Cohomology of vector bundles and non-pluriharmonic loci

TL;DR

This paper investigates the cohomology of vector bundles near non-pluriharmonic loci in Stein and projective manifolds, leading to variants of the Lefschetz hyperplane theorem.

Contribution

It introduces new results on cohomology groups of vector bundles in complex manifolds and derives variants of the Lefschetz hyperplane theorem.

Findings

Cohomology groups are characterized near non-pluriharmonic loci.

Variants of the Lefschetz hyperplane theorem are established.

Results apply to both Stein and projective manifolds.

Abstract

In this paper, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

Equations

$$H^0(X,F)\to \underset{{\bigcap }_{j=1}^m\mathrm{supp}i\partial \overline{\partial }{\phi }_j\subset V}{\underrightarrow{\lim }}{H}^0(V,F).$$

$$\underset{{\bigcap }_{j=1}^m\mathrm{supp}i\partial \overline{\partial }{\phi }_j\subset V}{\underrightarrow{\lim }}{H}^q(V,F)=0$$

$$H^q(X,F)\to \underset{{\bigcap }_{j=1}^m\mathrm{supp}{T}_j\subset V}{\underrightarrow{\lim }}{H}^q(V,F)$$

$$H^q(X,F)\to \underset{Y\subset V}{\underrightarrow{\lim }}{H}^q(V,F)$$

$$H^q(X,\mathbb{C})\to \underset{{\bigcap }_{j=1}^m\mathrm{supp}(i\partial \overline{\partial }{\phi }_j)\subset V}{\underrightarrow{\lim }}{H}^q(V,\mathbb{C})$$

$$H^q(X,\mathbb{C})\to \underset{{\bigcap }_{j=1}^m\mathrm{supp}{T}_j\subset V}{\underrightarrow{\lim }}{H}^q(V,\mathbb{C})$$

$$\Vert u{\Vert }_{H,\omega }^2={\int }_X|u{|}_{H,\omega }^{2}d{V}_{\omega }<+\infty .$$

$$\Vert v{\Vert }_{H,\omega }^2\le {\int }_X⟨[i\Theta (F,H),\Lambda {]}^{-1}u,u{⟩}_{H,\omega }d{V}_{\omega }$$

$${\int }_D|u{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }\le C_{q,\delta }{\int }_D|f{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }.$$

$${\int }_{D_k}|u_k{|}_{H^{\ast },\omega }^2{e}^{-N\psi -\kappa \circ \rho }d{V}_{\omega }\le {\int }_{D_k}⟨[i\Theta (F^{\ast },H^{\ast }{e}^{-N\psi -\kappa \circ \rho }),\Lambda {]}^{-1}f,f{⟩}_{H^{\ast },\omega }{e}^{-N\psi -\kappa \circ \rho }d{V}_{\omega }.$$

$${\int }_{D_k}|u_k{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }={\int }_{D_k}|u_k{e}^{\delta \varphi }{|}_{H^{\ast },\omega }^2{e}^{-N\psi -\kappa \circ \rho -\delta \varphi }d{V}_{\omega }$$

$$\le$$

$$+(1+t){\delta }^2{\int }_{D_k}⟨[i\Theta (F^{\ast },H^{\ast }{e}^{-N\psi -\kappa \circ \rho -\delta \varphi }),\Lambda {]}^{-1}\overline{\partial }\varphi \wedge u_k,\overline{\partial }\varphi \wedge u_k{⟩}_{H^{\ast },\omega }{e}^{-\xi }d{V}_{\omega }$$

$$(1+t){\delta }^2⟨[i\Theta (F^{\ast },H^{\ast }{e}^{-N\psi -\kappa \circ \rho -\delta \varphi }),\Lambda {]}^{-1}\overline{\partial }\varphi \wedge u_k,\overline{\partial }\varphi \wedge u_k{⟩}_{H^{\ast },\omega }\le (1+t)\frac{\delta }{q}|u_k{|}_{H^{\ast },\omega }^2.$$

$${\int }_{D_k}|u_k{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }\le C_{q,\delta }{\int }_{D_k}|f{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }\le C_{q,\delta }{\int }_D|f{|}_{H^{\ast },\omega }^2{e}^{-\xi }d{V}_{\omega }.$$

$${\int }_D|\beta {|}_{H,\omega }^2{e}^{\xi }d{V}_{\omega }\le C_{q,\delta }{\int }_D|\alpha {|}_{H,\omega }^2{e}^{\xi }d{V}_{\omega }.$$

$${\int }_{U\cap D}|\alpha {|}_{H,\omega }^2{e}^{\xi }d{V}_{\omega }\le C{\int }_{U\cap D}|\alpha {|}_{H,i\partial \overline{\partial }\psi }^2|\phi {|}^{\delta -2}(i\partial \overline{\partial }\psi {)}^n<+\infty .$$

$$\frac{{\kappa }_j^{(k)}(t)}{({\kappa }_j(t){)}^{k+1}}<C_k$$

$$({\kappa }_j(t){)}^{k+1}\ge \sum _{l=0}^j(\genfrac{}{}{0ex}{}{l+k}{l})t^l=\frac{1}{k!}\sum _{l=0}^j\frac{(l+k)!}{l!}{t}^l\ge \frac{1}{k!}{\kappa }_j^{(k)}(t).$$

$$\sum _{|K|=k}{\int }_U|D^K(\chi {\beta }_j){|}_{H^{\prime },\omega }^2{e}^{{\kappa }_j\circ \rho /2^k}d{V}_{\omega }<C_k$$

$$\sum _{|K|=k}{\int }_U|\vartheta D^K(\chi {\beta }_j){|}_{H^{\prime },\omega }^2{e}^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }<C_{k+1}$$

$$\ast \partial \ast (\chi {\beta }_j)=\ast (\partial \chi \wedge \ast {\beta }_j)-\ast {\Gamma }_{H^{\prime }}\ast (\chi {\beta }_j)-\partial {\kappa }_j\circ \rho \wedge (\chi {\beta }_j).$$

