 1.Dnis called the unit open polydisk. — Definition 1.3 (Compact continuous family of complex manifolds) : M:= (M, X, π, S) is said to be a compact continuous family (of ndimensional complex manifolds), if it satisfies the followings. (1) Mand Xare compact Hausdorff spaces. πis a continuous surjection form Mto X. (2) S(6=∅) is a set. For any ϕ∈S,ϕis a map from an open set Mϕ(6=∅) of Mto the unit open polydisk Dn. For ϕ∈S, let π|ϕ:= π↾Mϕ. For any ϕ∈S, there exists an open set Xϕ(6=∅) of Xsuch that (ϕ, π|ϕ) is a homeomorphism from Mϕto Dn×Xϕ. (3) For ϕ∈Sand t∈Xϕ, let Mϕ,t := Mϕ∩π−1({t}), ϕ|t:= ϕ↾Mϕ,t . 3 For any ϕ1, ϕ2∈Sand t∈Xϕ1∩Xϕ2, the coordinate transformation ϕ2|t◦ϕ1|−1 t:ϕ1(Mϕ1,t ∩Mϕ2,t)→ϕ2(Mϕ1,t ∩Mϕ2,t) is holomorphic. (4) M=∪ϕ∈SMϕholds. Remark : (1) Each fiber of a compact continuous family is a compact complex manifold. (2) Suppose that (M, B, π) is a complex analytic family of compact complex manifolds. Then, for any compact subset X(6=∅) of B,π−1(X) is a compact continuous family. — Unlike complex analytic families, it seems that continuous families are not subjects of strong interest. Therefore, we have to confirm very primitive facts of compact continuous families on our own. Definition 1.4 (Local trivialization coordinate neighborhood) : When M:= (M, X, π, S) is a compact continuous family, Mis called the total space, Xis called the base space, πis called the projection and S is called the system of local trivialization coordinate neighborhoods. — Since the total space of a compact continuous family is compact, the system of local trivialization coordinate neighborhoods can be assumed to be a finite set. Definition 1.5 (Finite continuous family) : A compact continuous family whose system of local trivialization coordinate neighborhoods is a finite set is called a finite continuous family. — Definition 1.6 (Compact coordinate neighborhood) : Let (M, X, π, S) be a finite continuous family. Then, ( {(M′ ϕ, X′ ϕ)}ϕ∈S, r0) is said to be a system of compact coordinate neighborhoods of M, if it satisfies the followings. (1) M′ ϕis an open set of M,X′ ϕis an open set of Xand M′ ϕ⊂Mϕ, X′ ϕ⊂Xϕ hold. (2) r0∈(0,1) holds. (3) The homeomorphism (ϕ, π|ϕ) from Mϕto Dn×Xϕmaps M′ ϕonto Dn r0×X′ ϕ. That is, M′ ϕ= (ϕ, π|ϕ)−1(Dn r0×X′ ϕ) holds. 4 (4) M=∪ϕ∈SM′ ϕholds. Remark : In the above definition, neither M′ ϕ6=∅nor X′ ϕ6=∅is required. — Definition 1.7 (Bump system) : Let (M, X, π, S) be a finite continuous family. Then, ( {(ρϕ, M′ ϕ, X′ ϕ)}ϕ∈S, r0) is said to be a bump system of M, if it satisfies the followings. (1) ( {(M′ ϕ, X′ ϕ)}ϕ∈S, r0) is a system of compact coordinate neighborhoods of M. (2) ρϕis a continuous function from Mto [0,+∞) and supp(ρϕ)⊂M′ ϕ holds. (3) The function ρϕ◦(ϕ, π|ϕ)−1:Dn×Xϕ→[0,+∞) is C∞-class with respect to the variable (x, y)∈Dn(⊂R2n). For any multiindexes αand β, the function ∂xα∂yβ(ρϕ◦(ϕ, π|ϕ)−1) : Dn×Xϕ→R is continuous. (4) For any p∈M, there exists ϕ∈Ssuch that ρϕ(p)6= 0 holds. Remark : In the above definition, Pϕ∈Sρϕ= 1 is not required. — Lemma 1.8 (Existence of a bump system) : Suppose that Mis a finite continuous family. Then, there exists a bump system of M. Proof : We give it in Section 5.  Definition 1.9 (Continuous family with bumps) : M:= (M, X, π, S, {(ρϕ, M′ ϕ, X′ ϕ)}ϕ∈S, r0) is said to be a continuous family with bumps, if (M, X, π, S) is a finite continuous family and ({(ρϕ, M′ ϕ, X′ ϕ)}ϕ∈S, r0) is a bump system of M. — Hereafter, the continuous family M(= (M, X, π, S, {(ρϕ, M′ ϕ, X′ ϕ)}ϕ∈S, r0)) with bumps is fixed to be one. 5 Definition 1.10 (Real tangent bundle) : Let Nbe an n-dimensional complex manifold. Then, for q∈N, its real tangent space (TR)q(N) is a complex linear space by the almost complex structure of N. That is, when z=x+√−1yis a holomorphic local coordinate system of N, set √−1 ( ∂ ∂xk )q:= ( ∂ ∂yk )q,√−1 ( ∂ ∂yk )q:= −(∂ ∂xk )q. In addition, set TR(N) := ∪q∈N(TR)q(N). TR(N) is a holomorphic vector bundle on N. Remark : The real tangent bundle TR(N) is holomorphically isomorphic to the holomorphic tangent bundle T′(N) by (∂ ∂xk )q∈TR(N)7→ (∂ ∂zk )q:= 1 2(( ∂ ∂xk )q−√−1( ∂ ∂yk )q)∈T′(N). — Lemma 1.11 : Let Tbe a topological space. Let Ube an open set of Cn×T. Suppose that fis a complex valued continuous function on Uand for any t∈T, the map z7→ f(z, t) is holomorphic. Then, for any multi-index γ, ∂zγf:U→C is continuous. Proof : We give it in Section 6.  