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710: １３２人目の素数さん [] 2023/07/16(日) 06:08:57.49 ID:Gig56QD8 References [A] Asserda, S., The Levi problem on projective manifolds, Math. Z. 219 (1995), no. 4, 631-636. [B-P] B locki, Z. and Pflug, P., Hyperconvexity and Bergman completeness, Nagoya Math. J., 151 (1998), 221-225. [C] Chen, B.-Y., Bergman completeness of hyperconvex manifolds, Nagoya Math. J. 175 (2004), 165-170. [C-D] Coltoiu, M. and Diederich, K., The Levi problem for Riemann domains over Stein spaces with isolated singularities, Math. Ann. 338 (2007), 2, 283-289. [C-L] Coeure, G. and Loeb, J. J., A counterexample to the Serre problem with a bounded domain of C 2 as fiber, Ann. of Math. 122 (1985), 329-334. [Dm] Demailly, J.-P., Estimations L 2 pour l’op´erateur ∂¯¯ d’un fibr´e vectoriel holomorphe semi-positif au-dessus d’une vari´et´e k¨ahl´erienne compl`ete, Ann. Sci. Ecole Norm. Sup. (4) ´ 15 (1982), no. 3, 457-511. [F] Fornaess, J.E., A counter-example for the Levi problem for branched Riemann domains over C n, Math. Ann. 234 (1978), 275-277. [G-1] Grauert, H., On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460-472. [G-2] ——, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81 (1963), 377-391. [H] Herbort, G., The Bergman metric on hyperconvex domains, Math. Z., 232 (1999), 183-196. [K-Oh] Kai, C. and Ohsawa, T., A note on the Bergman metric of bounded homogeneous domains, Nagoya Math. J. 186 (2007), 15 http://rio2016.5ch.net/test/read.cgi/math/1687778456/710

711: １３２人目の素数さん [] 2023/07/16(日) 06:11:57.12 ID:Gig56QD8 [M-2] ——, On quotients of bounded homogeneous domains by unipotent discrete groups, arXiv:2202.10283v2 [math.CV] [Mi] Miyazawa, K., A remark on the paper “On the infinite dimensionality of the middle L 2 cohomology of complex domains”, preprint (in Japanese) [Oh-1] Ohsawa, T., Vanishing theorems on complete K¨ahler manifolds, Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, 21-38. [Oh-2] ——, On the infinite dimensionality of the middle L 2 cohomology of complex domains, Publ. Res. Inst. Math. Sci. 25 (1989), 499-502. [Oh-3] ——, L2 ∂¯-cohomology with weights and bundle convexity of certain locally pseudoconvex domains, to appear in Kyoto J. M. [Oh-4] ——, On the Levi problem on K¨ahler manifolds under the negativity of canonical bundles on the boundary, Pure Appl. Math. Q. 18 (2022), no. 2, 763- 771. [Oh-5] ——, Bundle convexity and kernel asymptotics on a class of locally pseudoconvex domains, to appear. [Oh-6] ——, Geometry of analytic continuation on complex manifolds —— history, survey and report, preprint. [Oh-7] ——, On the Levi problem under the negativity of canonical bundles on the boundary, preprint. [Oh-8] ——, On hyperconvexity and towards bundle-valued kernel asymptotics on locally pseudoconvex domains, Rev. Roumaine Math. pure et appl. 67 (2023), 1-1, 169-189. [P] Pinney, K. R., Line bundle convexity of pseudoconvex domains in complex mannifolds, Math. Z. 206 (1991), no. 4, 605-615 http://rio2016.5ch.net/test/read.cgi/math/1687778456/711

712: １３２人目の素数さん [] 2023/07/16(日) 06:13:37.28 ID:Gig56QD8 [R] Rosay, J.-P., Extension of holomorphic bundles to the disc (and Serre’s problem on Stein bundles), Ann. Inst. Fourier, Grenoble 57 (2007), 517-523. [S] Siu, Y.-T., Holomorphic fiber bundles whose fibers are bounded Stein domains with zero first Betti number, Math. Ann. 219 (1976), 171-192. [St] Stehl´e, J.-L., Fonctions plurisousharmoniques et convexit´e holomorphe de certains fibr´es analytiques, Lecture Notes in Math., Vol. 474, Springer, Berlin, 1975, pp. 155-179. [T] Takayama, S., The Levi problem and the structure theorem for non-negatively curved complete K¨ahler manifolds, J. Reine Angew. Math. 504 (1998), 139-157. http://rio2016.5ch.net/test/read.cgi/math/1687778456/712

713: １３２人目の素数さん [] 2023/07/16(日) 08:25:34.67 ID:Gig56QD8 NOTES ON THE BUNDLE OF COEURE AND LOEB ´ Abstract A weak holomorphic-convexity property of a fiber bundle constructed by Coeur´e and Loeb will be proved after an observation that it admits a complete K¨ahler metric. Other remarks on their geometric and function theoretic properties will be presented, too. In particular, it will be shown that there exists a locally pseudoconvex branched Riemann domain over C2 which is holomorphically separable. http://rio2016.5ch.net/test/read.cgi/math/1687778456/713

