Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/335893633 Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems Preprint · September 2019 DOI: 10.13140/RG.2.2.34948.88967 CITATIONS 0 2 authors, including: Hiroki Yagisita Kyoto Sangyo University 10 PUBLICATIONS22 CITATIONS SEE PROFILE All content following this page was uploaded by Hiroki Yagisita on 18 September 2019. The user has requested enhancement of the downloaded file. Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems Hiroki Yagisita (Kyoto Sangyo University) Abstract: Let Abe a commutative Banach algebra. Let Mbe a complex manifold on A(an A-manifold). Then, we define an A-holomorphic vector bundle (∧kT∗)(M) on M. For an open set Uof M,ωis said to be an A-holomorphic differential k-form on U, if ωis an A-holomorphic section of (∧kT∗)(M) on U. So, if the set of all A-holomorphic differential k-forms on Uis denoted by Ωk M(U), then {Ωk M(U)}Uis a sheaf of modules on the structure sheaf OMof the A-manifold Mand the cohomology group Hl(M, Ωk M) with the coefficient sheaf {Ωk M(U)}Uis an OM(M)-module and therefore, in particular, an Amodule. There is no new thing in our definition of a holomorphic differential form. However, this is necessary to get the cohomology group Hl(M, Ωk M) as an A-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of C-manifolds (and also the one of an analytic family). Directing attention to a finite family of C-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of Cmanifolds. Also, we state a few related problems. One of them is the following. Let n∈N. Then, does there exist a Cn-manifold Nsuch that (1) for any C-manifolds M1, M2,··· , Mn−1and Mn,Ncan not be embedded in the direct product M1×M2×···×Mn−1×Mnas a Cn-submanifold but (2) there exists k∈Nsuch that Ncan be embedded in the k-dimensional Euclidean Rn-space (Rk)nas an Rn-submanifold ? So, we propose something that is likely to be a candidate for such a C2-manifold N. Keywords: additive Cousin problem, foliation, foliated manifold, K-group, Riemann-Roch theorem, Gelfand representation, von Neumann algebra, Radon measure, L∞space, coherent analytic sheaf, domain of holomorphy, holomorphically convex, ∂equation, Levi problem, pseudoconvex manifold, Stein manifold, Kahler manifold, immersion, Kodaira embedding theorem, Kodaira vanishing theorem, harmonic integral, relative de Rham resolution, Weierstrass preparation theorem, projective algebraic variety, Chow’s theorem, Serre’s GAGA, vector sheaf, non-Hausdorff manifold. 1 1 Definitions and problems From §2 of [8], we follow some terms (e.g., commutative Banach algebra, Banach A-module, etc.). Let Abe a commutative Banach algebra and be fixed. Definition 1 (Topological module) : Xis said to be a topological A-module, if Xis an A-module, it is a topological C-linear space, (c1A)u=cu (c∈C, u ∈X) holds and the map (a, u)∈A×X7→ au ∈X is continuous. Remark : If one is a Banach A-module, then it is a topological A-module. — Definition 2 (Linear mapping) : Let Xand Ybe A-modules. A mapping F:X→Yis said to be A-linear, if it satisfies F(u+v) = F(u) + F(v) ( u, v ∈X), F(fu) = fF(u) ( f∈A, u ∈X). Definition 3 (Continuous multilinear mapping) : Let X1, X2,··· , Xkand Ybe topological A-modules. A map ffrom X1×X2× ··· × Xkto Yis said to be (A, k)-linear, if fis A-linear