[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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973: 2023/07/31(月)10:12 ID:jznoxopE(23/50) AAS
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.

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974: 2023/07/31(月)10:13 ID:jznoxopE(24/50) AAS
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}.

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975: 2023/07/31(月)10:15 ID:jznoxopE(25/50) AAS
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}
.

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976: 2023/07/31(月)10:17 ID:jznoxopE(26/50) AAS
The definition of L^{p,q}_{(2),φ}(M, E) will be
naturally extended for continuous metrics and continuous weights.

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977: 2023/07/31(月)10:27 ID:jznoxopE(27/50) AAS
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.

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978: 2023/07/31(月)10:29 ID:jznoxopE(28/50) AAS
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

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979: 2023/07/31(月)10:30 ID:jznoxopE(29/50) AAS
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).

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980: 2023/07/31(月)10:32 ID:jznoxopE(30/50) AAS
Θ > 0 (resp. ≥0) for an E^∗ ⊗ E-valued (1,1)-form Θ will mean the positivity (resp.
semipositivity) in this sense.

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981: 2023/07/31(月)10:32 ID:jznoxopE(31/50) AAS
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.

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982: 2023/07/31(月)10:33 ID:jznoxopE(32/50) AAS
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. ¯∂
∗
).

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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
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).

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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 =
√
−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
).

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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.

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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
(
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).

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

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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
′
(
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).

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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
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).

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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.

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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),
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.

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

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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.

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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.

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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.

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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
n,q
(2),εφ
(M, E)
for q ≥ 1

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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
xµ for any µ. Then
dim H
n,0
(M, E) = ∞.

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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 + ∂
¯∂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).

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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=
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) = ∞

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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.

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