714: １３２人目の素数さん [] 2023/07/16(日) 08:28:27.84 ID:Gig56QD8 Introduction Given a complex manifold M and a holomorphic vector bundle E → M, the notion of E-convexity of M was introduced by Grauert [G-2] by generalzing holomorphic convexity. Then, for locally pseudoconvex smooth bounded domains Ω ⋐ M, Pinney [P] showed that Ω is Econvex in a suitably weakened sense if E is a line bundle euqipped with a fiber metric with positive curvature. Here the E-convexity of Pinney is ∀γ ∈ Ω^N s.t. γ(N) /⋐ Ω ∃s ∈ H^{0,0}(Ω, E) s.t. s(γ(N)) /⋐ E. http://rio2016.5ch.net/test/read.cgi/math/1687778456/714

715: １３２人目の素数さん [] 2023/07/16(日) 08:30:58.43 ID:Gig56QD8 The E-convexity in this sense was verified by Asserda [A] assuming the compactness of M and quite recently by [Oh-3,5] in other situations including the cases where ∂Ω is a proper analytic set of M, under the assumtion of the curvature positivity of E|∂Ω Compared to the classical holomorphically convex cases, not so much more has been known on E-convex domains. For instance, the precise boundary behavior of the bundle-valued Bergman kernels is not known (cf. [Oh-5]). On the other hand, in some cases the bundle-convexity is available to analyze multiplier ideal sheaves. As a result, Grauert’s characterization of Stein manifolds in [G-1] was extended by Takayama [T] as the holomorphic convexity of weakly 1-complete manifolds with negative cananical bundles. http://rio2016.5ch.net/test/read.cgi/math/1687778456/715

716: １３２人目の素数さん [] 2023/07/16(日) 08:33:22.39 ID:Gig56QD8 More recently, [T] was extended in [Oh-6] as the holomorphic convexity of weakly 1-complete manifolds whose canonical bundle is negative at infinity (i.e. on the complement of a compact subset of the manifold). Another extension of [T] was obtained in [Oh-4,7] asserting a similar conclusion under certan regularity or curvature assumptions of ∂Ω. In the latter extension, the conclusion is weaker than the genuine holomorhic convexity, because it only says that the domain is properly mapped onto a locally closed analytic set in some C^N. http://rio2016.5ch.net/test/read.cgi/math/1687778456/716

717: １３２人目の素数さん [] 2023/07/16(日) 08:35:09.19 ID:Gig56QD8 This seems to suggest that it may be worthwhile to study locally pseudoconvex domains in complex manifolds by focusing on the finer structures of function spaces. Since the well-known Coeur\'e-Loeb's counterexample to the Serre problem is a locally pseudoconvex domain of similar type which actually arises in nature, we would like to study here some of its function theoretic properties from this viewpoint. http://rio2016.5ch.net/test/read.cgi/math/1687778456/717

718: １３２人目の素数さん [] 2023/07/16(日) 08:38:32.54 ID:Gig56QD8 Let us recall that the Serre problem asks whether or not analytic fiber bundles over Stein manifolds with Stein fibers are Stein. Both positive and negative answers are known to contain significant contents. In particular, counterexamples sometimes share interesting function-theoretic properties with Stein manifolds, such as the Oka's principle (cf. [R]). After recalling Coeur\'e-Loeb's example, we shall prove the following. Theorem 1. There exists a logarithmically convex bounded Reinhardt domain F in C^2 and an analytic fiber bundle over C^*:= C-{0} with F as fibers which is not Stein but admits a complete K\"ahler metric. http://rio2016.5ch.net/test/read.cgi/math/1687778456/718

719: １３２人目の素数さん [] 2023/07/16(日) 17:32:03.53 ID:Gig56QD8 In [C-D] it was remarked that the total space of the above bundle can be realized as a branched Riemann domain over C^3 . In Theorem 1, F can be chosen in such a way that it admits a fixed point free automorphism generating an infinite cyclic subgroup Γ of AutF such thatthe bundle is C∗ ×ρ F for an isomorphism ρ between the fundamentalgroup of C∗ and Γ. As a by-product of Theorem 1 we shall show that F/Γ, which is a fortiori non-Stein, can be realized as a branched locally pseudoconvex Riemann domain over C^2. http://rio2016.5ch.net/test/read.cgi/math/1687778456/719

720: １３２人目の素数さん [] 2023/07/16(日) 17:33:15.54 ID:Gig56QD8 Coeure-Loeb’s bundle ´ Let Ω be a bounded Stein domain in C n , let AutΩ be the group of biholomorphic self-maps of Ω and let Γ be an infinite cyclic properly discontinuous subgroup of AutΩ generated (automatically) by a fixed point free element. It is known that the quotient manifold Ω/Γ is not always Stein. Before the construction of the domain F in Theorem 1, we shall discuss at first several questions related to this phenomenon. Recall that Ω/Γ is Stein if AutΩ is transitive (cf. [M-1]). http://rio2016.5ch.net/test/read.cgi/math/1687778456/720