Definition 1.12 (Trivial Hermitian metric) : Let ϕ∈S. For p∈Mϕ, define a complex inner product on (TR)p(π−1({π(p)})) as h˙p1,˙p2iϕ,p := h(D(ϕ|π(p)))p( ˙p1),(D(ϕ|π(p)))p( ˙p2)iCn. Here, D(ϕ|π(p)) is the differential of the holomorphic local coordinate system ϕ|π(p):= ϕ↾Mϕ,π(p)=ϕ↾Mϕ∩π−1({π(p)}) of π−1({π(p)}) and h·,·iCnis the canonical inner product (the canonical Hermitian metric) on Cn. Also, k·kϕ,p denotes the norm by this complex inner product. That is, let k˙pkϕ,p := qh˙p, ˙piϕ,p. 6 For t∈Xϕ, by the complex inner products, the complex manifold Mϕ,t := Mϕ∩π−1({t}) is a Hermitian manifold that is isomorphic to Dn. In addition, the topological space TR(Mϕ) := ∪p∈Mϕ(TR)p(π−1({π(p)})) is a continuous Hermitian vector bundle on the topological space Mϕ. — Definition 1.13 (Hermitian real tangent bundle) : For p∈M, define a complex inner product on (TR)p(π−1({π(p)})) as h˙p1,˙p2ip:= X ϕ∈S ρϕ(p)h˙p1,˙p2iϕ,p. Also, k·kpdenotes the norm by this complex inner product. That is, let k˙pkp:= qh˙p, ˙pip. For t∈X, by the complex inner products, the fiber π−1({t}) of the continuous family Mis a Hermitian manifold. In addition, the topological space TR(M) := ∪p∈M(TR)p(π−1({π(p)})) is a continuous Hermitian vector bundle on the total space Mof the continuous family. The projection TR(M)→Mof the vector bundle is denoted by . That is, for ˙p∈TR(M), ( ˙p)∈M, ˙p∈(TR)( ˙p)(π−1({π(( ˙p))})) hold. ( ˙p) is called the base point of a real tangent vector ˙p. Remark : In the definition of TR(M), Lemma 1.11 was used. That is, according to Lemma 1.11, for any local trivialization coordinate systems ψ1 and ψ2of Mthat are compatible with each other, the local trivialization coordinate system of TR(M) determined by ψ1and the one determined by ψ2are compatible with each other. — Lemma 1.14 : Let ϕ∈S. Let Kbe a compact subset of Mϕ. Then, there exist an open set Uof Mϕand C > 0 such that the followings hold. 7 (1) K⊂Uholds. (2) For any p∈Uand ˙p∈(TR)p(π−1({π(p)})), k˙pkϕ,p ≤Ck˙pkp,k˙pkp≤Ck˙pkϕ,p hold. Proof : Both the Hermitian metrics are continuous. So, it follows.  Definition 1.15 (Distance on a fiber) : Let t∈X. For a piecewise C∞-curve c: [a, b]→π−1({t}), let Lt(c) := Zb akd ds(c(s))kc(s)ds. For p, q ∈π−1({t}), let dt(p, q) := inf ( {Lt(c)|cis a piecewise C∞-curve with the start point pand the end point q}∪{1}). By dt,π−1({t}) is a metric space. Remark : Re (hz, wiCn) = hz, wiR2nand especially, kzkCn=kzkR2nhold. Therefore, the norm defined by a complex inner product is the norm defined by a real one. — Lemma 1.16 : Let ϕ∈S. Let Nbe an open set of Mϕ. Let Kbe a compact subset of N. Then, there exist an open set Uof N,δ > 0 and C > 0 such that the followings hold. (1) K⊂Uholds. (2) Suppose p∈Uand q∈π−1({π(p)}). Then, dπ(p)(q, p)< δ =⇒ q∈N, kϕ(q)−ϕ(p)kCn≤C dπ(p)(q, p) holds. Proof : We give it in Section 7.  Remark on Lemma 1.16 : Let ϕ∈S. Let Nbe an open set of Mϕ. Let Kbe a compact subset of N. Then, there exist an open set Uof N,δ > 0 and C > 0 such that the followings hold. 8 (1) K⊂Uholds. (2) Suppose that p∈Uand q∈Mϕ,π(p)(:= Mϕ∩π−1({π(p)})). Then, kϕ(q)−ϕ(p)kCn< δ =⇒ q∈N, dπ(p)(q, p)≤Ckϕ(q)−ϕ(p)kCn holds. Proof : We give it in Section 8. (However, this remark is not used.)  Definition 1.17 (Continuous section) : Let T(6=∅) be a subset of X.uis said to be a continuous section of M on T, if uis a continuous map from Tto Mand for any t∈T,π(u(t)) = t holds. The set of all continuous sections of Mon Tis denoted by Γ(M|T). — Definition 1.18 (Distance between continuous sections) : Let T(6=∅) be a subset of X. For u, v ∈Γ(M|T), let dT(u, v) := sup t∈T dt(u(t), v(t)). By dT, Γ(M|T) is a metric space. — 9 (4) For any r∈[0,1) and ε > 0, there exists δ > 0 such that for any t∈T,p∈Utand q∈π−1({t}), kΨ(p)ku(t)≤r, dt(q, p)< δ =⇒q∈Ut,kΨ(q)−Ψ(p)ku(t)< ε holds. — Definition 3.4 (Coordinate neighborhood of the set of all continuous sections defined by a holomorphic normal coordinate neighborhood) : Let T(6=∅) be a subset of X. Let u∈Γ(M|T). Let (U, Ψ) to be a holomorphic normal coordinate neighborhood of u. Then, let ˜ U:= {v∈Γ(M|T)| ∃r∈[0,1),∀t∈T:v(t)∈Ut,kΨ(v(t))ku(t)≤r}. A map ˜ Ψ : ˜ U→Γb(u∗(TR(M))) is defined as (˜ Ψ(v)) (t) := Ψ(v(t)) ( v∈˜ U, t ∈T). (˜ U, ˜ Ψ) is called the coordinate neighborhood of Γ(M|T) defined by (U, Ψ). — The next proposition is the technical main result of this note. This proof is given in the next section. Proposition 3.5 (Existence of a holomorphic normal coordinate neighborhood) : Let T(6=∅) be a normal subset of X. Let u∈Γ(M|T). Then, there exists a holomorphic normal coordinate neighborhood of u. Proof : It is given in the next section.  On the other hand, the main result of this note is as follows. Theorem 3.6 (Main result) : Let T(6=∅) be a normal subset of X. Then, the metric space Γ(M|T) becomes a Cb(T)-manifold according to all coordinate neighborhoods of Γ(M|T) defined by holomorphic normal coordinate neighborhoods. — For the proof of this theorem, Proposition 3.5 and the following lemma are used. Lemma 3.7 : Let ε > 0. Let Xand Ybe n-dimensional complex inner product spaces. Let fbe a holomorphic map from {z∈X|kzkX< ε }to {w∈Y|kwkY< 1}. Then, 16 k(Df)0kL(X;Y)≤4n2 ε holds and z∈X, kzkX≤1 4ε =⇒ kf(z)−(f(0) + (Df)0(z)) kY≤16n3 ε2kzk2 X holds. Proof : We give it in Section 9.  