with respect to each variable xi∈Xi. Let LA(X1, X2,··· , Xk;Y) denote the set of all continuous (A, k)-linear mappings from X1×X2×···×Xkto Y. LA(X1, X2,··· , Xk;Y) is an A-module. Remark : If one is (A, k)-linear, then it is (C, k)-linear. — Definition 4 (Norm of a multilinear mapping) : Let X1, X2,··· , Xkand Ybe Banach A-modules. For an (A, k)-linear mapping ffrom X1×X2×···×Xkto Y, let kfk:= sup {kf(x1, x2,··· , xk)kY| kx1kX1=kx2kX2=··· =kxkkXk= 1 }. — 2 Lemma 5 : Let X1, X2,··· , Xkand Ybe Banach A-modules. Then, for any (A, k)- linear mapping ffrom X1×X2×···×Xkto Y,fis continuous if and only if kfk<+∞holds. LA(X1, X2,··· , Xk;Y) is a Banach A-module. Proof : It is easy. ■ Definition 6 (Continuous antisymmetric form) : Let Xbe a topological A-module. A mapping ffrom Xkto Ais said to be an (A, k)-linear form of X, if fis (A, k)-linear. An (A, k)-linear form fof Xis said to be antisymmetric, if for any permutation σ,f(xσ(1), xσ(2),··· , xσ(k)) = sgn(σ)·f(x1, x2,··· , xk) holds. Let Ak A(X) denote the set of all continuous antisymmetric (A, k)-linear forms of X.Ak A(X) is an A-submodule of LA(X, X, ··· , X;A). Lemma 7 : Let Xbe a Banach A-module. Then, Ak A(X) is a Banach A-submodule of LA(X, X, ··· , X;A). Proof : It is easy. ■ Definition 8 (Pull back) : Let X1and X2be topological A-modules. For a map Ffrom X1to X2 and a map ffrom Xk 2to A, define a map F∗(f) from Xk 1to Aby (F∗(f)) (x1, x2,··· , xk) := f(F(x1), F(x2),··· , F(xk)). If F∈LA(X1;X2) and f∈Ak A(X2) hold, then F∗(f)∈Ak A(X1) holds. — Lemma 9 : Let X1and X2be topological A-modules. Suppose that Fis a bijection from X1to X2. Suppose that Fand F−1are continuous and A-linear. Then, F∗ ↾Ak A(X2)is a bijection to Ak A(X1). F∗ ↾Ak A(X2)and F∗ ↾Ak A(X2)−1are A-linear. Further, if X1and X2are Banach A-modules, then F∗ ↾Ak A(X2)and F∗ ↾Ak A(X2)−1 are continuous. Proof : It is easy. ■ Definition 10 (Banach-like module) : Xis said to be a Banach-like A-module, if Xis a topological A-module and there exist a Banach A-module Yand a bijection Ffrom Xto Ysuch that Fand F−1are continuous and A-linear. — Lemma 11 : Let Xbe a Banach-like A-module. Then, Ak A(X) is a Banach-like Amodule. Proof : It follows from Lemmas 7 and 9. ■ 3 Definition 12 (Differentiable mapping) : Let Xand Ybe Banach A-modules. Let fbe a mapping from an open set Uof Xto Y.fis said to be A-differentiable (on U), if fis Frechet differentiable and for any p∈U, the Frechet derivative (Df)pis A-linear. — Definition 13 (Manifold on a commutative Banach algebra) : Let Mbe a Hausdorff space. Let Sbe a set. Mis said to be an Amanifold with the system Sof coordinate neighborhoods, if the followings hold. For any φ∈S, there exists a Banach A-module Xsuch that φis a homeomorphism from an open set of Mto an open set of X. For any p∈M, there exists φ∈Ssuch that pbelongs to the domain of φ. For any φ1, φ2∈S, the coordinate transformation φ2◦φ−1 1:φ1(U1∩U2)→φ2(U1∩U2) is A-differentiable. Here, U1and U2are the domains of φ1and φ2, respectively. — Let Mbe an A-manifold with the system Sof coordinate neighborhoods and be fixed. For φ∈S, we denote the domain of φby Uφand the Banach A-module such that φ(Uφ) is an open set of it by Xφ. For p∈M, we denote the set of all φ∈Ssuch that p∈Uφholds by Sp. Definition 14 (Tangent space) : Let p∈M. As ˙p1∼˙p2indicates