https://rio2016.5ch.net/test/read.cgi/math/1674744315/2 www.ma.huji.ac.il/hart/puzzle/choice.pdf Choice Games Sergiu Hart November 4, 2013 ”A similar result, but now without using the Axiom of Choice.GAME2” で、選択公理なしで同じことが成り立つから、”選択公理”は、単なる目くらましってことも暗示している
https://www.アマゾン 現代複素解析への道標 レジェンドたちの射程 Tankobon Hardcover – November 24, 2017 書評 susumukuni 5.0 out of 5 stars 複素解析の語り部によるレジェンドたちの射程 Reviewed in Japan on December 17, 2017 「吹田予想」（ベルグマン核と対数容量との間で成立する最良不等式）解決の関わりは著者の前著『岡潔 多変数関数論の建設』でも触れられているが、本書の最終章では「スタイン多様体の変形族に現れるベルグマン計量の対数劣調和性から、吹田予想や最良L2評価式付きの正則関数の拡張定理の別証明が得られる」というベルントソンとレンペルトによる最新の興味深い結果が紹介されており素晴らしい。
Abstract. A theorem asserting the existence of proper holomorphic maps with connected fibers to an open subset of C N from a locally pseudoconvex bounded domain in a complex manifold will be proved under the negativity of the canonical bundle on the boundary. Related results of Takayama on the holomorphic embeddability and holomorphic convexity of pseudoconvex manifolds will be extended under similar curvature conditions.
Abstract. A theorem asserting the existence of proper holomorphic maps with connected fibers to an open subset of C^N from a locally pseudoconvex bounded domain in a complex manifold will be proved under the negativity of the canonical bundle on the boundary. Related results of Takayama on the holomorphic embeddability and holomorphic convexity of pseudoconvex manifolds will be extended under similar curvature conditions.
This is a continuation of [Oh-5] where the following was proved among other things. Theorem 1.1. Let M be a complex manifold and let Ω be a proper bounded domain in M with
1036 名前：C^2-smooth pseudoconvex boundary ∂Ω. Assume that M admits a K¨ahler metric and the canonical bundle K_M of M admits a fiber metric whose curvature form is negative on a neighborhood of ∂Ω. Then there exists a holomorphic map with connected fibers from Ω to C^N for some N ∈ ℕ which is proper onto the image. The main purpose of the present article is to strengthen it by removing the K¨ahlerness assumption (see §2). For that, the proof of Theorem 0.1 given in [Oh-5] by an application of the L^2 vanishing theorem on complete K¨ahler manifolds will be replaced by an argument which is more involved but also seems to be basic (see §1). []
More precisely, the proof is an application of the finite-dimensionality of L^2 ¯∂-cohomology groups on M with coefficients in line bundles whose curvature form is positive at infinity. Recall that the idea of exploiting the finite-dimensionality for producing holomorphic sections originates in a celebrated paper [G] of Grauert. Shortly speaking, it amounts to finding infinitely many linearly independent C^∞ sections s1, s2, . . . of the bundle in such a way that some nontrivial linear combination of ¯∂s1, ¯∂s2, . . . , say ?^N_{k=1} c_k¯∂sk(ck ∈ C), is equal to ¯∂u for some u which is more regular than ?^N_{k=1} cksk.
This works if one can attach mutually different orders of singularities to sk for instance as in [G] where the holomorphic convexity of strongly pseudoconvex domains was proved.
Although such a method does not directly work for the weakly pseudoconvex cases, the method of solving the ¯∂-equation with L^2 estimates is available to produce a nontrivial holomorphic section of the form Σ^N_{k=1} cksk －u by appropriately estimating u. More precisely speaking, instead of specifying singularities of sk, one finds a solution u which has more zeros than Σ^N_{k=1} ck¯∂sk. For that, finite-dimensionality of the L^2 cohomology with respect to singular fiber metrics would be useful.
However, this part of analysis does not seem to be explored a lot. For instance, the author does not know whether or not Nadel’s vanishing theorem as in [Na] can be extended as a finiteness theorem with coefficients in the multiplier ideal sheaves of singular fiber metrics under an appropriate positivity assumption of the curvature current near infinity.
So, instead of analyzing the L^2 cohomology with respect to singular fiber metrics, we shall avoid the singularities by simply removing them from the manifold and consider the L^2 cohomology of the complement, which turns out to have similar finite-dimensionality property because of the L^2 estimate on complete Hermitian manifolds. Such an argument is restricted to the cases where the singularities of the fiber metic are isolated. As a technique, it was first introduced in [D-Oh-3] to estimate the Bergman distances. It is useful for other purposes and applied also in [Oh-3,4,5,6],
1043 名前： but will be repeated here for the sake of the reader’s convenience. []
Once one has infinitely many linearly independent holomorphic sections of a line bundle L → M, one can find singular fiber metrics of L by taking the reciprocal of the sum of squares of the moduli of local trivializations of the sections. Very roughly speaking, this is the main trick to derive the conclusion of Theorem 0.1 from K_M|∂Ω < 0.
