[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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922: １３２人目の素数さん [] 2023/07/30(日) 19:25:23.60 ID:Rf2iGg9G >>601 >一つの箱に確率pで数が入れられるとする。また、一つの同値類内で考える >lemma 3：確率p＝0で、可算有限長さ一点コンパクト化の数列 sN+において、決定番号ωの確率1、ω未満(つまり有限n)の確率0 >lemma 4：確率p＝0で、可算有限長さの数列 sN = (s1，s2，s3 ，・・・)において、決定番号ω未満(つまり有限n)の確率0 >証明：lemma 3で、sN+からωを除いて、数列 sNとして適用すればよい lemma3は正しいが、lemma4は誤り　 sN+からΩを除いたら、決定番号ωとなる場合が存在しなくなる つまり、証明は誤り http://rio2016.5ch.net/test/read.cgi/math/1674527723/922

923: １３２人目の素数さん [] 2023/07/30(日) 19:30:49.15 ID:IpiBUMr/ >>921 文盲に分からないのは当たり前 http://rio2016.5ch.net/test/read.cgi/math/1674527723/923

924: １３２人目の素数さん [] 2023/07/30(日) 20:02:09.25 ID:Rf2iGg9G p=0とする nを自然数としたとき、snで尻尾同値な2列について、 n番目の項を除いても、s(n-1)で尻尾同値となる確率は0 sN+で尻尾同値な２列について、 ω番目の項を除いても、sNで尻尾同値となる確率は0 ただ、有限列の場合と異なるのは、 s(n-1)では、決定番号n-1となる確率は１だが sNでは、確率１となるような決定番号ω-1は存在しない、ということ 実際は、snの各項は0番目の項、1番目の項、・・・、n-1番目の項と名付けるべきで その場合には、決定番号の最大値はn-1となる （s(n-1)の決定番号の最大値はn-2) sN+は、s(ω+1)であって、決定番号の最大値はω sNは、sωであって、この場合、決定番号の最大値は存在しない なぜならωは極限順序数だから さらにωは(濃度の)始順序数でもある つまりωより小さな順序数は、濃度もωより小さい またn<ωとなる順序数について　nより大きな順序数の全体集合はωと同濃度である http://rio2016.5ch.net/test/read.cgi/math/1674527723/924

925: １３２人目の素数さん [] 2023/07/30(日) 20:07:21.88 ID:Rf2iGg9G >>417 >”固定”なるものは確率論でいう一つの試行でしかない 　箱入り無数目の確率計算ではそうなっていないので誤り 　正解は>>632の言う通り 　「 1〜100 のいずれかをランダムに選ぶ」 http://rio2016.5ch.net/test/read.cgi/math/1674527723/925

926: １３２人目の素数さん [] 2023/07/30(日) 20:21:51.15 ID:Rf2iGg9G 「箱入り無数目」で、箱の中身は各試行ごとに入れ替えたりしない、とすれば 「箱の中身の確率分布」は全く考える必要がない （実際、確率計算はそのような前提の上でなされている） また、箱の数を非可算個にしてしまえば、１００列用意する必要もない ただ、非可算個の箱の中から１個選べばいい 箱の番号（注：中身に非ず）は[0,1]の実数とする これで確率測度は定まった なお、[0,1]の実数は、整列定理により整列できるので問題ない さて、（可算番目）尻尾同値により、代表と異なる中身をもつ箱はたかだか可算個である したがって、そのような箱を選ぶ確率は０である ゆえに、適当に箱を選んでも当たる確率は１である http://rio2016.5ch.net/test/read.cgi/math/1674527723/926