$$\vartheta D^K(\chi {\beta }_j)=\ast \partial \ast D^K(\chi {\beta }_j)=D^K\ast \partial \ast (\chi {\beta }_j)+L{\beta }_j,$$

$$\sum _{k^{\prime }\le k,l\le k+1}\sum _{|K^{\prime }|=k^{\prime }}{t}_{K^{\prime },l}{\kappa }_j^{(l)}\circ \rho D^{K^{\prime }}{\beta }_j$$

$${\int }_U|\vartheta D^K(\chi {\beta }_j){|}_{H^{\prime },\omega }^2{e}^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }$$

$$\le$$

$${\int }_U(|Ag{|}_{\omega }^2+|Bg{|}_{\omega }^2)e^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }\le {\int }_U(2|\overline{\partial }g{|}_{\omega }^2+2|\vartheta g{|}_{\omega }^2+C|g{|}_{\omega }^2)e^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }.$$

$$\displaystyle\sum_{|I|=q-1}\sum_{l=1}^{n}\int_{U}\big{|}\frac{\partial g_{I}}{\partial\bar{\theta}_{l}}\big{|}^{2}e^{\kappa_{j}\circ\rho/2^{k+1}}dV_{\omega}-\sum_{|J|=q-2}\sum_{l,t}\int_{U}\left(\frac{\partial g_{lJ}}{\partial\bar{\theta}_{t}}\frac{\partial\bar{g}_{tJ}}{\partial\theta_{l}}-\frac{\partial g_{lJ}}{\partial\theta_{l}}\frac{\partial\bar{g}_{tJ}}{\partial\bar{\theta}_{t}}\right)e^{\kappa_{j}\circ\rho/2^{k+1}}dV_{\omega}.$$

$${\int }_U\left(\frac{{\partial }^2{g}_{lJ}}{\partial {\bar{\theta }}_t\partial {\theta }_l}-\frac{{\partial }^2{g}_{lJ}}{\partial {\theta }_l\partial {\bar{\theta }}_t}\right){\bar{g}}_{tJ}{e}^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }+{\int }_{U}R{e}^{{\kappa }_j\circ \rho /2^{k+1}}d{V}_{\omega }.$$

$$\displaystyle\int_{U}\big{|}g_{I_{1}}\frac{\partial g_{I_{2}}}{\partial\bar{\theta}_{t}}\big{|}\kappa_{j}^{(i)}\circ\rho\,e^{\kappa_{j}\circ\rho/2^{k+1}}dV_{\omega}$$

$$\le$$

Full text

1. Introduction

Let $X$ be a Stein manifold. Let $\varphi$ be an exhaustive plurisubharmonic function on $X$. The support of $i\partial\overline{\partial}\varphi$ has some interesting properties. In this paper, we study the cohomology of holomorphic vector bundles on open neighborhoods of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$. Here we denote by $\mathrm{supp}\,T$ the support of a current $T$. Let $F$ be a holomorphic vector bundle over $X$. Let $A\subset X$ be a closed set. For any open neighborhoods $V\subset U$ of $A$, the inclusion map induces $H^{q}(U,F)\to H^{q}(V,F)$. We define the direct limit $\underset{A\subset V}{\varinjlim}H^{q}(V,F)$ where $V$ runs through all open neighborhoods of $A$. Our first main result is the following:

Theorem 1**.**

Let $X$ be a Stein manifold of dimension $n$ ($n\geq 3$). Let $m$ be a positive integer which satisfies $1\leq m\leq n-2$. Let $\varphi_{1},\ldots,\varphi_{m}$ be non-constant plurisubharmonic functions on $X$ such that for every $r<\sup_{X}\varphi_{j}$ the sublevel set $\{z\in X\,|\,\varphi_{j}(z)\leq r\}$ is compact ($1\leq j\leq m$). Let $F$ be a holomorphic vector bundle over $X$. Then the natural map

[TABLE]

is an isomorphism and

[TABLE]

for $0<q<n-m-1$.

Let $X$ be a projective manifold. We have the Hodge decompotion $H^{2}(X,\mathbb{C})=H^{2,0}(X,\mathbb{C})\oplus H^{1,1}(X,\mathbb{C})\oplus H^{0,2}(X,\mathbb{C})$. Define $H^{1,1}(X,\mathbb{R})=H^{1,1}(X,\mathbb{C})\cap H^{2}(X,\mathbb{R})$. Let $\mathcal{K}_{NS}\subset H^{1,1}(X,\mathbb{R})$ be the open cone generated by classes of ample divisors (see Section 6 of [4]). Our second main result is the following:

Theorem 2**.**

Let $X$ be a projective manifold of dimension $n$ ($n\geq 3$). Let $m$ be a positive integer which satisfies $1\leq m\leq n-2$. Let $T_{1},\ldots,T_{m}$ be closed positive currents of type $(1,1)$ on $X$ whose cohomology classes belong to $\mathcal{K}_{NS}$. let $F$ be a holomorphic vector bundle over $X$. Then the natural map

[TABLE]

is an isomorphism for $0\leq q<n-m-1$ and is injective for $q=n-m-1$.

Let $t\in H^{0}(X,L)$ be a non-zero holomorphic section of the ample line bundle $L$ over $X$. We define the hypersurface $Y=\{z\in X\,|\,t(z)=0\}$. Let $h_{0}$ be a smooth hermitian metric of $L$ such that $\omega_{0}:=2\pi c_{1}(L,h_{0})$ is Kähler form. Here $c_{1}(L,h_{0})$ is the Chern form of $L$ associated to $h_{0}$. Then $Y=\mathrm{supp}\,(\omega_{0}+i\partial\overline{\partial}\log|t|^{2}_{h_{0}})$. Because of the vanishing theorem of cohomology groups with compact supports in the Stein manifold $X\setminus Y$, we have that the natural map

[TABLE]

is an isomorphism for $q<n-1$ and is injective for $q=n-1$. Theorem 2 is a counterpart of this result. Unfortunately, we do not know whether Theorem 1 and Theorem 2 hold in the case when the degree is $n-m$.

