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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