23/07/30 19:25:23.60 Rf2iGg9G.net
>>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
1001:１３２人目の素数さん
23/07/30 19:30:49.15 IpiBUMr/.net
>>921
文盲に分からないのは当たり前
1002:１３２人目の素数さん
23/07/30 20:02:09.25 Rf2iGg9G.net
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
1003:１３２人目の素数さん
23/07/30 20:07:21.88 Rf2iGg9G.net
>>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」
1004:１３２人目の素数さん
23/07/30 20:21:51.15 Rf2iGg9G.net
「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
1005:１３２人目の素数さん
23/07/30 20:34:51.20 Rf2iGg9G.net
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
1006:１３２人目の素数さん
23/07/30 20:37:58.57 Rf2iGg9G.net
>>927で、Xはいかなる集麹№ﾅもよい、とｂ｢うのがポインャg
（自明ｂﾅない問題とすｂ驍ｽめには、Xは2個以上の要素を持つとすればいい）
1007:１３２人目の素数さん
23/07/30 20:38:51.30 Rf2iGg9G.net
>>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）
1008:１３２人目の素数さん
23/07/30 20:42:51.17 Rf2iGg9G.net
今宵はこれまで
1009:１３２人目の素数さん
23/07/30 21:19:01.14 esnUGRo8.net
結局724は問題の体をなしていないことが分かった
1010:１３２人目の素数さん
23/07/30 21:24:02.69 IpiBUMr/.net
>>931
負け惜しみ乙
1011:１３２人目の素数さん
23/07/30 21:36:34.03 2UJHJvqn.net
>>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
1012:分かってないってことでは、ないでしょうか？ｗ
1013:１３２人目の素数さん
23/07/30 21:43:19.72 esnUGRo8.net
こういう結論でよいようですね
1014:１３２人目の素数さん
23/07/30 21:50:09.12 IpiBUMr/.net
>>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
1015:１３２人目の素数さん
23/07/30 21:52:44.46 IpiBUMr/.net
>>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
1016:１３２人目の素数さん
23/07/30 22:00:19.83 IpiBUMr/.net
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
1017:１３２人目の素数さん
23/07/30 22:24:38.99 IpiBUMr/.net
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
1018:１３２人目の素数さん
23/07/31 01:15:03.45 Cgy3PWyO.net
>>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
1019:１３２人目の素数さん
23/07/31 05:54:17.72 jznoxopE.net
>>938
わかったからもういい
1020:１３２人目の素数さん
23/07/31 07:45:17.26 KGw9oDo5.net
>>940
何を分かったの？
1021:１３２人目の素数さん
23/07/31 08:29:53.84 jznoxopE.net
おまえの言いたいことは分かった
1022:１３２人目の素数さん
23/07/31 08:43:16.22 4Almmw4D.net
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
ｽﾚﾘﾝｸ(math板)
1023:１３２人目の素数さん
23/07/31 08:43:35.85 4Almmw4D.net
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
ｽﾚﾘﾝｸ(math板)
1024:１３２人目の素数さん
23/07/31 08:43:48.98 4Almmw4D.net
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
ｽﾚﾘﾝｸ(math板)
1025:１３２人目の素数さん
23/07/31 08:45:49.69 4Almmw4D.net
>>943-945
1026:１３２人目の素数さん
23/07/31 08:47:47.93 4Almmw4D.net
現在数学板に複数ある「SET Aスレ」は統合化いたします
1027:１３２人目の素数さん
23/07/31 08:48:07.04 4Almmw4D.net
御協力お願い致します
1028:１３２人目の素数さん
23/07/31 08:49:40.88 4Almmw4D.net
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
1029:１３２人目の素数さん
23/07/31 08:50:01.11 4Almmw4D.net
御協力お願い致します
1030:１３２人目の素数さん
23/07/31 08:53:09.00 jznoxopE.net
では埋めよう
1031:１３２人目の素数さん
23/07/31 08:54:55.17 4Almmw4D.net
>>951 よろしくお願いします
1032:１３２人目の素数さん
23/07/31 08:57:02.76 4Almmw4D.net
「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
1033:１３２人目の素数さん
23/07/31 09:05:07.98 jznoxopE.net
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.
1034:１３２人目の素数さん
23/07/31 09:08:22.88 jznoxopE.net
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.
1035:１３２人目の素数さん
23/07/31 09:13:58.31 jznoxopE.net
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: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).
1037:１３２人目の素数さん
23/07/31 09:17:00.22 jznoxopE.net
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.
1038:１３２人目の素数さん
23/07/31 09:19:07.58 jznoxopE.net
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.
1039:１３２人目の素数さん
23/07/31 09:20:45.34 jznoxopE.net
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.
1040:１３２人目の素数さん
23/07/31 09:24:42.05 jznoxopE.net
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.