Let $T^{*}_{X}$ be the holomorphic cotangent bundle over $X$. If we take $F=\bigwedge^{p}T^{*}_{X}$, our main results imply the following variants of the Lefschetz hyperplane theorem (see Lemma 1 of [9]).

Corollary 1**.**

Let $X,\varphi_{1},\ldots,\varphi_{m}$ be as in Theorem 1. Then the natural map

[TABLE]

is an isomorphism for $q<n-m-1$ and is injective for $q=n-m-1$.

Corollary 2**.**

Let $X,T_{1},\ldots,T_{m}$ be as in Theorem 2. Then the natural map

[TABLE]

is an isomorphism for $q<n-m-1$ and is injective for $q=n-m-1$.

Corollary 1 generalizes the main theorem of [9]. The degree which appears in [9] is $\max\{n-4,1\}$. On the other hand, those which appear in our main results are $n-2$ when $m=1$. The improvement of the degree is due to the method of Lee and Nagata ([8]) and the estimate of the Sobolev norm.

We prove the case $m=1$ of Theorem 1, 2 in Section 3, 4, 5. By using Mayer-Vietoris sequence, we prove the general case in Section 6.

Acknowledgment. The author would like to thank Seungjae Lee and Yoshikazu Nagata for sending him their paper [8]. This work was supported by the Grant-in-Aid for Scientific Research (KAKENHI No.​ 17K14200).

2. Preliminaries

Let $X$ be a Kähler manifold and let $\omega$ be a Kähler metric on $X$. We assume that $X$ is weakly pseudoconvex, that is, there exists a smooth plurisubharmonic exhaustion function on $X$. Let $F$ be a holomorphic vector bundle over $X$ and let $H$ be a smooth hermitian metric of $F$. We denote by $L^{(p,q)}(X,F,H,\omega)$ the Hilbert space of $F$-valued $(p,q)$-forms $u$ which satisfy

[TABLE]

Here $dV_{\omega}=\frac{\omega^{n}}{n!}$. Let $i\Theta(F,H)$ be the Chern curvature tensor of $(F,H)$ and let $\Lambda$ be the adjoint of multiplication of $\omega$. Suppose that the operator $[i\Theta(F,H),\Lambda]$ acting on $(n,q)$-forms with values in $F$ is positive definite on $X$ ($q\geq 1$). Then, for any $\overline{\partial}$-closed form $u\in L^{(n,q)}(X,F,H,\omega)$ which satisfies $\int_{X}\langle[i\Theta(F,H),\Lambda]^{-1}u,u\rangle_{H,\omega}dV_{\omega}<+\infty$, there exists $v\in L^{(n,q-1)}(X,F,H,\omega)$ such that $\overline{\partial}v=u$ and that

[TABLE]

(cf. [3]). We note that $\omega$ is possibly non complete.

3. $L^{2}$-estimate

In [9], the surjectivity between the cohomology groups was proved by the Donnelly-Fefferman-Berndtsson type $L^{2}$-estimates for $(0,q)$-forms ([1], [5]). In [8], Lee and Nagata showed that $L^{2}$-Serre duality and $L^{2}$-estimates for not $(0,q)$ but $(n,q)$-forms improve the integrability condition of an $L^{2}$-estimate. By using the method of [8], we prove Proposition 1 below.

Let $X$ be a Stein manifold of dimension $n$ and let $D$ be a relatively compact subdomain in $X$. Assume that there exist negative plurisubharmonic functions $\varphi,\eta\in C^{\infty}(\overline{D})$ on $D$ such that $\max\{\varphi(z),\eta(z)\}\to 0$ when $z\to\partial D$. We assume that $\inf_{D}\eta<-1$. Define $\phi=-\log(-\varphi)$ and $\rho=\max_{\varepsilon}\{-\log(-\eta),0\}$. Here $\varepsilon>0$ is a small positive number and $\max_{\varepsilon}$ is a regularized max function (see Chapter I, Section 5 of [3]). Let $F$ be a holomorphic vector bundle over $X$ and $H$ be a smooth hermitian metric of $F$. We define $F^{*},H^{*}$ to be the dual of $F,H$. Let $\psi\in C^{\infty}(\overline{D})$ be a strictly plurisubharmonic function. We take a large positive integer $N$ such that the hermitian vector bundle $(F^{*},H^{*}e^{-(N-1)\psi})$ is Nakano positive on $\overline{D}$. Let $\delta>0$ be a positive number. Put $\omega=i\partial\overline{\partial}(\frac{\psi}{\delta}+\frac{\rho}{\delta}+\phi)$. Then $\omega$ is a complete Kähler metric on $D$. Let $\kappa\in C^{\infty}(\mathbb{R})$ such that $\kappa^{\prime}(t)\geq 1$, $\kappa^{\prime\prime}(t)\geq 0$ for $t\geq 0$. Put $\xi=N\psi+\kappa\circ\rho-\delta\phi$. The proof of the following lemma is completely similar to that of Lemma 3.1 of [8].

Lemma 1**.**

Let $f\in L^{(n,q)}(D,F^{*},H^{*}e^{-\xi},\omega)$ such that $\overline{\partial}f=0$. Assume that $\delta<q$. Then there exists $u\in L^{(n,q-1)}(D,F^{*},H^{*}e^{-\xi},\omega)$ such that $\overline{\partial}u=f$ and that

[TABLE]

Here $C_{q,\delta}$ depends only on $q$ and $\delta$.

Proof.

