Cartan-Thullen theorem for $\mathbb C^n$-holomorphic convexity and a related problem
Abstract
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=j$ implies $\frac{\partial f_i}{\partial z_j}=0$, where $(z_1,z_2,\cdots,z_n) \in \mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$ holds. We note that a $\mathbb C^n$-analytic mapping and a $\mathbb C^n$-analytic manifold can also be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for $\mathbb C^n$-analytic functions. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $\mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, \cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following very partial result. If an open $\mathbb C^n$-holomorphically convex set $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $\mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $\frac{\partial f_i}{\partial z_j}=0$. Kasuya suggested that a $\mathbb C^n$-analytic manifold and a $\mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.
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Cartan-Thullen theorem for Cn-holomorphic convexity and a related problem Hiroki Yagisita (Kyoto Sangyo University) Abstract: Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set Uof Ck, the following conditions are equivalent: (a) Uis a domain of existence, (b) U is a domain of holomorphy and (c) Uis holomorphically convex. On the other hand, when f( = (f1, f2,· · · , fn) ) is a Cn-valued function on an open set Uof Ck1×Ck2× · · · × Ckn,fis said to be Cn-analytic, if fis complex analytic and for any iand j,i6=jimplies ∂fi ∂zj= 0, where (z1, z2,· · · , zn)∈ Ck1×Ck2× · · · × Cknholds. We note that a Cn-analytic mapping and a Cn-analytic manifold can also be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for Cnanalytic functions. For n= 1, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open Cn-holomorphically convex set Uexist such that Uis not the direct product of any holomorphically convex sets U1, U2,· · · , Un−1and Un? As a corollary of our generalization, we give the following very partial result. If an open Cn-holomorphically convex set Uis convex, then Uis the direct product of some holomorphically convex sets. Also, fis said to be Cn-triangular, if fis complex analytic and for any i and j,i < j implies ∂fi ∂zj= 0. Kasuya suggested that a Cn-analytic manifold and a Cn-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation. Keywords: Stein space, pseudoconvex manifold. 1
1 Introduction First, we generalize the notion of a holomorphic function. Definition 1 (Structure sheaf) : Let k1, k2,· · · , kn−1, kn, l1, l2,· · · , ln−1and lnbe natural numbers. Let U be an open set of Cl1×Cl2× · · · × Cln. Let f( = (f1, f2,· · · , fn) ) be a map from Uto Ck1×Ck2× · · · × Ckn. Then, fis said to be Cn-holomorphic (Cnanalytic), if fis holomorphic and for any a∈Uand any i, j ∈ {1,2,· · · , n}, i6=jimplies ∂fi ∂zj(a) = 0, where (z1, z2,· · · , zn)∈Cl1×Cl2× · · · × Clnholds. Let Ol1,l2,···,ln(U) denote