In fact, for the bundles L with L|∂Ω > 0, the proof of dim H^{n,0}(Ω, L^m) = ∞ for m >> 1 will be given in detail here (see Theorem 1.4, Theorem 1.5 and Theorem 1.6). The rest is acturally similar as in the case K_M < 0. We shall also generalize the following theorems of Takayama.
Let M be a complex manifold. We shall say that M is a C^k pseudoconvex manifold if M is equipped with a C^k plurisubharmonic exhaustion function, say φ. C^∞ (resp. C^0) pseudoconvex manifolds are also called weakly 1-complete (resp. pseudoconvex) manifolds. The sublevel sets {x; φ(x) < c} will be denoted by Mc. Theorem 0.2 and Theorem 0.3 are respectively a generalization of Kodaira’s embedding theorem and that of Grauert’s characterization of Stein manifolds.
Our intension here is to draw similar conclusions by assuming the curvature conditions only on the complement of a compact subset of the manifold in quetion
Theorem 0.2 will be generalized as follows. Theorem 1.4. Let (M, φ) be a connected and noncompact C^2 pseudoconvex manifold which admits a holomorphic Hermitian line bundle whose curvature form is positive on M - Mc. Then there exists a holomorphic embedding of M - Mc into CP^N which extends to M meromorphically.
Theorem 0.3 will be extended to Theorem 1.5. A C^2 pseudoconvex manifold (M, φ) is holomorphically convex if the canonical bundle is negative outside a compact set.
This extends Grauert’s theorem asserting that strongly 1-convex manifold are holomorphically convex.
The proofs will be done by combining the method of Takayama with an L^2 variant of the Andreotti-Grauert theory [A-G] on complete Hermitian manifolds whose special form needed here will be recalled in§3. In §4 we shall extend Theorem 0.4 for the domains Ω as in Theorem 0.1. Whether or not Ω in Theorem 0.1 is holomorphically convex is still open.
Theorem 2.1. (cf. [Oh-4, Theorem 0.3 and Theorem 4.1]) Let M be a complex manifold, let Ω ⊊ M be a relatively compact pseudoconvex domain with a C^2-smooth boundary and let B be a holomorphic line bundle over M with a fiber metric h whose curvature form is positive on a neighborhood of ∂Ω. Then there exists a positive integer m0 such that for all m ≥ m0 dimH^{0,0}(Ω, B^m) = ∞ and that, for any compact set K ⊂ Ω and for any positive number R, one can find a compact set K˜ ⊂ Ω such that for any point x ∈ Ω -K˜ there exists an element s of H^{0,0}(Ω, B^m) satisfying sup_{K} |s|_h^m < 1 and |s(x)|_h^m > R.
We shall give the proof of Theorem 1.1 in this section for the convenience of the reader, after recalling the basic L^2 estimates in a general setting.
Let (M, g) be a complete Hermitian manifold of dimension n and let (E, h) be a holomorphic Hermitian vector bundle over M. Let C^{p,q}(M, E) denote the space of E-valued C^∞ (p, q)-
1056 名前：forms on M and letC^{p,q}_0(M, E) = {u ∈ C^{p,q}(M, E); suppu is compact}. []
Given a C^2 function φ : M → R, let L^{p,q}_{(2),φ}(M, E) (= L^{p,q}_{(2),g,φ}(M, E)) be the space of E-valued square integrable measurable (p, q)-forms on M with respect to g and he^{－φ} .
Recall that L^{p,q}_{(2),φ}(M, E) is identified with the completion of C^{p,q}_0(M, E) with respect to the L^2 norm ||u||φ := (∫_Me^{－φ}|u|^2_{g,h}dVg)1/2. Here dVg := 1/n!ω^n for the fundamental form ω = ω_g of g.