Proof of Theorem 3.6 : 1◦: Let u∈Γ(M|T). Let ˜ Bu:= {˙u∈Γb(u∗(TR(M))) |k˙uku<1}. Let (U, Ψ) be a holomorphic normal coordinate neighborhood of u. Then, we show ˜ Ψ( ˜ U)⊂˜ Bu. Let v∈˜ U.v:T→Uand Ψ : U→u∗(TR(M)) are continuous. Also, there exists r∈[0,1) such that supt∈TkΨ(v(t))ku(t)≤rholds. Thus, ˜ Ψ(v) = Ψ◦v∈˜ Buholds. 2◦: Let u∈Γ(M|T). Let (U, Ψ) be a holomorphic normal coordinate neighborhood of u. A map ˜ Ψ−1:˜ Bu→Γ(M|T) is defined as (˜ Ψ−1( ˙u)) (t) := Ψ−1( ˙u(t)) ( ˙u∈˜ Bu, t ∈T). Then, we show ˜ Ψ−1(˜ Bu)⊂˜ U. Let ˙u∈˜ Bu. ˙u:T→ ∪p∈u(T)Bp(TR) and Ψ−1:∪p∈u(T)Bp(TR)→ Uare continuous. Also, k˙uku∈[0,1) and k˙u(t)ku(t)≤ k˙ukuhold. Thus, ˜ Ψ−1( ˙u) = Ψ−1◦˙u∈˜ Uholds. 3◦: From 1◦and 2◦, ˜ Ψ−1◦˜ Ψ = 1˜ U,˜ Ψ◦˜ Ψ−1= 1 ˜ Bu hold. 4◦: We show that ˜ Ψ−1is Lipschitz continuous. Let ˙u0,˙u1∈˜ Bu. Then, for any s∈[0,1], ˙us:= ˙u0+s( ˙u1−˙u0) = (1 −s) ˙u0+s˙u1∈˜ Bu holds. Thus, 17 dt(( ˜ Ψ−1( ˙u0))(t),(˜ Ψ−1( ˙u1))(t)) =dt(Ψ−1( ˙u0(t)),Ψ−1( ˙u1(t))) ≤Z1 0kd ds(Ψ−1( ˙us(t))) kΨ−1( ˙us(t)) ds =Z1 0k(D(Ψ|−1 t)) ˙us(t)( ˙u1(t)−˙u0(t)) kΨ|−1 t( ˙us(t)) ds ≤(Z1 0k(D(Ψ|−1 t)) ˙us(t)kL((TR)u(t)(π−1({t}));(TR)Ψ|−1 t( ˙us(t))(π−1({t}))) ds )k˙u1(t)−˙u0(t)ku(t) holds. Therefore, dT(˜ Ψ−1( ˙u0),˜ Ψ−1( ˙u1)) ≤( sup t∈T, ˙p∈Bu(t)(TR)k(D(Ψ|−1 t))˙pkL((TR)u(t)(π−1({t}));(TR)Ψ|−1 t( ˙p)(π−1({t}))) )k˙u1−˙u0ku holds. 5◦: We show that ˜ Ψ is continuous. Let v∈˜ Uand ε > 0. Then, there exists r∈[0,1) such that for any t∈T, v(t)∈Ut,kΨ(v(t)) ku(t)≤r hold. Therefore, there exists δ > 0 such that for any t∈Tand q∈π−1({t}), dt(q, v(t)) < δ =⇒q∈Ut,kΨ(q)−Ψ(v(t))ku(t)<ε 2 holds. Let w∈˜ Uand dT(w, v)< δ. Then, because of dt(w(t), v(t)) < δ, kΨ(w(t)) −Ψ(v(t)) ku(t)<ε 2 holds. So, k˜ Ψ(w)−˜ Ψ(v)ku≤ε 2< ε holds. 6◦: We show that ˜ Uis an open set of Γ(M|T). Let v∈˜ U. Then, there exists r∈[0,1) such that for any t∈T, 18 v(t)∈Ut,kΨ(v(t)) ku(t)≤r hold. Thus, there exists ε > 0 such that for any t∈Tand q∈π−1({t}), dt(q, v(t)) < ε =⇒q∈Ut,kΨ(q)−Ψ(v(t))ku(t)<1−r 2 holds. Let w∈Γ(M|T) and dT(w, v)< ε. Then, because of dt(w(t), v(t)) < ε, w(t)∈Ut,kΨ(w(t)) −Ψ(v(t)) ku(t)<1−r 2 hold. Therefore, furthermore, kΨ(w(t)) ku(t) ≤ kΨ(w(t)) −Ψ(v(t)) ku(t)+kΨ(v(t)) ku(t) <1−r 2+r=1 + r 2 holds, while 1+r 2∈[0,1) holds. Hence, w∈˜ Uholds. 7◦:˜ Buis an open set of Γb(u∗(TR(M))). On the other hand, from Ψ(u(t)) = 0u(t), we obtain u∈˜ U. Therefore, by Proposition 3.5, Γ(M|T) is a topological manifold according to all coordinate neighborhoods defined by holomorphic normal coordinate neighborhoods. 8◦: We show that each coordinate transformation is Cb(T)-differentiable (i.e., it is Frechet differentiable and its Frechet derivatives are Cb(T)-linear). Let (U, Ψ) be a holomorphic normal coordinate neighborhood of u. Let (V, Φ) be a holomorphic normal coordinate neighborhood of v. Let ˙u∈ ˜ Ψ( ˜ U∩˜ V). Then, there exists ε > 0 such that for any ˙ h∈Γb(u∗(TR(M))), k˙ hku< ε =⇒˙u+˙ h∈˜ Ψ( ˜ U∩˜ V) holds. Also, because Tis normal, if k˙pku(t)< ε holds, then there exists ˙ h∈Γb(u∗(TR(M))) such that k˙ hku< ε, ˙ h(t) = ˙p hold. (For this, since we can take a local trivialization coordinate system of the vector bundle u∗(TR(M)) and make the Schmidt orthogonalization on 19 the canonical basis at each tto obtain an isomorphism to the product bundle of the complex inner product space, it is only necessary to paste the constant section through ˙pand the zero section by a partition of unity.) Hence, k˙pku(t)< ε =⇒˙u(t) + ˙p∈Ψ|t(Ut∩Vt) holds. So, by Lemma 3.7, (3.1) k(D(Φ|t◦Ψ|−1 t))˙u(t)kL((TR)u(t)(π−1({t}));(TR)v(t)(π−1({t}))) ≤4n2 ε holds and also, (3.2) k˙pku(t)≤1 4ε =⇒ k(Φ|t◦Ψ|−1 t)( ˙u(t) + ˙p)−( (Φ|t◦Ψ|−1 t)( ˙u(t)) + (D(Φ|t◦Ψ|−1 t))˙u(t)( ˙p) ) kv(t) ≤16n3 ε2k˙pk2 u(t) holds. Now, for ˙ h∈Γb(u∗(TR(M))) and t∈T, we define (˜ A(˙ h))(t) := (D(Φ|t◦Ψ|−1 t))˙u(t)(˙ h(t)). Then, ( ˜ A(˙ h))(t)∈(TR)v(t)(π−1({t})) holds. Furthermore, in virtue of Lemma 1.11, the section t∈T7→ (˜ A(˙ h))(t)∈v∗(TR(M)) is continuous. (This is because taking a respective local trivialization coordinate system of the vector bundle u∗(TR(M)) and v∗(TR(M)) and displaying the map ˙p7→ (Φ◦Ψ−1)( ˙p) in the local coordinates to be holomorphic on each fiber and to be continuous, so that the map ˙p7→ (D(Φ|π(( ˙p)) ◦Ψ|−1 π(( ˙p))))˙pis also continuous.) Also, while it is obvious that ˜ Ais Cb(T)-linear, from (3.1), ˜ Ais a continuous map from Γb(u∗(TR(M))) to Γb(v∗(TR(M))). However, from (3.2), ˜ Ais the Frechet derivative of ˜ Φ◦˜ Ψ−1at the point ˙u. Corollary 3.8 : Suppose that Tis a paracompact contractible subset of X. Then, Γ(M|T) is an n-dimensional Cb(T)-manifold. 