純粋・応用数学・数学隣接分野（含むガロア理論）13

[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)
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1(3): 2023/01/24(火)11:35 ID:7EkKRL+N(1/7) AAS
クレレ誌：
外部ﾘﾝｸ:ja.wikipedia.org
クレレ誌はアカデミーの紀要ではない最初の主要な数学学術誌の一つである（Neuenschwander 1994, p. 1533）。ニールス・アーベル、ゲオルク・カントール、ゴットホルト・アイゼンシュタインらの研究を含む著名な論文を掲載してきた。
(引用終り)

そこで
現代の純粋・応用数学・数学隣接分野（含むガロア理論）スレとして
新スレを立てる（＾＾；

＜前スレ＞
純粋・応用数学・数学隣接分野（含むガロア理論）12
2chｽﾚ:math
省16

983: 2023/07/31(月)10:34 ID:jznoxopE(33/50) AAS
We put
H
p,q
(2),φ
(M, E)(= H
p,q
(2),g,φ
(M, E)) =
Ker (
¯∂ : L
省24

984: 2023/07/31(月)10:34 ID:jznoxopE(34/50) AAS
Let Λ = Λg denote the adjoint of the exterior multiplication by ω.
Then Nakano’s formula
(2.2) ¯∂
¯∂
∗ + ¯∂
∗ ¯∂ − ∂h∂
∗ − ∂
∗
∂h =
√
省21

985: 2023/07/31(月)10:35 ID:jznoxopE(35/50) AAS
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.

986: 2023/07/31(月)10:36 ID:jznoxopE(36/50) AAS
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
(
省20

987: 2023/07/31(月)10:37 ID:jznoxopE(37/50) AAS
From Proposition 1.1 one infers

988: 2023/07/31(月)10:37 ID:jznoxopE(38/50) AAS
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
′
省21

989: 2023/07/31(月)10:38 ID:jznoxopE(39/50) AAS
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
省20

990: 2023/07/31(月)10:38 ID:jznoxopE(40/50) AAS
Recall that the following was proved in [H] by a basic argument of
functional analysis.

991: 2023/07/31(月)10:39 ID:jznoxopE(41/50) AAS
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),
省10

992: 2023/07/31(月)10:40 ID:jznoxopE(42/50) AAS
Hence we obtain

993: 2023/07/31(月)10:40 ID:jznoxopE(43/50) AAS
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.

994: 2023/07/31(月)10:41 ID:jznoxopE(44/50) AAS
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.

995: 2023/07/31(月)10:41 ID:jznoxopE(45/50) AAS
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.

996: 2023/07/31(月)10:42 ID:jznoxopE(46/50) AAS
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
省4

997: 2023/07/31(月)10:43 ID:jznoxopE(47/50) AAS
§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
省4

998: 2023/07/31(月)10:43 ID:jznoxopE(48/50) AAS
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 + ∂
省48

999: 2023/07/31(月)10:44 ID:jznoxopE(49/50) AAS
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=
省3

1000: 2023/07/31(月)10:45 ID:jznoxopE(50/50) AAS
This observation will be basic for the proofs of Theorems 0.4 and
0.5.

1001(1): 1001 ID:Thread(1/2) AAS
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1002(1): 1002 ID:Thread(2/2) AAS
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