There exist relatively compact weakly pseudoconvex subdomains $D_{1}\subset D_{2}\subset\cdots\subset D$ which exhaust $D$. Because of the Nakano positivity of $(F^{*},H^{*}e^{-(N-1)\psi})$, there exists $u_{k}$ in $L^{(n,q-1)}(D_{k},F^{*},H^{*}e^{-N\psi-\kappa\circ\rho})$ such that $\overline{\partial}u_{k}=f$ and that

[TABLE]

Since $\phi$ and $\overline{\partial}\phi$ are bounded in $D_{k}$, we have that $u_{k}e^{\delta\phi}$ is the minimal solution of $\overline{\partial}(u_{k}e^{\delta\phi})=(f+\delta\overline{\partial}\phi\wedge u_{k})e^{\delta\phi}$ in $L^{(n,q-1)}(D_{k},F^{*},H^{*}e^{-N\psi-\kappa\circ\rho-\delta\phi},\omega)$. Then

[TABLE]

for every $t>0$. We have that $\langle[i\Theta(F^{*},H^{*}e^{-N\psi-\kappa\circ\rho-\delta\phi}),\Lambda]v,v\rangle_{H^{*},\,\omega}\geq q\delta|v|^{2}_{H^{*},\,\omega}$ for any $F$-valued $(n,q)$-form $v$ since $i\partial\overline{\partial}(\psi+\kappa\circ\rho+\delta\phi)\geq\delta\omega$. Because $|\overline{\partial}\phi|_{\omega}\leq 1$, we have that

[TABLE]

By taking $t$ sufficiently small, we have that

[TABLE]

The constant $C_{q,\delta}$ does not depend on $k$. Hence we may choose a subsequence of $\{u_{k}\}_{k\in\mathbb{N}}$ converging weakly in $L^{(n,q-1)}(D,F^{*},H^{*}e^{-\xi},\omega)$ to $u$. Then $u$ is the $F$-valued $(n,q)$-form we are looking for. ∎

Lemma 2**.**

Let $\alpha\in L^{(0,q)}(D,F,He^{\xi},\omega)$ such that $\overline{\partial}\alpha=0$. Assume that $q\geq 1$ and that $\delta<n-q$. Then there exists $\beta\in L^{(0,q-1)}(D,F,He^{\xi},\omega)$ such that $\overline{\partial}\beta=\alpha$ and that

[TABLE]

Proof.

Let $\star_{F}$ be the Hodge-star operator $L^{(0,q)}(D,F,He^{\xi},\omega)\to L^{(n,n-q)}(D,F^{*},H^{*}e^{-\xi},\omega)$ as in [2]. Let $\overline{\partial}_{F^{*}}^{*}$ be the Hilbert space adjoint to $\overline{\partial}:L^{(n,n-q-1)}(D,F^{*},H^{*}e^{-\xi},\omega)\to L^{(n,n-q)}(D,F^{*},H^{*}e^{-\xi},\omega)$. We note that the formal adjoint of $\overline{\partial}$ is equal to $-\star_{F^{*}}\overline{\partial}\,\star_{F}$. Since $\omega$ is complete, we have that $\star_{F}\alpha\in L^{(n,n-q)}(D,F^{*},H^{*}e^{-\xi},\omega)$ is contained in the domain of $\overline{\partial}_{F^{*}}^{*}$ and $\overline{\partial}_{F^{*}}^{*}\star_{F}\alpha=0$. Lemma 1 shows that there exists $u\in L^{(n,n-q+1)}(D,F^{*},H^{*}e^{-\xi},\omega)$ such that $\overline{\partial}_{F^{*}}^{*}u=\star_{F}\alpha$ and that $\|u\|^{2}_{H^{*}e^{-\xi},\,\omega}\leq C_{q,\delta}\|\star_{F}\alpha\|^{2}_{H^{*}e^{-\xi},\,\omega}=C_{q,\delta}\|\alpha\|^{2}_{He^{\xi},\,\omega}$. By the completeness of $\omega$ again, we have that $\beta=\star_{F^{*}}u\in L^{(0,q-1)}(D,F,He^{\xi},\omega)$ is contained in the domain of $\overline{\partial}$ and $\overline{\partial}\beta=\alpha$. We have that $\|\beta\|^{2}_{He^{\xi},\,\omega}\leq C_{q,\delta}\|\alpha\|^{2}_{He^{\xi},\,\omega}$ and this completes the proof. ∎

We denote by $\Omega^{(0,q)}(D,F)$ (resp. $\Omega^{(0,q)}(\overline{D},F)$) the space of $F$-valued smooth $(0,q)$-forms on $D$ (resp. on a neighborhood of $\overline{D}$). Let $\partial^{\prime}D=\{z\in\partial D\,|\,\varphi(z)=0\}$.

Proposition 1**.**

Let $1\leq q\leq n-2$. Assume that $\varphi$ is pluriharmonic on a neighborhood of $\partial^{\prime}D$ and $d\varphi\neq 0$ on $\partial^{\prime}D$. Let $\alpha\in\Omega^{(0,q)}(\overline{D},F)$ such that $\overline{\partial}\alpha=0$ in $D$ and that $\mathrm{supp}\,\alpha\cap D\subset\{z\in D\,|\,\rho=0\}$. Then there exists $\beta\in\Omega^{(0,q-1)}(D,F)$ such that $\overline{\partial}\beta=\alpha$ and that $\mathrm{supp}\,\beta\subset\{z\in D\,|\,\rho\leq 1\}$.

Proof.

Let $1<\delta<2$. If $a\in\{z\in\partial D\,|\,\eta(z)\neq 0\}$, then $a\in\partial^{\prime}D$. There exists a small neighborhood $U\subset X$ of $a$ such that $\varphi$ is pluriharmonic and $i\partial\overline{\partial}\phi=i\frac{\partial\varphi\wedge\overline{\partial}\varphi}{\varphi^{2}}$ on $U\cap D$. Since $\eta(a)\neq 0$, we may assume that $i\partial\overline{\partial}\rho$ and $\kappa\circ\rho$ are bounded. Then $e^{\xi}dV_{\omega}\leq C|\varphi|^{\delta-2}(i\partial\overline{\partial}\psi)^{n}$ on $U\cap D$. Hence

[TABLE]

If $b\in\{z\in\partial D\,|\,\eta(z)=0\}$, there exists a small open neighborhood $U\subset X$ of $b$ such that $U\cap\mathrm{supp}\,\alpha=\emptyset$. Hence we have that $\alpha\in L^{(0,q)}(D,F,He^{\xi},\omega)$.