20 Proof : Let u∈Γ(M|T). Then, as a continuous complex vector bundle on T,u∗(TR(M)) is isomorphic to Cn×T. That is, there exists {˙uk}n k=1 such that ˙uk∈Γ(u∗(TR(M))) holds and for any t∈T,{˙uk(t)}n k=1 is a basis of (TR)u(t)(π−1({t})). We denote the Schmidt orthogonalization on {˙uk(t)}n k=1 at each t∈Tby {˙ek(t)}n k=1. Then, ˙ek∈Γb(u∗(TR(M))) holds and for any t∈T,{˙ek(t)}n k=1 is an orthonormal basis of (TR)u(t)(π−1({t})). We define a map F: (Cb(T))n→Γb(u∗(TR(M))) as F({˙z1(t)}t∈T,{˙z2(t)}t∈T,···,{˙zn(t)}t∈T) := {X k ˙zk(t)˙ek(t)}t∈T. Because of maxk|zk| ≤ kzkCn≤nmaxk|zk|, as in Example 2.9, Fand F−1 are continuous Cb(T)-linear maps.  Remark (Serre-Swan theorem) : By making a continuous complex vector bundle Mon Xcorrespond to the module Γ(M) of all continuous sections of Mon X, the category of continuous complex vector bundles on Xis equivalent to the one of finitely generated projective C(X)-modules ([2, 9, 11, 12]). — Remark (Manifold on the real commutative algebra C∞(X;R)) : Let M→Xbe a differentiable family of differentiable manifolds. The set of all differentiable sections of Mon Xbecomes a C∞(X;R)-manifold ([4, 6, 7, 8]). However, C∞(X;R) is not a real Banach space, but it is a real Frechet space. — Conjecture (Perhaps not too difficult) : For example, similarly, it seems that when Mis a compact continuous family of almost complex manifolds on X, an almost complex C(X;R)- manifold structure is introduced into Γ(M). — 21 4 Holomorphic linear connections Sprays are used in order to prove existence of a holomorphic normal coordinate neighborhood. However, the global sprays corresponding to LeviCivita connections may not be holomorphic. For each section, holomorphic sprays on its neighborhood are constructed as the ones corresponding to convex combinations of trivial connections. Definition 4.1 (Holomorphic linear spray) : Let Zbe a real C∞-vector field on the real tangent bundle TR(N) of an n-dimensional complex manifold N. Then, Zis said to be a holomorphic linear spray on N, if for any holomorphic local coordinate system z= (z1, z2,···, zn) of N, there exists a family {Γk i,j}i,j,k ∈{1,2,···,n}of holomorphic functions on the coordinate neighborhood of zsuch that Γk i,j = Γk j,i (i, j, k = 1,2,···, n ) holds and the ordinary differential equation that an integral curve {( ˙z(s), z(s)) }s∈(−ε,+ε)of the vector field Zshould satisfy is d ds ˙zk=−X i,j Γk i,j(z1, z2,···, zn) ˙zi˙zj, d dszk= ˙zk. — Again, recall that a continuous family M(= (M, X, π, S, {(ρϕ, M′ ϕ, X′ ϕ)}ϕ∈S, r0)) with bumps was fixed to be one. Definition 4.2 (Trivial spray) : (1) Let ϕ∈S. Let Nϕ:= (ϕ, π|ϕ)−1(Dn r0+1 2×X′ ϕ) =ϕ−1(Dn r0+1 2 )∩π−1(X′ ϕ). Nϕis an open set of π−1(X′ ϕ). Further, let t∈X. Let Nϕ,t := Nϕ∩π−1({t}) =(ϕ|−1 t(Dn r0+1 2 ) (t∈X′ ϕ), ∅(t∈X\X′ ϕ). 22 Nϕ,t is an open set of the n-dimensional complex manifold π−1({t}). (2) Let ϕ∈Sand t∈X. A real C∞-vector field Yϕ,t on the real tangent bundle TR(Nϕ,t) is determined as with respect to the holomorphic coordinate system p∈Nϕ,t 7→ z=ϕ(p)∈Cnof Nϕ,t, the ordinary differential equation that an integral curve ( ˙z, z) of Yϕ,t should satisfy is d ds ˙z= 0,d dsz= ˙z. — Proposition 4.3 : (1) The trivial spray Yϕ,t is a holomorphic linear one on Nϕ,t. (2) Let ψ∈S. With respect to the holomorphic local coordinate system p∈Nψ,t ∩Nϕ,t 7→ z=ψ(p)∈Cnof Nϕ,t, the ordinary differential equation that an integral curve ( ˙z, z) of the trivial spray Yϕ,t should satisfy is d ds ˙zk=−X i,j (Γ(ϕ, ψ))k i,j(z, t) ˙zi˙zj, d dszk= ˙zk, provided that ϕ|t(i)is the i-th component of ϕ|t,ψ|t(j)is the j-th component of ψ|tand (Γ(ϕ, ψ))k i,j(z, t) := −X l,m ((∂zl∂zm(ψ|t(k)◦ϕ|−1 t))((ϕ|t◦ψ|−1 t)(z))) ((∂zi(ϕ|t(l)◦ψ|−1 t))(z)) ((∂zj(ϕ|t(m)◦ψ|−1 t))(z)) holds. Proof : Simple calculation.  Lemma 4.4 : Let (Γ(ϕ, ψ))k i,j be the same as Proposition 4.3. Then, the function (z, t)∈(ψ, π|ψ)(Nϕ∩Nψ)7→ (Γ(ϕ, ψ))k i,j(z, t)∈C is continuous. Proof: According to Lemma 1.11.  23 Definition 4.5 (Weighting space) : Let R(6=∅) be a subset of S. Let WR:= {c∈[0,1]S|X ϕ∈S cϕ=X ϕ∈R cϕ= 1 }. WRis called the weighting space of Rand an element of WRis called a weighting of R. Remark : WRis a compact subset of [0,1]S.R1⊂R2implies WR1⊂WR2. — Definition 4.6 (Spray determined by a weighting) : (1) Let R(6=∅) be a subset of S. Let NR:= ∩ϕ∈RNϕ= ( ∩ϕ∈Rϕ−1(Dn r0+1 2 ) ) ∩π−1(∩ϕ∈RX′ ϕ). NRis an open set of π−1(∩ϕ∈RX′ ϕ). Further, let t∈X. Let NR t:= ∩ϕ∈RNϕ,t =NR∩π−1({t}) =(∩ϕ∈R(ϕ|−1 t(Dn r0+1 2 ) ) (t∈ ∩ϕ∈RX′ ϕ), ∅(t∈X\∩ϕ∈RX′ ϕ). NR tis an open set of the n-dimensional complex manifold π−1({t}). (2) For c∈WSand t∈X, a holomorphic linear spray Yc,t on Nc−1((0,1]) t is defined as Yc,t := X ϕ∈S cϕYϕ,t. For a subset R(6=∅) of S,c∈WRand t∈X, a holomorphic linear spray Yc,t Ron NR tis defined as Yc,t R:= Yc,t ↾NR t=X ϕ∈R cϕYϕ,t↾NR t. Remark : When c∈WRholds, c−1((0,1]) ⊂Rand NR t⊂Nc−1((0,1]) thold. — Definition 4.7 (Coordinate display of the spray) : Let ψ∈R⊂S. Let NR,ψ := (ψ, π|ψ)(NR), UR,ψ := Cn×NR,ψ ×WR. 24 NR,ψ is an open set of Dn r0+1 2×(∩ϕ∈RX′ ϕ) and UR,ψ is an open set of (Cn×Dn r0+1 