927: １３２人目の素数さん [] 2023/07/30(日) 20:34:51.20 ID:Rf2iGg9G 関数　f,g∈(0,1]→X　について ある正の実数ε>０が存在して 0<x<=εなら、f(x)=g(x)となるとき fとgは近傍同値とする その場合、任意の近傍同値類の関数fについて、同値類の代表関数Fとの間に ある正の実数Ε>0が存在して、0<x<=Εなら、f(x)=F(x)となる したがって、いかなる関数fについても、 近傍同値類とその代表関数Fがわかり しかもfに関してΕより小さいxを見つけることができれば f(x)=F(x)となるのでf(x)の値を当てられる ただそれだけの話 http://rio2016.5ch.net/test/read.cgi/math/1674527723/927

928: １３２人目の素数さん [] 2023/07/30(日) 20:37:58.57 ID:Rf2iGg9G >>927で、Xはいかなる集合でもよい、というのがポイント （自明でない問題とするためには、Xは2個以上の要素を持つとすればいい） http://rio2016.5ch.net/test/read.cgi/math/1674527723/928

929: １３２人目の素数さん [] 2023/07/30(日) 20:38:51.30 ID:Rf2iGg9G >>927で、Xはいかなる集合もよい、というのがポイント （自明でない問題とするためには、Xは2個以上の要素を持つとすればいい） http://rio2016.5ch.net/test/read.cgi/math/1674527723/929

930: １３２人目の素数さん [] 2023/07/30(日) 20:42:51.17 ID:Rf2iGg9G 今宵はこれまで http://rio2016.5ch.net/test/read.cgi/math/1674527723/930

931: １３２人目の素数さん [] 2023/07/30(日) 21:19:01.14 ID:esnUGRo8 結局724は問題の体をなしていないことが分かった http://rio2016.5ch.net/test/read.cgi/math/1674527723/931

932: １３２人目の素数さん [] 2023/07/30(日) 21:24:02.69 ID:IpiBUMr/ >>931 負け惜しみ乙 http://rio2016.5ch.net/test/read.cgi/math/1674527723/932

933: １３２人目の素数さん [] 2023/07/30(日) 21:36:34.03 ID:2UJHJvqn >>931 >結局724は問題の体をなしていないことが分かった ご苦労さまです スレ主です 724の出題者が、何にも分かってないってことでは、ないでしょうか？ｗ http://rio2016.5ch.net/test/read.cgi/math/1674527723/933

934: １３２人目の素数さん [] 2023/07/30(日) 21:43:19.72 ID:esnUGRo8 こういう結論でよいようですね http://rio2016.5ch.net/test/read.cgi/math/1674527723/934

935: １３２人目の素数さん [] 2023/07/30(日) 21:50:09.12 ID:IpiBUMr/ >>934 問題の体をなしてないことにすれば自尊心保てるよねｗ http://rio2016.5ch.net/test/read.cgi/math/1674527723/935

936: １３２人目の素数さん [] 2023/07/30(日) 21:52:44.46 ID:IpiBUMr/ >>933 サルは誤答してるよね>>741 「問題の体をなしてない（ｷﾘｯ）」にシレっと乗っかろうとしてるけどさｗ やはりサル知恵だねｗ http://rio2016.5ch.net/test/read.cgi/math/1674527723/936

937: １３２人目の素数さん [] 2023/07/30(日) 22:00:19.83 ID:IpiBUMr/ まあ「どんな実数を入れるかはまったく自由」の意味も分からない方にとっては問題の体をなしてないのでしょうｗ ご自分が理解できる・解ける問題じゃないと問題の体をなしてないようですねｗ http://rio2016.5ch.net/test/read.cgi/math/1674527723/937

938: １３２人目の素数さん [] 2023/07/30(日) 22:24:38.99 ID:IpiBUMr/ 0897１３２人目の素数さん 2023/07/30(日) 08:34:02.46ID:esnUGRo8 >>895 例えば ｎ番目の箱にｎ回サイコロを振って出た目の数を入れる というのは 許されるのかどうか 0899１３２人目の素数さん 2023/07/30(日) 09:12:34.73ID:esnUGRo8 >>898 つまりサイコロを振って入れるのでも構わないということ？ 0907１３２人目の素数さん 2023/07/30(日) 10:10:54.73ID:esnUGRo8 >>905 「まったく自由」の数学的な意味を明確化したい 「どんな実数を入れるかはまったく自由」の意味が分からないようだとさすがに箱入り無数目は無理ですね ご自身のレベルに合ったスレを見つける努力をされた方がよろしいのではないでしょうか http://rio2016.5ch.net/test/read.cgi/math/1674527723/938