that there exist φ1, φ2∈Spsuch that ˙p2= (D(φ2◦φ−1 1))φ1(p)( ˙p1) holds, ∼is an equivalence relation of ∪φ∈SpXφ. Let Tp(M) denote the quotient set (∪φ∈SpXφ)/∼. The tangent space Tp(M) is a Banach-like A-module. — Definition 15 (Cotangent exterior space) : Let p∈M. Let (∧kT∗)p(M) denote Ak A(Tp(M)). The cotangent exterior space (∧kT∗)p(M) is a Banach-like A-module. — Definition 16 (Holomorphic mapping between complex manifolds on a commutative Banach algebra) : Let M1be an A-manifold with a system S1of coordinate neighborhoods. Let M2be an A-manifold with a system S2of coordinate neighborhoods. Let f:M1→M2be a continuous mapping. fis said to be A-holomorphic, if for any φ1∈S1and φ2∈S2, the mapping φ2◦f◦φ−1 1:φ1(U1∩f−1(U2)) →X2 4 is A-differentiable. Here, φ1is a mapping from U1to a Banach A-module X1 and φ2is a mapping from U2to a Banach A-module X2. Remark : Let fbe a map from an open set of a Banach A-module to a Banach A-module. Then, fis A-holomorphic if and only if fis A-differentiable. — Definition 17 (Finite direct product of Banach modules) : Let X1, X2,··· , Xk−1and Xkbe Banach A-modules. Let k(x1, x2,··· , xk)k:= max lkxlkXl (x1∈X1, x2∈X2,··· , xk∈Xk). The finite direct product X1×X2×···×Xkis a Banach A-module. Definition 18 (Holomorphic vector bundle on a complex manifold on a commutative Banach algebra) : E:= (E, M, π, {(Uλ, Xλ, φλ)}λ∈Λ) is said to be an A-holomorphic vector bundle (on M), if it satisfies the followings. Eand Mare A-manifolds. πis an A-holomorphic surjection from Eto M. Each Uλis an open set of M.M=∪λ∈ΛUλholds. Each φλis a map from π−1(Uλ) to a Banach A-module Xλ. For λ∈Λ, let π|λ:= π↾π−1(Uλ). Each map (φλ, π|λ) : π−1(Uλ)→Xλ×Uλ is an A-biholomorphic map. For λ∈Λ and p∈Uλ, let φλ|p:= φλ↾π−1({p}). For any λ1, λ2∈Λ and p∈Uλ1∩Uλ2, the coordinate transformation φλ2|p◦φλ1|−1 p:Xλ1→Xλ2 is A-linear. — Definition 19 (Tangent bundle) : Let T(M) denote ∪p∈MTp(M). — Proposition 20 : The tangent bundle T(M)→Mis an A-holomorphic vector bundle. Proof : It is a corollary of Proposition 34 in Section 2. ■ 5 Definition 21 (Cotangent exterior bundle) : Let (∧kT∗)(M) denote ∪p∈M(∧kT∗)p(M). — Proposition 22 : The cotangent exterior bundle (∧kT∗)(M)→Mis an A-holomorphic vector bundle. Proof : It is a corollary of Proposition 38 in Section 2. ■ Theorem 23 (Probably well-known) : Let A=C. Let Mbe an n-dimensional complex manifold. For an open set Uof M, the set of all holomorphic sections of (∧kT∗)(M) on Uis denoted by Ωk M(U). Then, the sheaf {Ωk M(U)}Uon Mis isomorphic to the sheaf of germs of holomorphic differential k-forms on Mas a sheaf of modules on the sheaf OMof germs of holomorphic functions on M. Proof : As we define the correspondence Fkby F1(dzi) : n ∑ j=1 ˙zj∂ ∂zj∈Tp(M)7→ ˙zi∈C ( ˙z= ( ˙z1,˙z2,··· ,˙zn)∈Cn), Fk(dzi1∧dzi2∧···∧dzik) := ∑ σ∈Sk sgn(σ)F1(dziσ(1) )⊗F1(dziσ(2) )⊗···⊗F1(dziσ(k)), it is a somewhat difficult exercise of linear algebras. ■ The sheaf of germs of A-valued A-holomorphic mappings on Mis denoted by OM. From the above, we get the following definition. Definition 24 (Holomorphic differential form of a complex manifold on a commutative Banach algebra) : Let Ube an open set of M.