2 )×( ( ∩ϕ∈RX′ ϕ)×WR). Let {(Γ(ϕ, ψ))k i,j}ϕ,ψ,i,j,k be the same as Proposition 4.3. A map fR,ψ :UR,ψ →C2n is defined as fR,ψ k( ˙z, z, t, c) := −X i,j (X ϕ∈R cϕ(Γ(ϕ, ψ))k i,j(z, t) ) ˙zi˙zj(k= 1,2,···, n), fR,ψ k( ˙z, z, t, c) := ˙zk−n(k=n+ 1, n + 2,···,2n). — Definition 4.8 (Coordinate display of the exponential map) : Let ψ∈R⊂S. Then, the set of all ( ˙z, z, t, c)∈UR,ψ (= Cn×NR,ψ ×WR) such that there exists ( ˙w, w) : [0,1] →Cn×Dn r0+1 2 such that d ds( ˙w, w) = fR,ψ( ˙w, w, t, c) (s∈[0,1]), ( ˙w, w)(0) = ( ˙z, z) hold is denoted by IR,ψ. Also, a map (˙eR,ψ, eR,ψ) : IR,ψ →Cn×Dn r0+1 2 is defined by ( ˙eR,ψ, eR,ψ)( ˙z, z, t, c) = ( ˙w, w)(1). — Lemma 4.9 (ODE with a continuous parameter) : Let Λ be a topological space. Let Ube an open set of Rm×Λ. Suppose that f:U→Rmis a continuous map. Suppose that for any λ∈Λ, the map x7→ f(x, λ) is differentiable. Suppose that the map Dxf:U→L(Rm;Rm) is continuous. Further, let (x0, λ0)∈U. Suppose that u0: [0,1] →Rmis the solution of the initial value problem d dsu=f(u, λ0), u(0) = x0. Then, for any ε > 0, there exists an open set Vof Usuch that it satisfies the followings. 25 ( (D(ψ|t))u(t)( ˙p), ψ(u(t)), t, χ(t) ) ∈U′ R,ψ holds. Hence, ˙pbelongs to the domain of the exponential map for the spray Yχ(t),t Ron NR t. So, further, it does to the one for Yχ(t),t. Now, we define exp( ˙p) as the value of the exponential map for the spray Yχ(t),t at ˙p. Then, exp( ˙p)∈π−1({t})⊂π−1(T) holds. 3◦: We define U:= exp(∪p∈u(T)B′ p(TR)). Uis a subset of π−1(T). exp is a map whose domain is ∪p∈u(T)B′ p(TR) and whose range is U. 4◦: For t0∈T, we define Tt0:= ( ∩ψ∈Rt0(u−1(M′ ψ)) ) ∩(∩ψ∈S\Rt0(T\supp(χψ)) ). Tt0is an open set of T.t0∈Tt0holds. Thus, in particular, {Tt0}t0∈Tis an open covering of T. Also, t∈Tt0 =⇒χ(t)∈WRt0, u(t)∈ ∩ϕ∈Rt0M′ ϕ⊂KRt0 holds. Therefore, if ψ∈Rt0holds, then Tt0is a subset of X′ ψ,χ↾Tt0is a continuous map from Tt0to WRand ψ◦(u↾Tt0) is a continuous map from Tt0 to Dn r0such that for any t∈Tt0, ((ψ◦(u↾Tt0))(t), t)∈KRt0,ψ holds. 5◦: For a subset T′of T, we define UT′:= U∩π−1(T′) = exp(∪p∈u(T′)B′ p(TR)), exp|T′:= exp↾∪p∈u(T′)B′ p(TR). Let t0∈T. We examine the map exp|Tt0and its range UTt0and the map exp|{t0}and its range U{t0}. By 1◦, there exists ψ∈Rt0. In virtue of 4◦,u(Tt0)⊂ ∩ϕ∈Rt0M′ ϕ⊂Mψ holds. Now, because ∪p∈u(Tt0)B′ p(TR) is an open set of u∗(TR(M))|Tt0( = TR(M)|u(Tt0) = ∪p∈u(Tt0)(TR)p(π−1({π(p)})) ), the trivialization coordinate system ˙p∈u∗(TR(M))|Tt07→ ( (D(ψ|π(( ˙p))))( ˙p)( ˙p), π(( ˙p)) ) ∈Cn×Tt0 32 of the vector bundle u∗(TR(M))|Tt0on Tt0maps ∪p∈u(Tt0)B′ p(TR) to an open set of Cn×Tt0. That is, as we denote the image of the set ∪p∈u(Tt0)B′ p(TR) by Gt0,Gt0is an open set of Cn×Tt0. We show Gt0⊂U′ Rt0,ψ,χ↾Tt0,ψ◦(u↾Tt0). As t∈ Tt0and ˙p∈B′ u(t)(TR) hold, because of u(t)∈KRt0⊂(ψ, π|ψ)−1(Dn r0×X′ ψ), by Lemma 4.17, k(D(ψ|t))u(t)( ˙p)kCn≤Ca 1k˙pku(t)<min{δa 0,δb 0 α0} holds. So, in virtue of 4◦and Lemma 4.16, ( (D(ψ|t))u(t)( ˙p), t )∈U′ Rt0,ψ,χ↾Tt0,ψ◦(u↾Tt0) holds. Gt0⊂U′ Rt0,ψ,χ↾Tt0,ψ◦(u↾Tt0)holds. From the above, Gt0is an open set of U′ Rt0,ψ,χ↾Tt0,ψ◦(u↾Tt0). Hence, as we set Ht0:= {(eRt0,ψ,χ↾Tt0,ψ◦(u↾Tt0)( ˙z, t), t )∈Dn r0+1 2×Tt0|( ˙z, t)∈Gt0}, by Lemma 4.16, Ht0is an open set of Dn r0+1 2×Tt0and homeomorphically the map ( ˙z, t)7→ (eRt0,ψ,χ↾Tt0,ψ◦(u↾Tt0)( ˙z, t), t ) maps Gt0to Ht0. Therefore, UTt0is an open set of π−1(Tt0) and homeomorphically the map exp|Tt0maps to its domain ∪p∈u(Tt0)B′ p(TR) to its range UTt0. Further, exp|{t0}is a biholomorphic map from its domain B′ u(t0)(TR) to its range U{t0}. Also, u(t0) = exp|{t0}(0u(t0))∈U{t0} holds. Also, because for ˙p∈B′ u(t0)(TR), (D(exp|{t0}))˙p = (D(ψ|−1 t0))(eRt0,ψ,χ↾Tt0,ψ◦(u↾Tt0)((D(ψ|t0))u(t0)( ˙p),t0),t0) ·(D˙zeRt0,ψ,χ↾Tt0,ψ◦(u↾Tt0))((D(ψ|t0))u(t0)( ˙p), t0)·(D(ψ|t0))u(t0) holds, from Lemma 4.16 and Lemma 4.17, 33 sup ˙p∈B′ u(t0)(TR)k(D(exp|{t0}))˙pkL( (TR)u(t0)(π−1({t0})) ; (TR)exp|{t0}( ˙p)(π−1({t0})) ) ≤Cb 1·α0·Ca 1 holds. 6◦: For any t0∈T,Tt0is an open set of Tand t0∈Tt0holds. Hence, from 5◦,Uis an open set of π−1(T) and homeomorphically the map exp maps its domain ∪p∈u(T)B′ p(TR) to its range U. Further, for any t∈T, u(t) = exp(0u(t))∈Ut( := U∩π−1({t}) ) holds and the map exp|{t}is a biholomorphic map from its domain B′ u(t)(TR) to its range Ut. Also, sup t∈T, ˙p∈B′ u(t)(TR)k(D(exp|{t}))˙pkL( (TR)u(t)(π−1({t})) ; (TR)exp|{t}( ˙p)(π−1({t})) ) ≤α0Ca 1Cb 1<+∞ holds. 7◦: For q∈U, we define Ψ(q) := Ca 1 min{δa 0,δb 0 α0}exp−1(q). From 6◦, (U, Ψ) satisfies the conditions (1), (2) and (3) of the definition. 