Let $\kappa_{j}\in C^{\infty}(\mathbb{R})$ ($j=1,2,\cdots$) be functions which satisfies the following conditions:

• (i)

$\kappa_{j}(t)\leq\kappa_{j+1}(t)$ for any $j\in\mathbb{N}$ and $t\geq 0$,

• (ii)

$\kappa_{j}^{\prime}(t)\geq 1$ and $\kappa_{j}^{\prime\prime}(t)\geq 0$ for any $j\in\mathbb{N}$ and $t\geq 0$,

• (iii)

there exists a positive constant $C$ such that $\kappa_{j}(t)\leq C$ for any $j\in\mathbb{N}$ and $0\leq t\leq 1/2$,

• (iv)

$\lim_{j\to\infty}\kappa_{j}(t)=+\infty$ for any $t\geq 1$.

Let $\xi_{j}=N\psi+\kappa_{j}\circ\rho-\delta\phi$. We have that $\alpha\in L^{(0,q)}(D,F,He^{\xi_{j}},\omega)$. Since $q\leq n-2$, there exist $\beta_{j}\in L^{(0,q-1)}(D,F,He^{\xi_{j}},\omega)$ such that $\overline{\partial}\beta_{j}=\alpha$ and $\|\beta_{j}\|^{2}_{He^{\xi_{j}},\,\omega}\leq C_{q,\delta}\|\alpha\|^{2}_{He^{\xi_{j}},\omega}$ by Lemma 2. Because $\kappa_{j}\circ\rho\leq C$ on $\mathrm{supp}\,\alpha$, we have that $\|\alpha\|^{2}_{He^{\xi_{j}},\,\omega}$ does not depend on $j$. Take a weakly convergent subsequence $\{\beta_{j_{\nu}}\}_{\nu\in\mathbb{N}}$ in $L^{(0,q-1)}(D,F,He^{N\psi-\delta\phi},\omega)$ and there exists the weak limit $\beta\in L^{(0,q-1)}(D,F,He^{N\psi-\delta\phi},\omega)$ such that $\overline{\partial}\beta=\alpha$ and $\mathrm{supp}\,\beta\subset\{z\in D\,|\,\rho(z)\leq 1\}$. The regularity of $\beta$ will be discussed in Section 4. ∎

4. Estimate of the Sobolev norm

It is enough to consider the case $q\geq 2$. In the proof of Proposition 1, we take $\kappa_{j}\in C^{\infty}(\mathbb{R})$ ($j\in\mathbb{N}$) which satisfy four conditions. We add the following condition to $\{\kappa_{j}\}_{j\in\mathbb{N}}$:

• (v)

For any non-negative integer $k$, there exists positive constant $C_{k}$ which does not depend on $j$ and satisfies

[TABLE]

for any $j\in\mathbb{N}$ and $t\geq 0$.

Lemma 3**.**

There exist functions $\kappa_{j}\in C^{\infty}(\mathbb{R})$ ($j\in\mathbb{N}$) which satisfy the above five conditions.

Proof.

We define $\kappa_{j}(t)=\sum_{l=0}^{j}t^{l}$. It is easy to see that $\{\kappa_{j}\}_{j\in\mathbb{N}}$ satisfies the conditions (i), (ii), (iii), (iv). For $t\geq 0$, we have that

[TABLE]

Hence $\{\kappa_{j}\}_{j\in\mathbb{N}}$ satisfies (v). ∎

Assume $\{\kappa_{j}\}_{j\in\mathbb{N}}$ satisfies the above five conditions. Take $\{\beta_{j}\}_{j\in\mathbb{N}}$ and $\beta$ as in the proof of Proposition 1. We may assume that $\beta_{j}$ is orthogonal to the kernel of $\overline{\partial}:L^{(0,q-1)}(D,F,He^{\xi_{j}},\omega)\to L^{(0,q)}(D,F,He^{\xi_{j}},\omega)$ and $\beta_{j}$ is smooth (cf. [7]). Let $a\in D$ and let $\chi\in C^{\infty}(D)$ be a non-negative function such that $\chi=1$ on a neighborhood of $a$ and $\mathrm{supp}\,\chi$ is sufficiently small. Denote by $H^{\prime}=He^{N\psi-\delta\phi}$ the hermitian metric of $F$. Then $H^{\prime}e^{\kappa_{j}\circ\rho}=He^{\xi_{j}}$. We may assume that $\mathrm{supp}\,\chi$ is contained in a complex chart $U$ and that $F$ is trivialized there. Let $(x_{1},\ldots,x_{2n})$ be a local coordinates on $U$. Let $K=(k_{1},k_{2},\ldots,k_{2n})$ be the multi-index and let $|K|=k_{1}+\cdots+k_{2n}$. Define $D^{K}:\Omega^{(0,q-1)}(U,F)\to\Omega^{(0,q-1)}(U,F)$ by $D^{K}=\left(\frac{\partial}{\partial x_{1}}\right)^{k_{1}}\cdots\left(\frac{\partial}{\partial x_{2n}}\right)^{k_{2n}}$.

Lemma 4**.**

Let $k$ be a non-negative integer. Then

[TABLE]

for any $j\in\mathbb{N}$ where $C_{k}$ is a constant which does not depend on $j$.

