[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）13 (1002ﾚｽ)

純粋・応用数学・数学隣接分野（含むガロア理論）13 http://rio2016.5ch.net/test/read.cgi/math/1674527723/

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967: １３２人目の素数さん [] 2023/07/31(月) 09:59:24.57 ID:jznoxopE 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 http://rio2016.5ch.net/test/read.cgi/math/1674527723/967

968: １３２人目の素数さん [] 2023/07/31(月) 10:01:42.55 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/968

969: １３２人目の素数さん [] 2023/07/31(月) 10:02:51.03 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/969

970: １３２人目の素数さん [] 2023/07/31(月) 10:05:12.95 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/970

971: １３２人目の素数さん [] 2023/07/31(月) 10:05:50.76 ID:jznoxopE The proof of the desired improvement of Theorem 0.1 will rely on the following. http://rio2016.5ch.net/test/read.cgi/math/1674527723/971

972: １３２人目の素数さん [] 2023/07/31(月) 10:10:31.08 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/972

973: １３２人目の素数さん [] 2023/07/31(月) 10:12:10.83 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/973

974: １３２人目の素数さん [] 2023/07/31(月) 10:13:51.75 ID:jznoxopE 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)-forms on M and letC^{p,q}_0(M, E) = {u ∈ C^{p,q}(M, E); suppu is compact}. http://rio2016.5ch.net/test/read.cgi/math/1674527723/974

975: １３２人目の素数さん [] 2023/07/31(月) 10:15:52.02 ID:jznoxopE 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} . http://rio2016.5ch.net/test/read.cgi/math/1674527723/975

976: １３２人目の素数さん [] 2023/07/31(月) 10:17:16.60 ID:jznoxopE The definition of L^{p,q}_{(2),φ}(M, E) will be naturally extended for continuous metrics and continuous weights. http://rio2016.5ch.net/test/read.cgi/math/1674527723/976

977: １３２人目の素数さん [] 2023/07/31(月) 10:27:13.55 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/977

978: １３２人目の素数さん [] 2023/07/31(月) 10:29:53.47 ID:jznoxopE 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 http://rio2016.5ch.net/test/read.cgi/math/1674527723/978

979: １３２人目の素数さん [] 2023/07/31(月) 10:30:48.69 ID:jznoxopE 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 = ?n 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 ?( ? µ hµκ¯Θ µ νjk¯ )(x)ξ νj ξ κk is positive definite (resp. positive semidefinite). http://rio2016.5ch.net/test/read.cgi/math/1674527723/979

980: １３２人目の素数さん [] 2023/07/31(月) 10:32:02.15 ID:jznoxopE Θ > 0 (resp. ≥0) for an E^∗ ⊗ E-valued (1,1)-form Θ will mean the positivity (resp. semipositivity) in this sense. http://rio2016.5ch.net/test/read.cgi/math/1674527723/980

981: １３２人目の素数さん [] 2023/07/31(月) 10:32:55.60 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/981

982: １３２人目の素数さん [] 2023/07/31(月) 10:33:40.62 ID:jznoxopE 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. ¯∂ ∗ ). http://rio2016.5ch.net/test/read.cgi/math/1674527723/982

983: １３２人目の素数さん [] 2023/07/31(月) 10:34:15.81 ID:jznoxopE 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). http://rio2016.5ch.net/test/read.cgi/math/1674527723/983

984: １３２人目の素数さん [] 2023/07/31(月) 10:34:53.86 ID:jznoxopE 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 ). http://rio2016.5ch.net/test/read.cgi/math/1674527723/984

985: １３２人目の素数さん [] 2023/07/31(月) 10:35:28.01 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/985

986: １３２人目の素数さん [] 2023/07/31(月) 10:36:40.94 ID:jznoxopE 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). http://rio2016.5ch.net/test/read.cgi/math/1674527723/986

987: １３２人目の素数さん [] 2023/07/31(月) 10:37:12.65 ID:jznoxopE From Proposition 1.1 one infers http://rio2016.5ch.net/test/read.cgi/math/1674527723/987

988: １３２人目の素数さん [] 2023/07/31(月) 10:37:44.26 ID:jznoxopE 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). http://rio2016.5ch.net/test/read.cgi/math/1674527723/988

989: １３２人目の素数さん [] 2023/07/31(月) 10:38:25.31 ID:jznoxopE 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). http://rio2016.5ch.net/test/read.cgi/math/1674527723/989

990: １３２人目の素数さん [] 2023/07/31(月) 10:38:55.22 ID:jznoxopE Recall that the following was proved in [H] by a basic argument of functional analysis. http://rio2016.5ch.net/test/read.cgi/math/1674527723/990

991: １３２人目の素数さん [] 2023/07/31(月) 10:39:37.54 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/991

992: １３２人目の素数さん [] 2023/07/31(月) 10:40:12.07 ID:jznoxopE Hence we obtain http://rio2016.5ch.net/test/read.cgi/math/1674527723/992

993: １３２人目の素数さん [] 2023/07/31(月) 10:40:52.24 ID:jznoxopE Theorem 2.3. In the situation of Proposition 1.2, dimH n,q (2),φ (M, E) < ∞ and H n,q φ (M, E) ∼= H n,q (2),φ (M, E) hold for all q ≥ 1. http://rio2016.5ch.net/test/read.cgi/math/1674527723/993

994: １３２人目の素数さん [] 2023/07/31(月) 10:41:28.54 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/994

995: １３２人目の素数さん [] 2023/07/31(月) 10:41:57.49 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/995

996: １３２人目の素数さん [] 2023/07/31(月) 10:42:29.70 ID:jznoxopE 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 Θh − 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 http://rio2016.5ch.net/test/read.cgi/math/1674527723/996

997: １３２人目の素数さん [] 2023/07/31(月) 10:43:00.40 ID:jznoxopE §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) = ∞. http://rio2016.5ch.net/test/read.cgi/math/1674527723/997

998: １３２人目の素数さん [] 2023/07/31(月) 10:43:37.26 ID:jznoxopE 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 = ?cµ ¯∂sµ, which is in the range of L n,0 (2),φ (M′ , E) ∂¯ −→ L n,1 (2),g′ ,φ (M′ , E). http://rio2016.5ch.net/test/read.cgi/math/1674527723/998

999: １３２人目の素数さん [] 2023/07/31(月) 10:44:29.89 ID:jznoxopE Then take u ∈ L n,0 (2),φ (M′ , E) satisfying ¯∂u = v and put s = ?cµsµ − 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) = ∞ http://rio2016.5ch.net/test/read.cgi/math/1674527723/999

1000: １３２人目の素数さん [] 2023/07/31(月) 10:45:25.12 ID:jznoxopE This observation will be basic for the proofs of Theorems 0.4 and 0.5. http://rio2016.5ch.net/test/read.cgi/math/1674527723/1000

1001: 1001 [] ID:Thread このスレッドは１０００を超えました。 新しいスレッドを立ててください。 life time: 187日 23時間 10分 2秒 http://rio2016.5ch.net/test/read.cgi/math/1674527723/1001

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