the set of all Cn-valued Cn-holomorphic functions on U. Then, {Ol1,l2,···,ln(U)}Uis called the sheaf of germs of Cn-holomorphic functions. Example 2 : (1) Let πj(U) := {zj∈Clj|∃z1, z2,· · · , zj−1, zj+1, zj+2,· · · , zn: (z1, z2,· · · , zn)∈U}. Let fjbe a holomorphic function on πj(U). Then, (f1, f2,· · · , fn) is a Cnholomorphic function on U. (2) Let εbe a small positive number. Let U:= ∪θ∈R({z1∈C| |z1−e√−1θ|< ε } × { z2∈C| |z2−θ|< ε }). Then, (log z1,0) is a C2-holomorphic function on U. However, log z1is a multivalued function on π1(U). Remark 3 : (1) The composition of Cn-holomorphic mappings is Cn-holomorphic. So, a Cn-analytic manifold can be easily defined with its structure sheaf. (2) For n= 1, {Ol(U)}Uis the sheaf of germs of holomorphic functions. (3) (f1, f2,· · · , fn) is Cn-holomorphic, if and only if (f1,0,0,· · · ,0,0,0), (0, f2,0,· · · ,0,0,0), · · · , (0,0,0,· · · ,0, fn−1,0) and (0,0,0,· · · ,0,0, fn) are Cn-holomorphic. Also, (f1, f2,· · · , fn),(g1, g2,· · · , gn)∈Ol1,l2,···,ln(U) implies (f1g1, f2g2,· · · , fngn)∈Ol1,l2,···,ln(U). Further, if a sequence {fm}∞ m=1 in Ol1,l2,···,ln(U) uniformly convergences to g∈(Ol1+l2+···+ln(U))non compact sets, then g∈Ol1,l2,···,ln(U) holds. So, Ol1,l2,···,ln(U) is the direct product of closed C-subalgebras of the usual one Ol1+l2+···+ln(U). (4) When Ais a commutative Banach algebra, Lorch ([2]) gave a definition that an A-valued function on an open set of Ais A-holomorphic. With the norm maxj=1,2,···,n |zj|,Cnis a locally compact one. We did a little study on A-analytic manifolds ([3, 4]). — 2
Since the structure sheaf {Ol1,l2,···,ln(U)}Uwas defined, we define Cnexistence, Cn-holomorphy and Cn-holomorphic convexity. Just in case, we state uniqueness theorem. Proposition 4 : Let Ube a connected open set of Cl1×Cl2× · · · × Cln. Let f, g ∈ Ol1,l2,···,ln(U). Let a∈U. If for any multi-index α,∂|α|f ∂zα(a) = ∂|α|g ∂zα(a) holds, then f=gholds. Proof : It is an easy corollary of the usual uniqueness theorem. ■ Definition 5 (Existence, Holomorphy) : Let Ube an open set of Cl1×Cl2× · · · × Cln. (1) Uis said to be a domain of Cn-existence, if the following holds. There exists f∈Ol1,l2,···,ln(U) such that for any open sets Vand Wof Cl1×Cl2× · · · × Cln, if Vis connected and ∅ 6=V\Uand ∅ 6=W⊂U∩V hold, then for any g∈Ol1,l2,···,ln(V), f↾W6=g↾Wholds. (2) Uis said to be a domain of Cn-holomorphy, if the following holds. For any open sets Vand Wof Cl1×Cl2× · · · × Cln, if Vis connected and ∅ 6=V\Uand ∅ 6=W⊂U∩Vhold, then there exists f∈Ol1,l2,···,ln(U) such that for any g∈Ol1,l2,···,ln(V), f↾W6=g↾Wholds. Lemma 6 : Cn-existence implies Cn-holomorphy. Proof : It is obvious. ■ Definition 7 (Holomorphic convexity) : Let |{wk}m k=1|denote maxk=1,2,···,m |wk|for w1, w2,· · · , wm∈C. Let Ube an open set of Cl1×Cl2× · · · × Cln. (1) Let Kbe a compact subset of U. Let b KU l1,l2,···,ln:= {z∈U| ∀ f∈Ol1,l2,···,ln(U) : |f(z)| ≤ sup w∈K |f(w)| }. Then, b KU l1,l2,···,lnis called the