Let $W^{k}_{(0,q-1)}(U,F)$ be the Sobolev space of $F$-valued $(0,q-1)$-forms whose derivatives up to order $k$ are in $L^{(0,q-1)}(U,F,H^{\prime},\omega)$. Lemma 4 implies that the subset $\{\chi\beta_{j}\}_{j\in\mathbb{N}}$ is bounded in $W^{k}_{(0,q-1)}(U,F)$. By taking weakly convergent subsequence of $\{\chi\beta_{j_{\nu}}\}_{\nu}$, we have that $\beta\in W^{k}_{(0,q-1)}(U^{\prime},F)$ where $U^{\prime}$ is a sufficiently small open neighborhood of $a$ contained in $U$. By the Sobolev lemma (cf. [3]), it follows that $\beta$ is smooth in $D$.

The proof of Lemma 4 proceeds by induction on $k$. We have seen that $\|\beta_{j}\|^{2}_{H^{\prime}e^{\kappa_{j}\circ\rho},\,\omega}$ does not depend on $j$, and the case $k=0$ holds. Let $\nabla_{j}$ be the hermitian connection on $F$ which is compatible with $H^{\prime}e^{\kappa_{j}\circ\rho}$. By the trivialization of $F$ in $U$, we have that $\nabla_{j}=d+\Gamma_{H^{\prime}}+\partial\kappa_{j}\circ\rho\,\mathrm{Id}_{F}$ where $\Gamma_{H^{\prime}}$ is the connection form defined by $H^{\prime}$. Let $\ast$ be the Hodge-star operator from $F$-valued $(p,q)$-form to $F$-valued $(n-q,n-p)$-form. Define $\vartheta=\ast\partial*:\Omega^{(0,q-1)}(U,F)\to\Omega^{(0,q-2)}(U,F)$. Then $-\vartheta$ is the formal adjoint of $\overline{\partial}:L^{(0,q-1)}(U,F,H_{Id},\omega)\to L^{(0,q)}(U,F,H_{Id},\omega)$. Here $H_{Id}$ is the flat metric of the trivial vector bundle $F$ over $U$.

Lemma 5**.**

Under the hypothesis that Lemma 4 holds for $\leq k$, we have that

[TABLE]

for any $j\in\mathbb{N}$ where $C_{k+1}$ is a constant which does not depend on $j$.

Proof.

We denote by $\overline{\partial}_{j}^{*}$ the Hilbert space adjoint to $\overline{\partial}:L^{(0,q-2)}(D,F,H^{\prime}e^{\kappa_{j}\circ\rho},\omega)\to L^{(0,q-1)}(D,F,H^{\prime}e^{\kappa_{j}\circ\rho},\omega)$. The formal adjoint of $\overline{\partial}$ is given by $-\ast(\partial+\Gamma_{H^{\prime}}+\partial\kappa_{j}\circ\rho\,\mathrm{Id})\ast$. Since $\beta_{j}$ is orthogonal to the kernel of $\overline{\partial}$ in $L^{(0,q-1)}(D,F,H^{\prime}e^{\kappa_{j}\circ\rho},\omega)$, we have that $\overline{\partial}_{j}^{*}\beta_{j}=0$. Then $\ast\partial\ast\beta_{j}=-\ast\Gamma_{H^{\prime}}\ast\beta_{j}-\ast\partial\kappa_{j}\circ\rho\wedge\beta_{j}$. Hence

[TABLE]

Let $K=(k_{1},\ldots,k_{2n})$ such that $k=k_{1}+\cdots+k_{2n}$. It follows that the order of the differential operator $\ast D^{K}-D^{K}\ast$ is $k-1$. Then

[TABLE]

where $L$ is the differential operator of order $k$. The term $D^{K}\ast\partial\ast(\chi\beta_{j})$ is written as

[TABLE]

where $t_{K^{\prime},l}$ is the function which does not depend on $j$. It follows that

[TABLE]

where $C$ does not depend on $j$. Since $\sup_{r\geq 0}r^{2(l+1)}e^{-r/2^{k+1}}<+\infty$, the condition (v) of $\{\kappa_{j}\}_{j\in\mathbb{N}}$ shows that the last term of the above inequality bounded from above by the constant which does not depend on $j$. ∎

A standard calculus shows that Lemma 5 implies Lemma 4 (cf. Chapter 5 of [7]).

Proof of Lemma 4.

We may assume that there exists an orthonormal frame $(\theta_{1},\ldots,\theta_{n})$ of $T^{*}_{X}$ on $U$. For any $f\in C^{\infty}(U)$, we define $\partial f/\partial\theta_{t}$ and $\partial f/\partial\bar{\theta}_{t}$ by $df=\sum_{t=1}^{n}\partial f/\partial\theta_{t}\,\theta_{t}+\partial f/\partial\bar{\theta}_{t}\,\bar{\theta}_{t}$. Let $I=(i_{1},\ldots,i_{q-1})$ be a multi-index with $i_{1}<\cdots<i_{q-1}$ and $\bar{\theta}_{I}=\bar{\theta}_{i_{1}}\wedge\cdots\wedge\bar{\theta}_{i_{q-1}}$. Let $g=\sum_{|I|=q-1}g_{I}\bar{\theta}_{I}$ be a smooth $(0,q-1)$-form on $U$ with compact support. Define $Ag=\sum_{|I|=q-1}\sum_{l=1}^{n}\frac{\partial g_{I}}{\partial\bar{\theta_{l}}}\bar{\theta}_{l}\wedge\bar{\theta}_{I}$ and $Bg=\sum_{|J|=q-2}\sum_{l}\frac{\partial g_{lJ}}{\partial\theta_{l}}\bar{\theta}_{J}$. Then $\overline{\partial}g-Ag$ and $-\vartheta g-Bg$ have no term where $g_{I}$ is differentiated. Hence we have that

[TABLE]

The left-hand side of the above inequality is equal to

[TABLE]

By integrating by parts, the second integral of the above is equal to

[TABLE]