More explicitly, when E is given by a system of transition functions eαβ with respect to a trivializing covering {Uα} of M and h is given as a system of C∞ positive definite Hermitian matrix valued functions hα on Uα satisfying hα =t eβαhβeβα on Uα ∩ Uβ, |u|2 g,hdVg is defined by tuαhα ∧ ∗uα, where u = {uα} with uα = eαβuβ on Uα ∩ Uβ and ∗ stands for the Hodge’s star operator with respect to g. We put ∗¯u = ∗u so that tuαhα ∧ ∗uα =tuαhα ∧ ∗¯uα
Let us denote by ¯∂ (resp. ∂) the complex exterior derivative of type (0, 1) (resp. (1, 0)). Then the correspondence uα 7→ ¯∂uα defines a linear differential operator ¯∂ : C p,q(M, E) → C p,q+1(M, E). The Chern connection Dh is defined to be ¯∂ + ∂h, where ∂h is defined by uα 7→ h －1 α ∂(hαuα). Since ¯∂ 2 = ∂ 2 h = ∂ ¯∂ + ¯∂∂ = 0, there exists a E ∗ ⊗ E-valued (1, 1)-form Θh such that D2 hu = Θh ∧u holds for all u ∈ C p,q(M, E). Θh is called the curvature form of h. Note that Θhe－φ = Θh+IdE ⊗∂ ¯∂φ. Θh is said to be positive (resp. semipositive) at x ∈ M if Θh = 馬 j,k=1 Θjk¯dzj ∧ dzk in terms of a local coordinate (z1, . . . , zn) LEVI PROBLEM UNDER THE NEGATIVITY 5 around x and (Θjk¯(x))j,k = (Θµ νjk¯ (x))j,k,µ,ν is positive (semipositive) in the sense (of Nakano) that the quadratic form (
Whenever there is no fear of confusion, as well as the Levi form ∂¯∂φ of φ, Θ_h will be identified with a Hermitian form along the fibers of E ⊗ TM, where TM stands for the holomorphic tangent bundle of M.
By an abuse of notation, ¯∂ (resp. ∂he－φ ) will also stand for the maximal closed extension of ¯∂|C p,q 0 (M,E) (resp. ∂he－φ |C p,q 0 (M,E) ) as a closed operator from L p,q (2),φ (M, E) to L p,q+1 (2),φ (M, E) (resp. L p+1,q (2),φ (M, E)). The adjoint of ¯∂ (resp. ∂he－φ ) will be denoted by ¯∂ ∗ = ¯∂ ∗ g,he－φ (resp. ∂ ∗ he－φ ). We recall that ∂ ∗ he－φ = －∗¯∂∗¯ holds as a differential operator acting on C p,q(M, E), so that ∂ ∗ he－φ will be also denoted by ∂ ∗ . By Dom¯∂ (resp. Dom¯∂ ∗ ) we shall denote the domain of ¯∂ (resp. ¯∂ ∗ ).
We put H p,q (2),φ (M, E)(= H p,q (2),g,φ (M, E)) = Ker ( ¯∂ : L p,q (2),φ (M, E) → L p,q+1 (2),φ (M, E) ) Im ( ¯∂ : L p,q－1 (2),φ (M, E) → L p,q (2),φ (M, E) ) and H p,q φ (M, E) = Ker ¯∂ ∩ Ker ¯∂ ∗ ∩ L p,q (2),φ (M, E).
Let Λ = Λg denote the adjoint of the exterior multiplication by ω. Then Nakano’s formula (2.2) ¯∂ ¯∂ ∗ + ¯∂ ∗ ¯∂ － ∂h∂ ∗ － ∂ ∗ ∂h = √ －1(ΘhΛ － ΛΘh) holds if dω = 0. Here Θh also stands for the exterior multiplication by Θh from the left hand side. Hence, for any open set Ω ⊂ M such that dω|Ω = 0 and for any u ∈ C n,q 0 (Ω, E), one has (2.3) k ¯∂uk 2 φ + k ¯∂ ∗uk 2 φ ≥ ( √ －1(Θh + IdE ⊗ ∂ ¯∂φ)Λu, u)φ. Here (u, w)φ stands for the inner product of u and v with respect to (g, he－φ ).
Here (u, w)φ stands for the inner product of u and v with respect to (g, he－φ ). The following direct consequence of (1.3) is important for our purpose.
Proposition 2.1. Let M, E, g, h and φ be as above. Assume that there exists a compact set K ⊂ M such that dωg = 0 holds on M \ K. Then there exist a compact set K′ containing K and a constant C such that K′ and C do not depend on the choice of φ and ( √ －1(Θh+IdE⊗∂ ¯∂φ)Λu, u)φ ≤ C ( k ¯∂uk 2 φ + k ¯∂ ∗uk 2 φ + ∫ K′ e －φ |u| 2 g,hdVg ) holds for any u ∈ C n,q 0 (M, E) (q ≥ 0).
Proposition 2.2. Let (M, E, g, h, φ, K) and (K′ , C) be as above. Assume moreover that one can find a constant C0 > 0 such that C0(Θh + IdE ⊗∂ ¯∂φ)－IdE ⊗g ≥ 0 holds on M \K. Then there exists a constant C ′ depending only on C, K′ and C0 such that kuk 2 φ ≤ C ′ ( k ¯∂uk 2 φ + k ¯∂ ∗uk 2 φ + ∫ K′ e －φ |u| 2 g,hdVg ) holds for any u ∈ C n,q 0 (M, E) (q ≥ 1).