Cn-holomorphically convex hull of K. (2) Uis said to be Cn-holomorphically convex, if for any compact subset Kof U,b KU l1,l2,···,lnis compact. — The following is the main result. We note that for n= 1, it is CartanThullen theorem ([1]) itself. Theorem 8 : Let Ube an open set of Cl1×Cl2×· · ·×Cln. Then, the following conditions are equivalent: (a) Uis a domain of Cn-existence, (b) Uis a domain of Cnholomorphy and (c) Uis Cn-holomorphically convex. 3
Remark 9 : Let Uj(6=∅) be a connected open set of Clj(j= 1,2,· · · , n). Let U:= U1×U2× · · · × Un. (1) Let Kjbe a compact subset of Uj(j= 1,2,· · · , n). Then, \ (K1×K2× · · · × Kn)U l1,l2,···,ln=c K1 U1 l1×c K2 U2 l2× · · · × c Kn Un ln holds. (2) Uis Cn-holomorphically convex, if and only if U1, U2,· · · , Un−1and Unare holomorphically convex. Proof : (1) U1×U2×· · ·×Uj−1×Uj+1 ×Uj+2 ×· · ·×Unis connected. Hence, if (f1, f2,· · · , fn)∈Ol1,l2,···,ln(U) holds, then for any aj∈Uj, the function (z1, z2,· · · , zj−1, zj+1, zj+2,· · · , zn)7→ fj(z1, z2,· · · , zj−1, aj, zj+1, zj+2,· · · , zn) is constant. So, Ol1,l2,···,ln(U) = Ol1(U1)×Ol2(U2)×· · ·×Oln(Un) holds. For any (z1, z2,· · · , zn)∈U, ∀f∈Ol1,l2,···,ln(U) : |f(z1, z2,· · · , zn)| ≤ sup w∈K1×K2×···×Kn |f(w)| ⇐⇒ ∀(f1, f2,· · · , fn)∈Ol1(U1)×Ol2(U2)× · · · × Oln(Un) : max i=1,2,···,n |fi(zi)| ≤ max i=1,2,···,n( sup wi∈Ki |fi(wi)|) ⇐⇒ ∀i∈ {1,2,· · · , n},∀fi∈Oli(Ui) : |fi(zi)| ≤ sup wi∈Ki |fi(wi)| holds. (2) Suppose that Uis Cn-holomorphically convex. We show that Uj is holomorphically convex. Let Kjbe a compact subset of Uj. There exists (a1, a2,· · · , an)∈U. From (1), \ ({a1}×{a2} × · · · × {aj−1} × Kj× {aj+1}×{aj+2} × · · · × {an})U l1,l2,···,ln =d {a1}U1 l1×d {a2}U2 l2×· · ·× \ {aj−1}Uj−1 lj−1×c Kj Uj lj×\ {aj+1}Uj+1 lj+1 ×\ {aj+2}Uj+2 lj+2 ×· · ·×[ {an}Un ln holds. Hence, πj(\ ({a1}×{a2} × · · · × {aj−1} × Kj× {aj+1}×{aj+2} × · · · × {an})U l1,l2,···,ln) = c Kj Uj lj 4
holds. Because Uis Cn-holomorphically convex, c Kj Uj ljis compact. Ujis holomorphically convex. Suppose that U1, U2,· · · , Un−1and Unare holomorphically convex. We show that Uis Cn-holomorphically convex. Let Kbe a compact subset of U. Then, there exists {Kj}n j=1 such that Kjis a compact subset of Ujand K⊂K1×K2× · · · × Knholds. So, from (1), b KU l1,l2,···,ln⊂c K1 U1 l1×c K2 U2 l2× · · · × c Kn Un ln(⊂U) holds. Because U1, U2,· · · , Un−1and Unare holomorphically convex, b KU l1,l2,···,ln is compact. Uis Cn-holomorphically convex. ■ Our generalization is considered natural. On the other hand, our result is partial, because we do not answer the following question. Question : Does a connected Cn-holomorphically convex open set Uexist such that U is not the direct product of any holomorphically convex ones U1, U2,· · · , Un−1 and Un? — Now, we can give the following very partial result. Corollary 10 : Let Ube a convex open set of Cl1×Cl2× · · · × Cln. (1) Let f∈Ol1,l2,···,ln(U). Then, there exists g∈Ol1,l2,···,ln(π1(U)× π2(U)× · · · × πn(U)) such that