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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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940: １３２人目の素数さん [] 2023/07/31(月) 05:54:17.72 ID:jznoxopE >>938 わかったからもういい http://rio2016.5ch.net/test/read.cgi/math/1674527723/940
942: １３２人目の素数さん [] 2023/07/31(月) 08:29:53.84 ID:jznoxopE おまえの言いたいことは分かった http://rio2016.5ch.net/test/read.cgi/math/1674527723/942
951: １３２人目の素数さん [] 2023/07/31(月) 08:53:09.00 ID:jznoxopE では埋めよう http://rio2016.5ch.net/test/read.cgi/math/1674527723/951
954: １３２人目の素数さん [] 2023/07/31(月) 09:05:07.98 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/954
955: １３２人目の素数さん [] 2023/07/31(月) 09:08:22.88 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/955
956: １３２人目の素数さん [] 2023/07/31(月) 09:13:58.31 ID:jznoxopE 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 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). http://rio2016.5ch.net/test/read.cgi/math/1674527723/956
957: １３２人目の素数さん [] 2023/07/31(月) 09:17:00.22 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/957
958: １３２人目の素数さん [] 2023/07/31(月) 09:19:07.58 ID:jznoxopE 訂正 ¯∂s1,¯∂s2, . . . , say ΣN_{k=1} c_k¯∂sk(ck ∈ C), is equal to ¯∂u for some u which is more regular than ΣN_{k=1} cksk. http://rio2016.5ch.net/test/read.cgi/math/1674527723/958
959: １３２人目の素数さん [] 2023/07/31(月) 09:20:45.34 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/959
960: １３２人目の素数さん [] 2023/07/31(月) 09:24:42.05 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/960
961: １３２人目の素数さん [] 2023/07/31(月) 09:26:19.46 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/961
962: １３２人目の素数さん [] 2023/07/31(月) 09:28:09.44 ID:jznoxopE 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], but will be repeated here for the sake of the reader’s convenience. http://rio2016.5ch.net/test/read.cgi/math/1674527723/962
963: １３２人目の素数さん [] 2023/07/31(月) 09:41:53.67 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/963
964: １３２人目の素数さん [] 2023/07/31(月) 09:43:46.51 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/964
965: １３２人目の素数さん [] 2023/07/31(月) 09:45:14.41 ID:jznoxopE Theorem 1.2. (cf. [T-1]) Weakly 1-complete manifolds with positive line bundles are embeddable into CP^N (N >> 1). Theorem 1.3. (cf. [T-2]) Pseudoconvex manifolds with negative canonical bundles are holomorphically convex. http://rio2016.5ch.net/test/read.cgi/math/1674527723/965
966: １３２人目の素数さん [] 2023/07/31(月) 09:58:43.90 ID:jznoxopE 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. http://rio2016.5ch.net/test/read.cgi/math/1674527723/966
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
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