ωis said to be an A-holomorphic differential k-form on U, if it is an A-holomorphic section of (∧kT∗)(M) on U. Let Ωk M(U) denote the set of all A-holomorphic differential k-forms on U. Then, {Ωk M(U)}Uis a sheaf of modules on the sheaf OMof germs of A-valued Aholomorphic mappings on M. The cohomology group Hl(M, Ωk M) with the coefficient sheaf {Ωk M(U)}Uis an OM(M)-module and, in particular, an Amodule. Remark : In the coefficient sheaf {Ωk M(U)}U,Umay be limited only to all open sets of Min an appropriate (weak) topology depending on the problem. — 6 Problem 25 : Let Nbe a compact continuous family of connected n-dimensional Cmanifolds on a compact Hausdorff space X. Let Γ(N) denote the set of all continuous sections of Non X. Then, Γ(N) is a C(X)-manifold. (see [8].) So, if Mis an open set of Γ(N), then the cohomology group Hl(M, Ωk M) is an OM(M)-module and, in particular, a C(X)-module. (1) Let Mbe a connected component of Γ(N). Is OM(M) = C(X) ? Seek more specific indications of the sheaf Ωk Mand the cohomology group Hl(M, Ωk M). When is Hl(M, Ωk M) a finitely generated projective C(X)-module ? Define the Euler characteristic χ(M, OM) = ∑l(−1)l[Hl(M, OM)] ∈ K(X) = K(C(X)) and seek its Riemann-Roch indication. (2) Let M1and M2be connected components of Γ(N). Then, when are Hl(M1,Ωk M1) and Hl(M2,Ωk M2) isomorphic as C(X)-modules ? Problem 26 : (1) Define a differential (p, q)-form and the Dolbeault operator ∂of an A-manifold. (For the case of an infinite-dimensional C-manifold, see [3].) (2) Let D1, D2,··· , Dn−1and Dnbe open disks of A. Then, is for any l≥1, Hl(D1×D2×···×Dn, OD1×D2×···×Dn) = 0 ? (3) Let Ube a connected open set of An. Let Fbe an OU-module. Then, when is for any l≥1, Hl(U, F) = 0 ? For example, when A= C({0,1,2,··· , m −1}) = Cmholds, define that a connected open set Uof Cmn is Cm-Stein. Also, define a Cm-coherent analytic sheaf on U. (4) Let Mbe an A-real analytic manifold. Then, define the sheaf of germs of A-real valued A-real continuous mappings on Mand the one of A-real valued A-real hyperfunctions on M. Remark : Related to some of Problems 25 and 26, see Appendix 1. Perhaps, it may be meaningful to have Ekas the sheaf of germs of C∞-functions on Rkin 0→En,x ×{0y} → En,x ×Em,y → {0x}×Em,y →0. Apparently, a resolution like Dolbeault ones and de Rham ones intertwined seems to be a fine one of the structure sheaf of a Cm-manifold. In Appendix 2, we tried to define an analog of singular homology theory for a continuous family of topological spaces. In Appendix 3, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of Cmanifolds. In Appendix 4, we try to define the structure sheaf of a manifold that is locally a continuous family of C-manifolds (and also the one of an analytic family). — 7 Additional Problem : Consider the following 1-dimensional C2-manifold N. It consists of four coordinate neighborhoods φk:Wk→C2(k= 1,2,3,4). That is, N=∪4 k=1Wk. Let φ1(W1) : = C× {z2∈C|Im z2>0}, φ2(W2) : = C× {z2∈C|Im z2<0}, φ3(W3) : = {z1∈C||z1|<1} × C, φ4(W4) : = {z1∈C||z1|<1} × C. Let W1∩W2=∅and W3∩W4=∅. Let c1and c2be real numbers such that 0< c1< c2<+∞ holds. Let φ3(W1∩W3) = {z1∈C||z1|<1} × {z2∈C|Im z2>+c1}, (φ1◦(φ−1 3)) (z1, z2)=(z1+ (+2), z2+ (−c1√−1) ). Let φ4(W1∩W4) = {z1∈C||z1|<1} × {z2∈C|Im z2>+c2}, (φ1◦(φ−1 4)) (z1, z2)=(z1+ (−2), z2+ (−c2√−1) ). Let φ3(W2∩W3) = {z1∈C||z1|<1} × {z2∈C|Im z2<−c1}, (φ2◦(φ−1 3)) (z1, z2)=(z1+ (+2), z2+ (+c1√−1) ). Let φ4(W2∩W4) = {z1∈C||z1|<1} × {z2∈C|Im