f=g↾Uholds. (2) Suppose that Uis Cn-holomorphically convex. Then, U=π1(U)× π2(U)× · · · × πn(U) holds. Proof : (1) Let f= (f1, f2,· · · , fn). For any aj∈πj(U), U∩π−1 j({aj}) is convex, so, it is connected and the function (z1, z2,· · · , zj−1, zj+1, zj+2,· · · , zn)∈U∩π−1 j({aj}) 7→ fj(z1, z2,· · · , zj−1, aj, zj+1, zj+2,· · · , zn)∈C is constant. From this, it follows. (2) From Theorem 8, Uis a domain of Cn-existence. Hence, from (1), it follows. ■ Comment : A map fis said to be Cn-triangular, if fis holomorphic and for any iand j,i < j implies ∂fi ∂zj= 0. Kasuya suggested that a Cn-analytic manifold and aCn-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation. — 5
2 Proof of main result The proof of Theorem 8 is almost the same as Cartan-Thullen theorem. Perhaps, it seems to be also proved as a consequence of some general theory. However, for the sake of confirmation, we describe it. That is, we choose a proof that works in our case. In fact, it is extremely easy as we see below. When a reader believes that some proof which he knows works, he should skip the following proof. Lemma 11 : Let Kbe a compact subset of U. Then, b KU l1,l2,···,lnis bounded. Proof : Let 1 ≤k≤lj. Then, (0,0,· · · ,0, zj,k,0,0,· · · ,0) ∈Ol1,l2,···,ln(U) holds. Here, zj= (zj,1, zj,2,· · · , zj,lj) holds. Hence, z∈b KU l1,l2,···,lnimplies |zj,k| ≤ supw∈K|wj,k|(<+∞). ■ Lemma 12 : Let Kbe a compact subset of U. Suppose that b KU l1,l2,···,lnis not compact. Then, there exists b∈(Cl1×Cl2× · · · × Cln)\U such that inf a∈ b KU l1,l2,··· ,ln |a−b|= 0 holds. Proof : From Lemma 11, b KU l1,l2,···,lnis not a closed set of Cl1×Cl2×· · ·×Cln. So, there exist a sequence {am}∞ m=1 in b KU l1,l2,···,lnand b∈(Cl1×Cl2× · · · × Cln)\b KU l1,l2,···,lnsuch that limm→∞ am=bholds. Because b KU l1,l2,···,lnis a closed set of U,b6∈ Uholds. ■ Lemma 13 : Let Kbe a compact subset of U. Let r:= inf z∈K, w∈(Cl1×Cl2×···×Cln)\U |z−w|. Then, for any a∈b KU l1,l2,···,lnand f∈Ol1,l2,···,ln(U), there exists g∈Ol1,l2,···,ln({z∈ Cl1×Cl2×· · ·×Cln| |z−a|< r }) such that for any multi-index α,∂|α|f ∂zα(a) = ∂|α|g ∂zα(a) holds. 6
Proof : Let s∈(0, r). Then, from Cauchy inequality, there exists c∈ (0,+∞) such that for any multi-index α, ∂|α|f ∂zα(a)≤sup z∈K ∂|α|f ∂zα(z)≤cα! s|α| holds. Hence, g:z7→ Pα 1 α! ∂|α|f ∂zα(a)(z−a)α∈Ol1,l2,···,ln({z∈Cl1×Cl2× · · · × Cln| |z−a|< r }) holds. ■ Lemma 14 : Cn-holomorphy implies Cn-holomorphic convexity. Proof : Suppose that Uis not Cn-holomorphically convex. Then, we show that Uis not a domain of Cn-holomorphy. There exists a compact subset K of Usuch that b KU l1,l2,···,lnis not compact. Let r:= inf z∈K, w∈(Cl1×Cl2×···×Cln)\U |z−w|. Then, from Lemma 12, there exist a∈b KU l1,l2,···,lnand b∈(Cl1×Cl2× · · · × Cln)\Usuch that |a−b|<r 2 holds. Hence, from Lemma 13 and Proposition 4, Uis not a domain of Cn-holomorphy. ■ Lemma 15 : Let {Km}∞ m=0 be a sequence of compact subsets of U. Let {pm}∞ m=1 be a sequence in U. Suppose that U=∪∞ m=0 (Km◦) holds and for