8◦: We show that the condition (4) of the definition is satisfied. Let 0 ≤r < 1 and ε > 0. We set δ:= min {δ2,δb 0 C2 (1 −r),min{δa 0,δb 0 α0} β0Ca 1Cb 1C2 (1 −r),min{δa 0,δb 0 α0} β0Ca 1Cb 1C2 ε}. δ > 0 holds. Suppose t∈T,p∈Ut,q∈π−1({t}), kΨ(p)ku(t)≤rand dt(q, p)< δ. We show q∈Utand kΨ(q)−Ψ(p)ku(t)< ε. From 1◦, there exists ψsuch that ψ∈Rt⊂Sholds. On the other hand, (4.1) kexp−1(p)ku(t)≤min{δa 0,δb 0 α0} Ca 1 r holds. So, in virtue of 1◦and Lemma 4.17, 34 k(D(ψ|t))u(t)(exp−1(p)) kCn≤min{δa 0,δb 0 α0}r holds and for any s∈[0,1], ks(D(ψ|t))u(t)(exp−1(p)) kCn< δa 0 holds. Hence, in virtue of 1◦and Lemma 4.14, eRt,ψ((D(ψ|t))u(t)(exp−1(p)), ψ(u(t)), t, χ(t)) ∈Dn r0+1 2 , keRt,ψ((D(ψ|t))u(t)(exp−1(p)), ψ(u(t)), t, χ(t)) −ψ(u(t)) kCn =keRt,ψ((D(ψ|t))u(t)(exp−1(p)), ψ(u(t)), t, χ(t)) −eRt,ψ(0, ψ(u(t)), t, χ(t)) kCn ≤α0k(D(ψ|t))u(t)(exp−1(p)) kCn≤δb 0r hold. Now, because of eRt,ψ((D(ψ|t))u(t)(exp−1(p)), ψ(u(t)), t, χ(t)) = ψ(exp(exp−1(p))), p∈Mψ, ψ(p)∈Dn r0+1 2 , (4.2) kψ(p)−ψ(u(t)) kCn≤δb 0r hold. Since from p∈Ut(= U∩π−1({t})) and 1◦,π(p) = t=π(u(t)) ∈X′ ψ holds, (ψ(p), π(p)) ∈Dn r0+1 2×X′ ψholds. Therefore, by dt(q, p)< δ ≤δ2and Lemma 4.18, q∈Mψand (4.3) kψ(q)−ψ(p)kCn≤C2dt(q, p) hold. From (4.2) and (4.3), kψ(p)−ψ(u(t)) kCn< δb 0, kψ(q)−ψ(u(t)) kCn≤C2dt(q, p) + δb 0r < C2δ+δb 0r≤C2 δb 0 C2 (1 −r) + δb 0r=δb 0 hold. So, for s∈[0,1], as we set c(s) := (1 −s)ψ(p) + sψ(q) = ψ(p) + s(ψ(q)−ψ(p)), 35 kc(s)−ψ(u(t)) kCn< δb 0 holds. Therefore, in virtue of 1◦, Lemma 4.14 and (4.3), (c(s), ψ(u(t)), t, χ(t)) ∈V′ Rt,ψ, khRt,ψ(ψ(q), ψ(u(t)), t, χ(t)) −hRt,ψ(ψ(p), ψ(u(t)), t, χ(t))kCn ≤β0kψ(q)−ψ(p)kCn≤β0C2dt(q, p) hold. Hence, further, from p∈Ut=U∩π−1({t}), hRt,ψ(ψ(p), ψ(u(t)), t, χ(t)) = (D(ψ|t))u(t)(exp−1(p)) holds. Now, we set ˙z:= hRt,ψ(ψ(q), ψ(u(t)), t, χ(t)). Then, k˙z−(D(ψ|t))u(t)(exp−1(p)) kCn≤β0C2dt(q, p) holds. In virtue of 1◦and Lemma 4.17, (4.4) k(D(ψ|t))−1 u(t)( ˙z)−exp−1(p)ku(t)≤β0Cb 1C2dt(q, p) holds. From (4.1) and (4.4), k(D(ψ|t))−1 u(t)( ˙z)ku(t) ≤β0Cb 1C2dt(q, p) + min{δa 0,δb 0 α0} Ca 1 r < β0Cb 1C2δ+min{δa 0,δb 0 α0} Ca 1 r ≤β0Cb 1C2 min{δa 0,δb 0 α0} β0Ca 1Cb 1C2 (1 −r) + min{δa 0,δb 0 α0} Ca 1 r =min{δa 0,δb 0 α0} Ca 1 holds. That is, (D(ψ|t))−1 u(t)( ˙z)∈B′ u(t)(TR) holds. Therefore, 36 q= exp((D(ψ|t))−1 u(t)( ˙z)) ∈U∩π−1({t}) = Ut holds. Further, from (4.4), kΨ(q)−Ψ(p)ku(t) =Ca 1 min{δa 0,δb 0 α0}k(D(ψ|t))−1 u(t)( ˙z)−exp−1(p)ku(t) ≤β0Ca 1Cb 1C2 min{δa 0,δb 0 α0}dt(q, p) <β0Ca 1Cb 1C2 min{δa 0,δb 0 α0}δ ≤β0Ca 1Cb 1C2 min{δa 0,δb 0 α0} min{δa 0,δb 0 α0} β0Ca 1Cb 1C2 ε=ε holds.  Problem : When dose there exist a global holomorphic linear connection (or a global holomorphic spray) on a complex manifold ? For example, is a closed Riemann surface admitting a global holomorphic linear connection an elliptic curve ? Also, related to this, dose there exist a nontrivial example of (sheaves of) manifolds modeled on (locally free sheaves of) the modules of all holomorphic sections of holomorphic vector bundles ? — Remark : Let Λ be a topological space. Let Nbe an open subset of Rm×Λ. Let Kbe a compact subset of N. Then, there exists δ > 0 such that for any x, y ∈Rmand λ∈Λ, (x, λ)∈K, ky−xkRm< δ =⇒(y, λ)∈N holds. Proof : (We do not use this remark, but we use argument under this proof to prove other lemmas. As a preliminary announcement, we describe it here.) 37 For (x, λ)∈K, there exist δx,λ >0 and an open set Λx,λ of Λ such that λ∈Λx,λ holds and for any y∈Rmand µ∈Λ, ky−xkRm<2δx,λ, µ ∈Λx,λ =⇒(y, µ)∈N holds. Then, there exists a finite subset Lof Ksuch that (x, λ)∈K=⇒ ∃ (x′, λ′)∈L:kx−x′kRm< δx′,λ′, λ ∈Λx′,λ′ holds. Now, we set δ:= min({δx,λ}(x,λ)∈L∪{1}). δ > 0 holds. Let (x, λ)∈K and ky−xkRm< δ. Then, there exists (x′, λ′)∈Lsuch that kx−x′kRm< δx′,λ′, λ ∈Λx′,λ′ hold. However, ky−x′kRm≤ ky−xkRm+kx−x′kRm < δ +δx′,λ′≤2δx′,λ′ holds. (y, λ)∈Nholds.  38 5 Proof of Lemma 1.8 Mis normal and {Mϕ}ϕ∈Sis a finite open covering of M. Hence, there exists an open covering {Gϕ}ϕ∈Sof Msuch that for any ϕ∈S, Gϕ⊂Mϕ holds. For ϕ∈S, we set X′ ϕ:= π(Gϕ). Then, X′ ϕ=π(Gϕ) = π(Gϕ)⊂π(Mϕ) = Xϕ holds. In addition, since Gϕis an open set of Mϕ,X′ ϕis an open set of Xϕ. So, X′ ϕis an open set of X. On the other hand, since ∪ϕ∈Sϕ(Gϕ) is a compact subset of Dn(:= Dn 1), there exists r0∈(0,1) such that ∪ϕ∈Sϕ(Gϕ)⊂Dn r0 holds. Now, we set M′ ϕ:= (ϕ, π|ϕ)−1(Dn r0×X′ ϕ). M′ ϕis an open set of M. Also, Dn r0×X′ ϕis a compact subset of Dn×Xϕ, M′ ϕ= (ϕ, π|ϕ)−1(Dn r0×X′ ϕ)⊂Mϕ holds. From (ϕ, π|ϕ) (Gϕ)⊂ϕ(Gϕ)×π(Gϕ)⊂Dn r0×X′ ϕ, Gϕ⊂M′ ϕholds. Therefore, M=∪ϕ∈SGϕ=∪ϕ∈SM′ ϕholds. ( {(M′ ϕ, X′ ϕ)}ϕ∈S, r0) is a system of compact coordinate neighborhoods of M. Similarly, there exist {(M′′ ϕ, X′′ ϕ)}ϕ∈Sand r1∈(0, r0) such that M′′ ϕis an open set of M,X′′ ϕis an open set of Xand M′′ ϕ⊂M′ ϕ, X′′ ϕ⊂X′ ϕ, M′′ ϕ= (ϕ, π|ϕ)−1(Dn r1×X′′ ϕ), M=∪ϕ∈SM′′ ϕ 39 hold. Further, similarly, there exist {(M′′′ ϕ, X′′′ ϕ)}ϕ∈Sand r2∈(0, r1) such that M′′′ ϕis an open set of M,X′′′ ϕis an open set of Xand M′′′ ϕ⊂M′′ ϕ, X′′′ ϕ⊂X′′ ϕ, M′′′ ϕ= (ϕ, π|ϕ)−1(Dn r2×X′′′ ϕ), M=∪ϕ∈SM′′′ ϕ hold. Now, since Xis normal, for ϕ∈S, there exists a non-negative continuous function ρ1,ϕ on Xsuch that X′′′ ϕ⊂ {t∈X|ρ1,ϕ(t)6= 0 } ⊂ X′′ ϕ holds. Also, there exists a non-negative C∞-function ρ2on C(= C1=R2) such that {(a, b)∈R2|ρ2(a, b)6= 0 }=D1 r1 holds. Now, we define a non-negative function ρϕon Mas ρϕ(p) := ρ1,ϕ(π(p)) (Qk= 1,2,···,n ρ2(ϕ(k)(p))) (p∈Mϕ), 0 (p∈M\Mϕ). Here, ϕ(k)is the k-th component of ϕ. Now, because of {p∈M|ρϕ(p)6= 0 } ⊂ {p∈Mϕ|(ϕ, π|ϕ)(p)∈Dn r1×X′′ ϕ}=M′′ ϕ, ρϕis a continuous function such that supp (ρϕ)⊂M′′ ϕ⊂M′ ϕ holds. However, because M′′′ ϕ={p∈Mϕ|(ϕ, π|ϕ)(p)∈Dn r2×X′′′ ϕ} ⊂ {p∈M|ρϕ(p)6= 0 } also holds, M=∪ϕ∈SM′′′ ϕ=∪ϕ∈S{p∈M|ρϕ(p)6= 0 } holds.  