1041:１３２人目の素数さん
23/07/31 09:26:19.46 jznoxopE.net
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.
1042:１３２人目の素数さん
23/07/31 09:28:09.44 jznoxopE.net
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: but will be repeated here for the sake of the reader’s convenience.
1044:１３２人目の素数さん
23/07/31 09:41:53.67 jznoxopE.net
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.
1045:１３２人目の素数さん
23/07/31 09:43:46.51 jznoxopE.net
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.
1046:１３２人目の素数さん
23/07/31 09:45:14.41 jznoxopE.net
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.
1047:１３２人目の素数さん
23/07/31 09:58:43.90 jznoxopE.net
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.
1048:１３２人目の素数さん
23/07/31 09:59:24.57 jznoxopE.net
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
1049:１３２人目の素数さん
23/07/31 10:01:42.55 jznoxopE.net
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.
1050:１３２人目の素数さん
23/07/31 10:02:51.03 jznoxopE.net
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.
1051:１３２人目の素数さん
23/07/31 10:05:12.95 jznoxopE.net
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.
1052:１３２人目の素数さん
23/07/31 10:05:50.76 jznoxopE.net
The proof of the desired improvement of Theorem 0.1 will rely on
the following.
1053:１３２人目の素数さん
23/07/31 10:10:31.08 jznoxopE.net
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.
1054:１３２人目の素数さん
23/07/31 10:12:10.83 jznoxopE.net
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.
1055:１３２人目の素数さん
23/07/31 10:13:51.75 jznoxopE.net
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:forms on M and letC^{p,q}_0(M, E) = {u ∈ C^{p,q}(M, E); suppu is compact}.
1057:１３２人目の素数さん
23/07/31 10:15:52.02 jznoxopE.net
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}
.
1058:１３２人目の素数さん
23/07/31 10:17:16.60 jznoxopE.net
The definition of L^{p,q}_{(2),φ}(M, E) will be
naturally extended for continuous metrics and continuous weights.
1059:１３２人目の素数さん
23/07/31 10:27:13.55 jznoxopE.net
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.
1060:１３２人目の素数さん
23/07/31 10:29:53.47 jznoxopE.net
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
1061:１３２人目の素数さん
23/07/31 10:30:48.69 jznoxopE.net
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
(
µ
hµκ¯Θ
µ
νjk¯
)(x)ξ
νj ξ
κk
is positive definite (resp. positive semidefinite).
1062:１３２人目の素数さん
23/07/31 10:32:02.15 jznoxopE.net
Θ > 0 (resp. ≥0) for an E^∗ ⊗ E-valued (1,1)-form Θ will mean the positivity (resp.
semipositivity) in this sense.
1063:１３２人目の素数さん
23/07/31 10:32:55.60 jznoxopE.net
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.
1064:１３２人目の素数さん
23/07/31 10:33:40.62 jznoxopE.net
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. ¯∂
∗
).
1065:１３２人目の素数さん
23/07/31 10:34:15.81 jznoxopE.net
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).
1066:１３２人目の素数さん
23/07/31 10:34:53.86 jznoxopE.net
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
).
1067:１３２人目の素数さん
23/07/31 10:35:28.01 jznoxopE.net
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.
1068:１３２人目の素数さん
23/07/31 10:36:40.94 jznoxopE.net
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).
1069:１３２人目の素数さん
23/07/31 10:37:12.65 jznoxopE.net
From Proposition 1.1 one infers
1070:１３２人目の素数さん
23/07/31 10:37:44.26 jznoxopE.net
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).
1071:１３２人目の素数さん
23/07/31 10:38:25.31 jznoxopE.net
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).
1072:１３２人目の素数さん
23/07/31 10:38:55.22 jznoxopE.net
Recall that the following was proved in [H] by a basic argument of
functional analysis.
1073:１３２人目の素数さん
23/07/31 10:39:37.54 jznoxopE.net
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.
1074:１３２人目の素数さん
23/07/31 10:40:12.07 jznoxopE.net
Hence we obtain
1075:１３２人目の素数さん
23/07/31 10:40:52.24 jznoxopE.net
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.
1076:１３２人目の素数さん
23/07/31 10:41:28.54 jznoxopE.net
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.
1077:１３２人目の素数さん
23/07/31 10:41:57.49 jznoxopE.net
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.
1078:１３２人目の素数さん
23/07/31 10:42:29.70 jznoxopE.net
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: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
1080:１３２人目の素数さん
23/07/31 10:43:00.40 jznoxopE.net
§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) = ∞.
1081:１３２人目の素数さん
23/07/31 10:43:37.26 jznoxopE.net
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).
1082:１３２人目の素数さん
23/07/31 10:44:29.89 jznoxopE.net
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) = ∞
1083:１３２人目の素数さん
23/07/31 10:45:25.12 jznoxopE.net
This observation will be basic for the proofs of Theorems 0.4 and
0.5.
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