Here $R$ is written as the sum of $s_{1}\,g_{I_{1}}\kappa_{j}^{(i)}\circ\rho\,\partial g_{I_{2}}/\partial\bar{\theta}_{t}$ and $s_{2}\,g_{I_{1}}g_{I_{2}}\kappa^{(i)}\circ\rho$ ($i=0,1,2$) where $s_{1}$ and $s_{2}$ are the smooth functions which depend on neither $g$ nor $j$. The order of the differential operator $\partial^{2}/\partial\theta_{l}\partial\bar{\theta}_{t}-\partial^{2}/\partial\bar{\theta}_{t}\partial\theta_{l}$ is one. The condition (v) of $\{\kappa_{j}\}_{j\in\mathbb{N}}$ shows that

[TABLE]

for any $\varepsilon>0$. Hence the left-hand side of (1) is bounded below by

[TABLE]

where $C$ does not depend on $j$. Because the order of the differential operator $\partial^{2}/\partial\theta_{l}\partial\bar{\theta}_{l}-\partial^{2}/\partial\bar{\theta}_{l}\partial\theta_{l}$ is one, we have that

[TABLE]

for any $\varepsilon>0$. Since $F$ is trivialized in $U$ and $H^{\prime}$ is equivalent to the flat metric $H_{\mathrm{Id}}$, we can replace $g$ by $D^{K}(\chi\beta_{j})$ in the above calculus, and we obtain

[TABLE]

Note that $\overline{\partial}(\chi\beta_{j})=\chi\alpha+\overline{\partial}\chi\wedge\beta_{j}$. The induction hypothesis and Lemma 5 show that the last term of the above inequality does not depend on $j$. ∎

5. Extension of closed $F$-valued forms in manifolds

In this section, we extend closed $F$-valued forms as in [9].

Let $X$ be a Stein manifold and let $\varphi$ be a non-constant plurisubharmonic function on $X$ such that for every $r<\sup_{X}\varphi$ the sublevel set $\{z\in X\,|\,\varphi(z)<r\}$ is compact. Let $F$ be a holomorphic vector bundle over $X$. Let $\psi\in C^{\infty}(X)$ be an exhaustive strictly plurisubharmonic function. We define $D_{\varphi}(r)=\{z\in X\,|\,\varphi(z)<r\}$.

Lemma 6**.**

Let $U\subset X$ be an open neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$ and let $u\in\Omega^{(0,q-1)}(U,F)$ ($1\leq q\leq n-2$) such that $\overline{\partial}u=0$. Let $r<\sup\varphi$ such that $d\varphi\neq 0$ on $\partial D_{\varphi}(r)\setminus\mathrm{supp}\,i\partial\overline{\partial}\varphi$ (Note that $\varphi$ is smooth on $X\setminus\mathrm{supp}\,i\partial\overline{\partial}\varphi$). Then there exists $v\in\Omega^{(0,q-1)}(D_{\varphi}(r),F)$ such that $u=v$ on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi\cap D_{\varphi}(r)$.

Proof.

We take $\chi\in C^{\infty}(X)$ such that $0\leq\chi\leq 1$, $\chi=1$ on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$ and that $\mathrm{supp}\,\chi\subset U$. Then $\alpha:=\overline{\partial}(\chi u)$ is $\overline{\partial}$-closed. By Lemma 7 below, there exists a smooth plurisubharmonic function $\varphi_{\varepsilon}$ on a neighborhood of $\overline{D_{\varphi}(r)}$ such that $\varphi_{\varepsilon}\geq\varphi$ on $\overline{D_{\varphi}(r)}$, $\varphi_{\varepsilon}>\varphi$ on $\mathrm{supp}\,i\partial\overline{\partial}\varphi\cap\overline{D_{\varphi}(r)}$, and that $\varphi_{\varepsilon}=\varphi$ on $\mathrm{supp}\,\alpha$.

Let $\eta=\varphi_{\varepsilon}-\varphi$. It follows that $\eta$ is plurisubharmonic on $D_{\varphi}(r)\setminus\mathrm{supp}\,i\partial\overline{\partial}\varphi$ and that $\eta=0$ on $\mathrm{supp}\,\alpha$. Since $\eta$ is lower semi-continuous on $\overline{D_{\varphi}(r)}$, there exists $c>0$ such that $\eta>c$ on $\overline{D_{\varphi}(r)}\cap\mathrm{supp}\,i\partial\overline{\partial}\varphi$. Let $D^{\prime}=\{z\in D\,|\,\varphi(z)-r<0,\eta(z)-c/2<0\}$. Note $\eta\in C^{\infty}(\overline{D^{\prime}})$ since $\varphi\in C^{\infty}(D\setminus\mathrm{supp}\,i\partial\overline{\partial}\varphi)$. By Proposition 1, there exist $c^{\prime}<\frac{c}{2}$ and $\beta\in\Omega^{(0,q-1)}(D^{\prime},F)$ such that $\overline{\partial}\beta=\alpha$ and that $\mathrm{supp}\,\beta\subset\{z\in D^{\prime}\,|\,\eta(z)\leq c^{\prime}\}$. (If $D^{\prime}$ is a disjoint union of bounded pseudoconvex domains, we apply Proposition 1 to each component.) We extend $\beta$ by [math] on $D_{\varphi}(r)\setminus D^{\prime}$, and we consider $\beta$ as an element of $\Omega^{(0,q-1)}(D_{\varphi}(r),F)$. We have that $\overline{\partial}\beta=\alpha$ on $D_{\varphi}(r)$. Then $v=\chi u-\beta$ is the form we are looking for. ∎

Lemma 7**.**

Let $V\subset X$ be an open neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$. Let $t<\sup\varphi$. There exists a smooth plurisubharmonic function $\varphi_{\varepsilon}$ on $D_{\varphi}(t)$ such that $\varphi_{\varepsilon}\geq\varphi$ on $D_{\varphi}(t)$, $\varphi_{\varepsilon}>\varphi$ on $\mathrm{supp}\,i\partial\overline{\partial}\varphi$, and that $\varphi_{\varepsilon}=\varphi$ on $D_{\varphi}(t)\setminus V$.