z2<−c2}, (φ2◦(φ−1 4)) (z1, z2)=(z1+ (−2), z2+ (+c2√−1) ). Then, answer the following question. Do C-manifolds M1and M2exist such that Nis embedded in M1×M2as a C2-manifold ? (If we allow that M1and M2are not Hausdorff, what about ?) — 8 Proposition 38 : The Hausdorff space (∧kT∗)(M) is an A-manifold with {(φ∧kT∗, φ ◦ π|φ)}φ∈Sas the system of coordinate neighborhoods. The A-manifold (∧kT∗)(M) is an A-holomorphic vector bundle on Mwith {(φ∧kT∗, π|φ)}φ∈Sas the system of local trivialization coordinate neighborhoods. Proof : It follows from Proposition 28 and Lemmas 30, 37. ■ Acknowledgment: This work was supported by JSPS KAKENHI Grant Number JP16K05245. ArXiv does not seem to accept frequent revisions. The revised version may have been put in “https://www.researchgate.net/profile/Hiroki_Yagisita”. 15 Appendix 1: Let Xbe a compact Hausdorff space. Let Ube a convex open set of Rn. Let Fbe a C1-map from C(X;U) to C(X;R). Suppose that for any u∈C(X;U), the Frechet derivative F′(u) of Fat uis C(X;R)-linear. Then, for any u0, u1∈C(X;U) and x∈X,u0(x) = u1(x) implies (F(u0))(x) = (F(u1))(x). So, there exists a function ffrom X×Uto Rsuch that for any u∈C(X;U) and x∈X, (F(u))(x) = f(x, u(x)) holds. Proof : From F(u1)−F(u0) = ∫1 0 (F′((1 −t)u0+tu1)) (u1−u0)dt =∫1 0 (F′((1 −t)u0+tu1)) ( n ∑ k=1 (u(k) 1−u(k) 0)ek)dt = n ∑ k=1 ∫1 0 (u(k) 1−u(k) 0) (F′((1 −t)u0+tu1))(ek)dt, (F(u1))(x)−(F(u0))(x) = n ∑ k=1 ∫1 0 (u(k) 1(x)−u(k) 0(x)) ((F′((1 −t)u0+tu1))(ek))(x)dt = n ∑ k=1 ∫1 0 0 ((F′((1 −t)u0+tu1))(ek))(x)dt = 0 holds. ■ 2: I try to define an analog of singular homology theory for a continuous family of topological spaces, but I do not know whether it will work or not. Let πbe a continuous mapping from a topological space Mto one X. Let ∆kdenote the standard k-simplex. Let Tπ kbe the set of all continuous mappings σ: ∆k×X→Msuch that for any (x, t)∈∆k×X, (π◦σ)(x, t) = t holds. Then, let Sπ kbe the free Z-module such that Tπ kis a base of Sπ k. In an appropriate class, it appears to be isomorphic to the usual singular homology. — 16 3: I am considering the following in the interview, but I do not know whether it will work or not. It might be related to a complex foliation. [Title] Dolbeault theorem for a topological sum of complex structures. [Abstract] For an open set Uof Cn×Rm, let O(U) denote the ring of all C-valued continuous functions f(z, t) on Usuch that f(z, t) is holomorphic with respect to the several complex variables z∈Cn. Then, {O(U)}Uis a sheaf of commutative rings on Cn×Rm. We show Dolbeault theorem Hq(U, OU)∼ =Hq ∂z(U, OU). Further, we show that if Dis an open polydisk of Cn,Tis an open set of Rmand U=D×Tand q≥1 hold, then this cohomology is vanishing. So, it is shown that for a continuous family of additive Cousin data on an open polydisk, there exists a continuous family of solutions of the first problem. [Comments] An open set Uof Cn×Rmis a very simple example of a continuous sum of C-manifolds. On the other hand, an appropriate Banach C-manifold is a continuous product. A projection π:U→Rmis a simple example of a continuous family. It seems that a continuous sum and a continuous family have not been fully studied yet. A finite sum and a finite product are familiar, but it seems that a finite family is not paying much attention. Although a continuous sum and