939: １３２人目の素数さん [sage] 2023/07/31(月) 01:15:03.45 ID:Cgy3PWyO >>920 丸めて蕎麦屋酒と読み替える語彙も無いのか此のガキ爺は (𓁹‿𓁹)ﾆﾁｬｧ http://rio2016.5ch.net/test/read.cgi/math/1674527723/939

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

941: １３２人目の素数さん [] 2023/07/31(月) 07:45:17.26 ID:KGw9oDo5 >>940 何を分かったの？ http://rio2016.5ch.net/test/read.cgi/math/1674527723/941

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

943: １３２人目の素数さん [sage] 2023/07/31(月) 08:43:16.22 ID:4Almmw4D 本スレは以下のスレに統合します ガロア第一論文と乗数イデアル他関連資料スレ5 https://rio2016.5ch.net/test/read.cgi/math/1687778456/ http://rio2016.5ch.net/test/read.cgi/math/1674527723/943

944: １３２人目の素数さん [sage] 2023/07/31(月) 08:43:35.85 ID:4Almmw4D 本スレは以下のスレに統合します ガロア第一論文と乗数イデアル他関連資料スレ5 https://rio2016.5ch.net/test/read.cgi/math/1687778456/ http://rio2016.5ch.net/test/read.cgi/math/1674527723/944

945: １３２人目の素数さん [sage] 2023/07/31(月) 08:43:48.98 ID:4Almmw4D 本スレは以下のスレに統合します ガロア第一論文と乗数イデアル他関連資料スレ5 https://rio2016.5ch.net/test/read.cgi/math/1687778456/ http://rio2016.5ch.net/test/read.cgi/math/1674527723/945

946: １３２人目の素数さん [sage] 2023/07/31(月) 08:45:49.69 ID:4Almmw4D >>943-945 http://rio2016.5ch.net/test/read.cgi/math/1674527723/946

947: １３２人目の素数さん [sage] 2023/07/31(月) 08:47:47.93 ID:4Almmw4D 現在数学板に複数ある「SET Aスレ」は統合化いたします http://rio2016.5ch.net/test/read.cgi/math/1674527723/947

948: １３２人目の素数さん [sage] 2023/07/31(月) 08:48:07.04 ID:4Almmw4D 御協力お願い致します http://rio2016.5ch.net/test/read.cgi/math/1674527723/948

949: １３２人目の素数さん [sage] 2023/07/31(月) 08:49:40.88 ID:4Almmw4D 現在数学板に複数ある「SET Aスレ」は一つに統合いたします http://rio2016.5ch.net/test/read.cgi/math/1674527723/949

950: １３２人目の素数さん [sage] 2023/07/31(月) 08:50:01.11 ID:4Almmw4D 御協力お願い致します http://rio2016.5ch.net/test/read.cgi/math/1674527723/950

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

952: １３２人目の素数さん [sage] 2023/07/31(月) 08:54:55.17 ID:4Almmw4D >>951　よろしくお願いします http://rio2016.5ch.net/test/read.cgi/math/1674527723/952

953: １３２人目の素数さん [sage] 2023/07/31(月) 08:57:02.76 ID:4Almmw4D 「SET Aスレ」統合化に御協力お願いします SET A氏設立のスレッドは複数ありますが、 どこでも同様の展開となっているため 様々な無駄が発生しております スレを１つにすることで無駄を削減できます 何卒、統合化に御協力お願いいたします http://rio2016.5ch.net/test/read.cgi/math/1674527723/953

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

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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