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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