a continuous family are unexplored places, they may remain unexplored for a while in the future or may be stepped on in a moment. The C(X)-manifold Mcorresponding to a continuous family π:U→X and the total space Udo not seem to reflect all of the important structures of π. It is desirable to construct a consistent structure as a whole by adding some additional information to Mor U. For example, it seems that the sheaf {M|D}Don Xreflects most information on π. Here, for an open set Dof X, M|Dis the Cb(D)-manifold corresponding to π|D. — 4: We try to define the structure sheaf of a manifold that is locally a continuous family of C-manifolds (and also the one of an analytic family). [Continuous family] Let π:M→Xbe a continuous family of C-manifolds. Then, for an open set Uof M, let Oπ(U) be the set of all pairs (f, g) of C-valued continuous functions on Usuch that for any t∈X,f↾π−1({t})is locally constant and g↾π−1({t})is C-holomorphic. [Analytic family] Let π:M→Xbe an analytic family of C-manifolds. Then, for an open set Uof M, let Oπ(U) be the set of all pairs (f, g) of C-holomorphic functions on Usuch that for any t∈X,f↾π−1({t})is locally constant. — 17 5: A holomorphic vector bundle is a complex one, and a complex one is a real one. So, similarly, define a complex foliation. That is, a holomorphic foliation is a complex one, and a complex one is a real one. — 6: Let M→Xbe an analytic family of compact C-manifolds. Let Ube a relatively compact open set of X. Let Γ(M|U) denote the set of all continuous sections of Mon U. Then, Γ(M|U) is a Cb(U)-manifold. (see [8].) In some cases, a stronger structure may be introduced into the Cb(U)-manifold. The following problem is related to it. Let π1:M→Xand π2:M→X be analytic families. Suppose that Xis Stein and Γ(M)∼ =(Cb(X))nholds. Then, does π1∼ =π2hold ? (Oka-Grauert principle) — 7: Let Xbe a compact Hausdorff space. Then, define a C(X)-analytic set of a Banach C(X)-module. Also, define a C(X)-algebraic set of a finitely generated projective C(X)-module. — 8: Define a locally direct product space of complex analytic spaces and its underlying complex analytic space. Define a locally direct product space of complex algebraic varieties (or, schemes etc.) and its underlying complex algebraic variety (or, scheme etc.). However, the rudimentary general theory of a ringed space may bring a locally direct product space. On the other hand, an underlying space may be more difficult. — References [1] S. Araki, Topological K-theory (Japanese), Sugaku, 22 (1970), 60-76. [2] L. Hormander, L2estimates and existence theorems for the ∂operator, Acta Math., 113 (1965), 89-152. [3] L. Lempert, The Dolbeault complex in infinite dimensions, J. Amer. Math. Soc., 11 (1998), 485-520. [4] A. Mallios and E. E. Rosinger, Space-time foam dense singularities and de Rham cohomology, Acta Appl. Math., 67 (2001), 59-89. [5] T. Ohsawa and K. Takegoshi, On the extension of L2holomorphic functions, Math. Z., 195 (1987), 197-204. [6] H. Ozeki, Vector bundles and projective modules (Japanese), Sugaku, 18 (1967), 223-233. [7] M. H. Papatriantafillou, Partitions of unity on A-manifolds, Internat. J. Math., 9 (1998), 877-883. [8] H. Yagisita, Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds, arXiv.org. 18 View publication statsView publication stats