By a theorem of Gaffney, the estimate in Proposition 1.2 implies the following. Proposition 2.3. In the situation of Proposition 1.2, kuk 2 φ ≤ C ′ ( k ¯∂uk 2 φ + k ¯∂ ∗uk 2 φ + ∫ K′ e －φ |u| 2 g,hdVg ) holds for all u ∈ L n,q (2),φ (M, E) ∩ Dom¯∂ ∩ Dom¯∂ ∗ (q ≥ 1).
Theorem 2.2. (Theorem 1.1.2 and Theorem 1.1.3 in [H]) Let H1 and H2 be Hilbert spaces and let T : H1 → H2 be a densely defined closed operator. Let H3 be another Hilbert space and let S : H2 → H3 be a densely defined closed operator such that ST = 0. Then a necessary and sufficient condition for the ranges RT , RS of T, S both to be closed is that there exists a constant C such that (2.4) kgkH2 ≤ C(kT ∗ gkH1 +kSgkH3 ); g ∈ DT ∗ ∩DS, g⊥(NT ∗ ∩NS), where DT ∗ and DS denote the domains of T ∗ and S, respectively, and NT ∗ = KerT ∗ and NS = KerS. Moreover, if one can select a strongly convergent subsequence from every sequence gk ∈ DT ∗ ∩DS with kgkkH2 bounded and T ∗ gk → 0 in H1, Sgk → 0 in H3, then NS/RT ∼= NT ∗ ∩NS holds and NT ∗ ∩ NS is finite dimensional.
It is an easy exercise to deduce from Theorem 1.3 that every strongly pseudoconvex manifold is holomorphically convex (cf. [G] or [H]). We are going to extend this application to the domains with weaker pseudoconvexity.
For any Hermitian metric g on M, a C 2 function ψ : M → R is called g-psh (g-plurisubharmonic) if g + ∂ ¯∂ψ ≥ 0 holds everywhere. Then Theorem 1.3 can be restated as follows.
Theorem 2.4. Let (M, g) be an n-dimensional complete Hermitian manifold and let (E, h) be a Hermitian holomorphic vector bundle over M. Assume that there exists a compact set K ⊂ M such that �
1079 名前：ｦh － IdE ⊗ g ≥ 0 and dωg = 0 hold on M \ K. Then, for any g-psh function φ on M and for any ε ∈ (0, 1), dim H n,q (2),εφ (M, E) < ∞ and H n,q εφ (M, E) ∼= H n,q (2),εφ (M, E) for q ≥ 1 []
§2 Infinite dimensionality and bundle convexity theorems By applying Theorem 1.4, we shall show at first the following. Theorem 2.5. Let (M, E, g, h) be as in Theorem 1.4 and let xµ (µ = 1, 2, . . .) be a sequence of points in M without accumulation points. Assume that there exists a (1 － ε)g-psh function φ on M \ {xµ} ∞ µ=1 for some ε ∈ (0, 1) such that e －φ is not integrable on any neighborhood of xµ for any µ. Then dim H n,0 (M, E) = ∞.
Proof. We put M′ = M \{xµ} ∞ µ=1 and let ψ be a bounded C ∞ ε 2 g-psh function on M′ such that g ′ := g + ∂ ¯∂ψ is a complete metric on M′ . Take sµ ∈ C n,0 (M, E) (µ ∈ N) in such a way that |sµ(xν)|g,h = δµν and ∫ M′ e －φ | ¯∂sµ| 2 g,hdVg < ∞. Since ∫ M′ e －φ－ψ | ¯∂sµ| 2 g ′ ,hdVg ′ ≤ ∫ M′ e －φ－ψ | ¯∂sµ| 2 g,hdVg and dim H n,1 (2),g′ ,φ (M′ , E) < ∞ by Theorem 1.4, one can find a nontrivial finite linear combination of ¯∂sµ, say v = 把µ ¯∂sµ, which is in the range of L n,0 (2),φ (M′ , E) ∂¯ －→ L n,1 (2),g′ ,φ (M′ , E).
Then take u ∈ L n,0 (2),φ (M′ , E) satisfying ¯∂u = v and put s = 把µsµ － u. Clearly s extends to a nonzero element of Hn,0 (M, E) which is zero at xµ except for finitely many µ. Hence, one can find mutually disjoint finite subsets Σν 6= ϕ (ν = 1, 2, . . .) of N and nonzero holomorphic sections sν of E such that sν(xµ) = 0 if µ /∈ Σν, so that dim Hn,0 (M, E) = ∞