40 6 Proof of Lemma 1.11 From the induction, it suffices to show it in the case of |γ|= 1. Furthermore, when |γ|= 1 holds, even if we assume n= 1, we do not lose generality. Let n= 1. Let (w, s)∈Uand ε > 0. Then, there exist r > 0 and an open set T′of Tsuch that s∈T′and |z−w|<2r, t ∈T′=⇒(z, t)∈U hold. Further, for any τ∈[0,2π], there exist δτ∈(0, r) and an open set Tτ of T′such that s∈Tτand |θ−τ|< δτ,|z−w|< δτ, t ∈Tτ =⇒ |f(z+re√−1θ, t)−f(w+re√−1τ, s)|<1 2rε hold. There exists a finite subset K(6=∅) of [0,2π] such that θ∈[0,2π] =⇒ ∃τ∈K:|θ−τ|< δτ holds. Now, we set δ:= min τ∈Kδτ, T ′′ := ∩τ∈KTτ. δ > 0 and s∈T′′ hold. T′′ is an open set of T. Now, let |z−w|< δ and t∈T′′. Then, for any θ∈[0,2π], there exists τ∈Ksuch that |f(z+re√−1θ, t)−f(w+re√−1θ, s)| ≤ |f(z+re√−1θ, t)−f(w+re√−1τ, s)|+|f(w+re√−1τ, s)−f(w+re√−1θ, s)| < rε holds. So, |(∂zf)(z, t)−(∂zf)(w, s)| =|1 2πr Zθ∈[0,2π] e−√−1θ(f(z+re√−1θ, t)−f(w+re√−1θ, s) ) dθ | < ε holds.  41 |s−s′|< δs′ holds, from λ∈Λ′⊂Λs′and kxkRm≤δ′≤δs′<2δs′, C1C2k(Dxf)(u0(s) + A(s)x, λ)−(Dxf)(u0(s), λ0)kL(Rm;Rm) ≤C1C2k(Dxf)(u0(s) + A(s)x, λ)−(Dxf)(u0(s′), λ0)kL(Rm;Rm) +C1C2k(Dxf)(u0(s), λ0)−(Dxf)(u0(s′), λ0)kL(Rm;Rm) ≤1 4+1 4≤1 2 holds. That is, (10.1) s∈[0,1], x ∈Rm,kxkRm≤δ′, λ ∈Λ′ =⇒ (u0(s) + A(s)x, λ)∈U, C1C2k(Dxf)(u0(s) + A(s)x, λ)−(Dxf)(u0(s), λ0)kL(Rm;Rm)≤1 2 holds. Now, let ε > 0. Then, for s∈[0,1], there exist δ′′ s>0 and an open set Λ′′ sof Λ such that λ0∈Λ′′ sand s′′ ∈[0,1],|s′′ −s|< δ′′ s, λ ∈Λ′′ s =⇒ (u0(s′′), λ)∈U, kf(u0(s′′), λ)−f(u0(s), λ0)kRm≤1 8C1min{δ′,ε 2C2} hold. There exists a finite subset L′′ (6=∅) of [0,1] such that s∈[0,1] =⇒ ∃s′′ ∈L′′ :|s−s′′|< δ′′ s′′ holds. Now, we set Λ′′′ := ∩s∈L′′ Λ′′ s. λ0∈Λ′′′ holds. Λ′′′ is an open set of Λ. Now, as s∈[0,1] and λ∈Λ′′′ hold, because there exists s′′ ∈L′′ such that |s−s′′|< δ′′ s′′ holds, from λ∈Λ′′′ ⊂Λ′′ s′′ , 48 C1kf(u0(s), λ)−f(u0(s), λ0)kRm ≤C1kf(u0(s), λ)−f(u0(s′′), λ0)kRm +C1kf(u0(s), λ0)−f(u0(s′′), λ0)kRm ≤1 8min{δ′,ε 2C2}+1 8min{δ′,ε 2C2}=1 4min{δ′,ε 2C2} holds. That is, (10.2) s∈[0,1], λ ∈Λ′′′ =⇒ (u0(s), λ)∈U, C1kf(u0(s), λ)−f(u0(s), λ0)kRm≤1 4min{δ′,ε 2C2} holds. Now, we set V:= {x∈Rm|kx−x0kRm<1 4min{δ′,ε 2C2}} × (Λ′∩Λ′′′). Then, (x0, λ0)∈Vholds. Also, Vis an open set of Rm×Λ. However, from u0(0) = x0,A(0) = 1Rmand (10.1), V⊂Uholds. Vis an open set of U. Well, we show that (2) is established. For v∈C([0,1]; Rm), we set kvkC([0,1];Rm):= sup s∈[0,1] kv(s)kRm. We set ˜ X:= {v∈C([0,1]; Rm)|kvkC([0,1];Rm)≤min{δ′,ε 2C2}}. ˜ Xis a complete metric space. ˜ X6=∅holds. Let (x, λ)∈V. Then, from (10.1), for any v∈˜ Xand s∈[0,1], (u0(s) + A(s)v(s), λ)∈Uholds. Now, for v∈˜ Xand s∈[0,1], we set (F(v))(s) := (x−x0) + Zs 0 (A(σ))−1 (f(u0(σ) + A(σ)v(σ), λ)−(f(u0(σ), λ0) + ((Dxf)(u0(σ), λ0))A(σ)v(σ)) ) dσ. 49 We show that Fis a contraction mapping of ˜ X. As v∈˜ X,s∈[0,1], σ∈[0,1] and τ∈[0,1] hold, because of kτ v(σ)kRm≤τmin{δ′,ε 2C2} ≤ δ′, from (10.1) and (10.2), (u0(σ) + τA(σ)v(σ), λ)∈U, f(u0(σ) + A(σ)v(σ), λ)−f(u0(σ), λ) = ( Z1 0 (Dxf)(u0(σ) + τA(σ)v(σ), λ)dτ )A(σ)v(σ), k(A(σ))−1( (f(u0(σ)+A(σ)v(σ), λ)−f(u0(σ), λ)) −((Dxf)(u0(σ), λ0))A(σ)v(σ) ) kRn =k(A(σ))−1(Z1 0 ((Dxf)(u0(σ)+τA(σ)v(σ), λ)−(Dxf)(u0(σ), λ0)) dτ )A(σ)v(σ)kRm ≤1 2kv(σ)kRm≤1 2kvkC([0,1];Rm), k(F(v))(s)kRm ≤ kx−x0kRm+kZs 0 (A(σ))−1(f(u0(σ), λ)−f(u0(σ), λ0)) dσ kRm+1 2kvkC([0,1];Rm) ≤1 4min{δ′,ε 2C2}+1 4min{δ′,ε 2C2}+1 2min{δ′,ε 2C2}= min{δ′,ε 2C2} hold. So, F(˜ X)⊂˜ Xholds. Next, as v∈˜ X,v′∈˜ X,s∈[0,1], σ∈[0,1] and τ∈[0,1] hold and we set cτ:= (1 −τ)v+τv′=v+τ(v′−v), because of kcτ(σ)kRm≤(1 −τ) min{δ′,ε 2C2}+τmin{δ′,ε 2C2} ≤ δ′, from (10.1), 50 (u0(σ) + A(σ)cτ(σ), λ)∈U, f(u0(σ) + A(σ)v′(σ), λ)−f(u0(σ) + A(σ)v(σ), λ) = ( Z1 0 (Dxf)(u0(σ) + A(σ)cτ(σ), λ)dτ )A(σ) (v′(σ)−v(σ)), k(F(v′))(s)−(F(v))(s)kRm =kZs 0 (A(σ))−1 ( (f(u0(σ) + A(σ)v′(σ), λ)−f(u0(σ) + A(σ)v(σ), λ)) −((Dxf)(u0(σ), λ0))A(σ)(v′(σ)−v(σ)) ) dσ kRn =kZs 0 (A(σ))−1(Z1 0 ((Dxf)(u0(σ) + A(σ)cτ(σ), λ)−(Dxf)(u0(σ), λ0)) dτ ) A(σ) (v′(σ)−v(σ)) dσ kRm ≤1 2kv′−vkC([0,1];Rm) hold. Fis a contraction mapping of ˜ X. There exists v∈˜ Xsuch that v=F(v) holds. We set u:= u0+Av. Then, d dsu=d dsu0+ ( d dsA)v+A(d dsv) =f(u0, λ0) + ((Dxf)(u0, λ0))Av +A(d dsF(v)) =f(u0, λ0)+((Dxf)(u0, λ0))Av+AA−1(f(u, λ)−(f(u0, λ0)+((Dxf)(u0, λ0))Av)) =f(u, λ), u(0) = u0(0) + A(0)v(0) = x0+ (F(v))(0) = x0+ (x−x0) = x, ku−u0kC([0,1];Rm)=kAvkC([0,1];Rm)≤C2min{δ′,ε 2C2} ≤ ε 2< ε hold.  51 11 Proof of Lemma 4.10 Let (x0, y0)∈V(⊂Cm=R2m).We denote the Frechet derivative of w for the variable (x, y)∈Vat a point (s, x0, y0) by A(s). That is, we set A(s) := (D(x,y)w)(s, x0, y0). Then, A: [0,1] →L(R2m;R2m) satisfies d dsA= ((D(x,y)f)(w(s, x0, y0))) A, A(0) = 1R2m. On the other hand, since fis holomorphic, (D(x,y)f)(w(s, x0, y0)) is complex linear. Therefore, for any c∈Cand (p, q)∈Cm=R2m, both c((A(s))(p, q)) and (A(s)) (c(p, q)) are the solutions of the initial value problem d ds(u, v) = ((D(x,y)f)(w(s, x0, y0))) (u, v), (u, v)(0) = c(p, q) to coincide. (D(x,y)w)(s, x0, y0) is complex linear.  