721: １３２人目の素数さん [] 2023/07/16(日) 17:36:13.85 ID:Gig56QD8 Therefore, in particular, an analytic fiber bundle over the punctured disc D∗:= {z ∈ C; 0 < |z| < 1} whose fiber is biholomorphic to a bounded homogeneous domain Ω0 is Stein if it arises as the infinite cyclic quotient of the product D × Ω0 associated to a nontrivial homomorphism ρ : π_1(D∗) → AutΩ0. A theorem of Siu [S] says more generally that analytic fiber bundles over Stein manifolds with fibers equivalent to a bounded pseudoconvex domains in C^n with zero first Betti number is Stein. It is also proved in [S] that holomorphic functions separates the points of the bundle if the base is Stein and the fiber is a bounded pseudoconvex domain in Cn . http://rio2016.5ch.net/test/read.cgi/math/1687778456/721

722: １３２人目の素数さん [] 2023/07/16(日) 17:54:05.49 ID:Gig56QD8 If Ω is homogeneous, the Bergman metric on Ω defined as ∂¯∂ log K_Ω(z, z)　from the Bergman kernel KΩ(z, w) of Ω is invariant under the action of AutΩ, so that it is a complete K¨ahler metric on Ω. Furthermore, the function log K_Ω(z, z) has bounded gradient with respect to∂¯∂ log K_Ω(z, z) and limz→∂Ω log K_Ω(z, z) = ∞ (cf. [K-Oh]). Therefore Ω is hyperconvex in the sense of Stehl´e [St], i.e. Ω admits a strictly plurisubharmonic bounded exhaustion function. So the Steinness of analytic Ω bundles over Stein manifolds follows also from Stehl´e’s theorem in [St]. http://rio2016.5ch.net/test/read.cgi/math/1687778456/722

723: １３２人目の素数さん [] 2023/07/16(日) 19:08:26.44 ID:Gig56QD8 On the other hand, Coeur’e and Loeb [C-L] constructed a bounded pseudoconvex Reinhardt domain F in C^2 satisfying the following property; For the bounded domain Ω = {z ∈ C; |ζ| < 1}×F in C^3, there exists an element σ ∈ AutF such that the element ˆσ ∈ AutΩ defined by σˆ(ζ, z) := ((2i − 1)ζ + 1)/(−ζ + 1 + 2i), σ(z)) generates an infinite cyclic group Γ := {σˆµ; µ ∈ Z} by which Ω has a non-Stein quotient Ω := Ω ˆ /Γ. http://rio2016.5ch.net/test/read.cgi/math/1687778456/723

724: １３２人目の素数さん [] 2023/07/16(日) 19:09:15.56 ID:Gig56QD8 Therefore, one will have a non-Stein fiber bundle over D ∗ with fiber F in this way. By the construction, this fiber bundle is naturally extended to a bundle over C ∗ . By Stehl´e’s theorem F is not hyperconvex. In fact, the Bergman metric on F is not complete. ( It is due to [B-P], [H] and [C] that hyperconvex manifolds have complete Bergman metrics.) Nevertheless, F has a complete K¨ahler metric which is invariant by some nontrivial action of an infinite cyclic group. Since this is the main ingredient of Theorem 1, let us recall the construction of F below http://rio2016.5ch.net/test/read.cgi/math/1687778456/724

725: １３２人目の素数さん [] 2023/07/16(日) 19:11:03.67 ID:Gig56QD8 Construction of the domain F We put H = {z ∈ C; Imz > 0}, T = ( 1+√ 5 2 1− √ 5 2 1 1 ) , V = T(H2 ) and F = V/Z 2 . Here Z 2 acts on V by ( z1 z2 ) 7→ ( z1 + 1 z2 ) and ( z1 z2 ) 7→ ( z1 z2 + 1 ) . さすがにもう無理 http://rio2016.5ch.net/test/read.cgi/math/1687778456/725

726: １３２人目の素数さん [] 2023/07/16(日) 21:12:36.39 ID:Gig56QD8 It is easy to see by a direct computation that the set $\{\sigma^k_A(v_1,v_2); k\in \mathbb{Z}\}$ has no accumulation points in $F$, so that $\hat{F}$:=$F/\{\sigma^k_A; k\mathbb{Z}\}$ is a complex manifold. $\hat{F}$ is non-Stein since so is $\hat{\Omega}$. Since $F$ has a $\sigma_A$-invariant complete K\"ahler metric, $\hat{F}$ has also a complete K\"ahler metric. Combining this with the invariance of $du_1\wedge du_2$, the $\sigma_A$-invariance of the Bergman kernel function of $F$ follows. Thus $\hat{F}$ is a complete K\"ahler manifold with trivial canonical bundle which is positive. Therefore, by the $L^2$ method one can conclude that $\hat{F}$ is holomorphically separable. http://rio2016.5ch.net/test/read.cgi/math/1687778456/726

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