any nonnegative integer m,Km⊂Km+1 and pm+1 ∈Km+1 \d Km U l1,l2,···,lnhold. Then, there exists f∈Ol1,l2,···,ln(U) such that for any m∈N,m≤ |f(pm)|holds. Proof : From p16∈ c K0 U l1,l2,···,ln, there exists g1∈Ol1,l2,···,ln(U) such that supw∈K0|g1(w)|<|g1(p1)|holds. There exists c1∈(0,+∞) such that supw∈K0|c1g1(w)|<1<|c1g1(p1)|holds. Then, there exists k1∈N such that supw∈K0|(c1g1(w))k1| ≤ 1 20and 2 + P0 j=1 |(cjgj(p1))kj|(= 2) ≤ |(c1g1(p1))k1|hold. From p26∈ c K1 U l1,l2,···,ln, there exists g2∈Ol1,l2,···,ln(U) such that supw∈K1|g2(w)|<|g2(p2)|holds. There exists c2∈(0,+∞) such that supw∈K1|c2g2(w)|<1<|c2g2(p2)|holds. Then, there exists k2∈Nsuch that supw∈K1|(c2g2(w))k2| ≤ 1 21and 3 + P1 j=1 |(cjgj(p2))kj| ≤ |(c2g2(p2))k2|hold. Hereinafter, in the same manner, there exists a sequence {(gm, cm, km)}∞ m=1 such that for any m∈N,gm∈Ol1,l2,···,ln(U), 7
cm∈(0,+∞), km∈N, supw∈Km−1|(cmgm(w))km| ≤ 1 2m−1and 1 + m+ Pm−1 j=1 |(cjgj(pm))kj| ≤ |(cmgm(pm))km|hold. For any m∈N, supw∈Km−1(P∞ j=m|(cjgj(w))kj|)≤P∞ j=m(supw∈Kj−1|(cjgj(w))kj|)≤ P∞ j=m 1 2j−1=1 2m−2holds. So, f:= P∞ m=1((cmgm)km)∈Ol1,l2,···,ln(U) holds. For any m∈N, 1 + m+|(cmgm(pm))km| = 1 + m+f(pm)− m−1 X j=1 ((cjgj(pm))kj)!+ ∞ X j=m+1 ((cjgj(pm))kj)!! ≤1 + m+|f(pm)|+ m−1 X j=1 |(cjgj(pm))kj|!+ ∞ X j=m+1 |(cjgj(pm))kj|! ≤ ∞ X j=m+1 |(cjgj(pm))kj|!+|f(pm)|+|(cmgm(pm))km| and, so, 1 + m ≤ ∞ X j=m+1 |(cjgj(pm))kj|!+|f(pm)| ≤ ∞ X j=m+1 ( sup w∈Kj−1 |(cjgj(w))kj|)!+|f(pm)| ≤ ∞ X j=m+1 1 2j−1!+|f(pm)| =1 2m−1+|f(pm)| ≤1 + |f(pm)| hold. ■ Lemma 16 : Suppose that Uis Cn-holomorphically convex. Suppose U6=Cl1×Cl2× · · · × Cln. Let {ak}∞ k=1 be a sequence in U. For k∈N, let Bk:= {z∈U| |ak−z|<inf w∈(Cl1×Cl2×···×Cln)\U |ak−w| }. 8
Then, there exists f∈Ol1,l2,···,ln(U) such that for any k∈N, sup z∈Bk |f(z)|= +∞ holds. Proof : Let ((q1),(q2, q3),(q4, q5, q6),(q7, q8, q9, q10),· · · ) := ((a1),(a1, a2),(a1, a2, a3),(a1, a2, a3, a4),· · · ). Then, {qm}∞ m=1 is a sequence in Uand for any k∈Nand l∈N, there exists m∈Nsuch that ak=qmand l≤mhold. Let r0:= 1, R0:= 1 and K0 := ( ∩w∈(Cl1×Cl2×···×Cln)\U{z∈Cl1×Cl2× · · · × Cln|r0≤ |z−w| } ) ∩ { z∈Cl1×Cl2× · · · × Cln| |z| ≤ R0}. Then, K0is a compact subset of Uand, so, c K0 U l1,l2,···,lnis a compact subset of U. Hence, there exists p1∈U\c K0 U l1,l2,···,lnsuch that |q1−p1|< inf w∈(Cl1×Cl2×···×Cln)\U|q1−w|and inf w∈(Cl1×Cl2×···×Cln)\U|p1−w| ≤ 1 2r0 hold. Let r1:= inf w∈(Cl1×Cl2×···×Cln)\U|p1−w|, R1:= max{|p1|,2R0}and K1 := ( ∩w∈(Cl1×Cl2×···×Cln)\U{z∈Cl1×Cl2× · · · × Cln|r1≤ |z−w| } ) ∩ { z∈Cl1×Cl2× · · · × Cln| |z| ≤ R1}. Then, p1∈K1\c K0 U l1,l2,···,ln, 0 < r1≤1 2r0<+∞and 0 <2R0≤R1<+∞ hold. So, K1and c K1 U l1,l2,···,lnare compact subsets of U. Hence, there exists p2∈U\c K1 U l1,l2,···,lnsuch that |q2−p2|<inf w∈(Cl1×Cl2×···×Cln)\U|q2−w|and inf w∈(Cl1×Cl2×···×Cln)\U|p2−w| ≤ 1 2r1hold. Let r2:= inf w∈(Cl1×Cl2×···×Cln)\U|p2− w|, R2:= max{|p2|,2R1}and K2 := ( ∩w∈(Cl1×Cl2×···×Cln)\U{z∈Cl1×Cl2× · · · × Cln|r2≤ |z−w| } ) ∩ { z∈Cl1×Cl2× · · · × Cln| |z| ≤ R2}. 9