52 12 Proof of Lemma 4.12 This is the same as proofs of usual inverse function theorems. Just to be sure, we confirm it. When Θ = ∅holds, it is trivial. Let Θ 6=∅and ε > 0. We set ε′:= ε 1 + ε. 0< ε′<min{ε, 1}holds. For λ∈Θ, there exist δ′′ λ>0 and an open set Λ′′ λ of Λ such that λ∈Λ′′ λand x∈Rm,kxkRm≤δ′′ λ, λ′′ ∈Λ′′ λ =⇒(x, λ′′)∈U, k(Dxf)(x, λ′′)−1RmkL(Rm;Rm)≤ε′ hold. Then, there exists a finite subset L(6=∅) of Θ such that Θ⊂Λ′:= ∪λ∈LΛ′′ λ holds. Λ′is an open set of Λ. Now, we set δ′:= min λ∈Lδ′′ λ, E:= {x∈Rm|kxkRm≤δ′} × Λ′. δ′>0 holds. Also, as (x, λ)∈Eholds, because there exists λ′′ ∈Lsuch that λ∈Λ′′ λ′′ holds, from kxkRm≤δ′≤δ′′ λ′′ , (x, λ)∈Uand k(Dxf)(x, λ)− 1RmkL(Rm;Rm)≤ε′hold. So, {0}×Λ′⊂E⊂U, (12.1) sup (x,λ)∈Ek1Rm−(Dxf)(x, λ)kL(Rm;Rm)≤ε′ hold. Also, we set E′:= ∪λ∈Λ′({y∈Rm|ky−f(0, λ)kRm≤(1 −ε′)δ′}×{λ}), E′′ := {x∈Rm|kxkRm≤δ′} × E′. Then, 53 (x, λ)∈E=⇒(x, f(0, λ), λ)∈E′′, (x, y, λ)∈E′′ =⇒(x, λ)∈E hold. We define a map h:E′′ →Rmas h(x, y, λ) := x−f(x, λ) + y. As (x, y, λ)∈E′′ and s∈[0,1] hold, (sx, λ)∈Eand h(x, y, λ) = ((x−f(x, λ)) −(0 −f(0, λ))) + (y−f(0, λ)) = ( Z1 0 (1Rm−(Dxf)(sx, λ)) ds )x+ (y−f(0, λ)) hold. Hence, by (12.1), (12.2) (x, y, λ)∈E′′ =⇒ kh(x, y, λ)kRm≤ε′kxkRm+ky−f(0, λ)kRm≤δ′ holds. As (x, y, λ)∈E′′, (x′, y, λ)∈E′′ and s∈[0,1] hold and we set c(s) := (1 −s)x+sx′=x+s(x′−x), (c(s), λ)∈Eand h(x′, y, λ)−h(x, y, λ) = (x′−f(x′, λ)) −(x−f(x, λ)) = ( Z1 0 (1Rm−(Dxf)(c(s), λ)) ds ) (x′−x) hold. Hence, by (12.1), (12.3) (x, y, λ)∈E′′,(x′, y, λ)∈E′′ =⇒ kh(x′, y, λ)−h(x, y, λ)kRm≤ε′kx′−xkRm holds. For ℓ∈C(E′;Rm), we set kℓkC(E′;Rm):= sup (y,λ)∈E′kℓ(y, λ)kRm. 54 We set ˜ X:= {ℓ∈C(E′;Rm)|kℓkC(E′;Rm)≤δ′}. ˜ Xis a complete metric space. ˜ X6=∅holds. For any ℓ∈˜ Xand (y, λ)∈E′, (ℓ(y, λ), y, λ)∈E′′ holds. Now, for ℓ∈˜ Xand (y, λ)∈E′, we set (F(ℓ))(y, λ) := h(ℓ(y, λ), y, λ). As ℓ∈˜ Xholds, because of F(ℓ)∈C(E′;Rm), from (12.2), F(ℓ)∈˜ Xholds. Further, as ℓ∈˜ Xand ℓ′∈˜ Xhold, from (12.3), kF(ℓ′)−F(ℓ)kC(E′;Rm)≤ ε′kℓ′−ℓkC(E′;Rm)holds. Fis a contraction mapping of ˜ X. Therefore, there exists ℓ∈˜ Xsuch that ℓ=F(ℓ) holds. In particular, (y, λ)∈E′=⇒(ℓ(y, λ), λ)∈E, f(ℓ(y, λ), λ) = y holds. Now, we set V′:= ∪λ∈Λ′({y∈Rm|ky−f(0, λ)kRm<(1 −ε′)δ′}×{λ}), g:= ℓ↾V′, U′:= {(g(y, λ), λ)∈E|(y, λ)∈V′}. V′is an open set of Rm×Λ. g∈C(V′;Rm) and (y, λ)∈V′=⇒(g(y, λ), λ)∈U′, f(g(y, λ), λ) = y hold. On the other hand, as (x, λ)∈U′holds, because there exists y∈Rm such that (y, λ)∈V′and x=g(y, λ) hold, y=f(g(y, λ), λ) = f(x, λ), (f(x, λ), λ) = (y, λ)∈V′, x=g(y, λ) = g(f(x, λ), λ) hold. That is, 55 (x, λ)∈U′=⇒(f(x, λ), λ)∈V′, g(f(x, λ), λ) = x holds. Therefore, homeomorphically the map (x, λ)7→ (f(x, λ), λ) maps U′ to V′. Also, from (12.2), for any (y, λ)∈V′,kg(y, λ)kRm=kℓ(y, λ)kRm= k(F(ℓ))(y, λ)kRm=kh(ℓ(y, λ), y, λ)kRm≤ε′δ′+ky−f(0, λ)kRm< ε′δ′+ (1 −ε′)δ′=δ′holds. Therefore, U′⊂G:= {x∈Rm|kxkRm< δ′} × Λ′⊂E⊂U holds. So, (x, λ)∈U′=⇒(x, λ)∈G, (f(x, λ), λ)∈V′ holds. Conversely, if (x, λ)∈Gand (f(x, λ), λ)∈V′hold, then g(f(x, λ), λ) = ℓ(f(x, λ), λ) = (F(ℓ))(f(x, λ), λ) =h(ℓ(f(x, λ), λ), f(x, λ), λ) = h(g(f(x, λ), λ), f(x, λ), λ), x=h(x, f(x, λ), λ) hold and so, from (12.3), kg(f(x, λ), λ)−xkRm ≤ kh(g(f(x, λ), λ), f(x, λ), λ)−h(x, f(x, λ), λ)kRm ≤ε′kg(f(x, λ), λ)−xkRm, g(f(x, λ), λ) = x, (x, λ) = (g(f(x, λ), λ), λ)∈U′ hold. Thus, (12.4) (x, λ)∈G, (f(x, λ), λ)∈V′ ⇐⇒ (x, λ)∈U′ holds. That is, U′={(x, λ)∈G|(f(x, λ), λ)∈V′}holds. U′is an open set of U. Further, we set δ:= 1−ε′ 1 + ε′δ′. 56 δ > 0 and ∪λ∈Λ′({y∈Rm|ky−f(0, λ)kRm< δ }×{λ})⊂V′⊂Rm×Λ′ hold. As x∈Rm,kxkRm< δ,λ∈Λ′and s∈[0,1] hold, because of (sx, λ)∈ G⊂E, from (12.1) and (12.4), kf(x, λ)−f(0, λ)kRm≤(1 + ε′)kxkRm< (1 + ε′)δ= (1 −ε′)δ′,(f(x, λ), λ)∈V′and (x, λ)∈U′hold. Therefore, {x∈Rm|kxkRm< δ }×Λ′⊂U′⊂Rm×Λ′ holds. From (12.1), sup (x,λ)∈U′k(Dxf)(x, λ)−1RmkL(Rm;Rm)≤ε′<min{ε, 1} holds. From a usual inverse function theorem, for any λ∈Λ′, the map y7→ g(y, λ) is differentiable. Hence, for any (y, λ)∈V′, (Dyg)(y, λ) = ((Dxf)(g(y, λ), λ))−1 holds. The map Dyg:V′→L(Rm;Rm) is continuous. Also, sup (y,λ)∈V′k(Dyg)(y, λ)−1RmkL(Rm;Rm) = sup (x,λ)∈U′k((Dxf)(x, λ))−1−1RmkL(Rm;Rm) = sup (x,λ)∈U′k∞ X k=1 (1Rm−(Dxf)(x, λ))kkL(Rm;Rm) ≤∞ X k=1 ε′k=ε holds.  57