Proof.

Since $X$ is Stein, we may assume that $X$ is a submanifold of $\mathbb{C}^{m}$. By the theorem of Docquier and Grauert, there exists an open neighborhood $W\subset\mathbb{C}^{m}$ of $X$ and a holomorphic retraction $\mu:W\to X$ (cf. Chapter VIII of [6]). Let $h:\mathbb{C}^{m}\to\mathbb{R}^{+}$ be a smooth function depending only on $|z|$ ($z\in\mathbb{C}^{m}$) whose support is contained in the unit ball and whose integral is equal to one. Define $h_{\varepsilon}(z)=(1/\varepsilon^{2m})h(z/\varepsilon)$ for $\varepsilon>0$. Let $W^{\prime}\subset W$ be a relatively compact open neighborhood of $D_{\varphi}(t)$. If $\varepsilon>0$ is sufficiently small, we can define $(\varphi\circ\mu)_{\varepsilon}=(\varphi\circ\mu)*h_{\varepsilon}$ on $W^{\prime}$. Put $\varphi_{\varepsilon}=(\varphi\circ\mu)_{\varepsilon}$ on $D_{\varphi}(t)$. Then we have $\varphi_{\varepsilon}\geq\varphi$ on $D_{\varphi}(t)$ and $\varphi_{\varepsilon}>\varphi$ on $\mathrm{supp}\,i\partial\overline{\partial}\varphi$. Since $\mu^{-1}(\mathrm{supp}\,i\partial\overline{\partial}\varphi)\cap(W^{\prime}\setminus\mu^{-1}(V))=\emptyset$, there exists small $\varepsilon>0$ such that the ball of radius $\varepsilon$ centered at every point of $W^{\prime}\setminus\mu^{-1}(V)$ does not intersect $\mu^{-1}(\mathrm{supp}\,i\partial\overline{\partial}\varphi)$. Then $(\varphi\circ\mu)_{\varepsilon}=\varphi\circ\mu$ on $W^{\prime}\setminus\mu^{-1}(V)$. ∎

Lemma 8**.**

Let $\varphi$ be a plurisubharmonic function on $X$ as in Theorem 1. Let $r<s<t<\sup_{X}\varphi$ such that $D_{\varphi}(r)\subset\subset D_{\varphi}(s)\subset\subset D_{\varphi}(t)$. Then there exists a plurisubharmonic function $\phi$ on $X$ which satisfies the following conditions:

• (a)

$\varphi=\phi$* on $\{z\in D\,|\,\varphi(z)\geq t\}$.*

• (b)

$\mathrm{supp}\,i\partial\overline{\partial}\varphi\cup\overline{D_{\varphi}(r)}\subset\mathrm{supp}\,i\partial\overline{\partial}\phi$.

Proof.

Let $\psi$ be an exhaustive strongly plurisubharmonic function on $X$. We may assume that $\psi>0$ on $X$. Let $M=\sup_{\overline{D_{\varphi}(t)}}\psi$ and let $\tau(z)=r+\frac{(s-r)\psi}{M+1}$. Then $r<\tau<s$ on the closure of $D_{\varphi}(t)$. We define

[TABLE]

Then $\phi$ is a plurisubharmonic function on $X$. It is easy to see that $\phi$ satisfies the conditions of the lemma. ∎

Lemma 9**.**

Let $X$ and $\varphi$ be as in Theorem 1. Let $U\subset X$ be an open neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$ and let $u\in\Omega^{(0,q-1)}(U,F)$ ($1\leq q\leq n-2$) such that $\overline{\partial}u=0$. Let $s_{1}<s_{2}<\cdots<\sup_{X}\varphi$ such that $\lim_{j\to\infty}s_{j}=\sup_{D}\varphi$ and that

[TABLE]

Then there exist $V_{j}\subset X$ and $v_{j}\in\Omega^{(0,q-1)}(V_{j},F)$ ($j=3,4,\ldots$) which satisfy the following:

• (i)

$V_{j}$* is an open neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi\cap\overline{D_{\varphi}(s_{j})}$.*

• (ii)

$\overline{\partial}v_{j}=0$* on $V_{j}$.*

• (iii)

$v_{j+1}=v_{j}$* on $D_{\varphi}(s_{j-2})$.*

• (iv)

$v_{j}=u$* on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$.*

Proof.

By Lemma 6, there exists $v\in\Omega^{(0,q-1)}(D_{\varphi}(s_{5}),F)$ such that $\overline{\partial}v=0$ and that $v=u$ on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi\cap D_{\varphi}(s_{5})$. We may assume that $U$ is sufficiently small. Then $v$ on $D_{\varphi}(s_{4})$ and $u$ on $U$ can be glued together to give the form $v_{3}$ on $V_{3}:=U\cup D_{\varphi}(s_{4})$.

Assume that there exist $V_{j}$ and $v_{j}$ which satisfy the condition of the lemma ($j\geq 3$). By Lemma 8, there exists plurisubharmonic function $\phi$ on $X$ such that $\varphi=\phi$ on $\{z\in D\,|\,\varphi(z)\geq s_{j}\}$ and that $\mathrm{supp}\,i\partial\overline{\partial}\varphi\cup\overline{D_{\varphi}(s_{j-2})}\subset\mathrm{supp}\,i\partial\overline{\partial}\phi$. Then $v_{j}$ is defined on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\phi$. By Lemma 6, there exists $\tilde{v}\in\Omega^{(0,q-1)}(D_{\varphi}(s_{j+3}),F)$ such that $\overline{\partial}\tilde{v}=0$ and that $\tilde{v}=v_{j}$ on an open neighborhood of $D_{\varphi}(s_{j+3})\cap\mathrm{supp}\,i\partial\overline{\partial}\phi$. As in the case of $v_{3}$, we can glue $\tilde{v}$ and $v_{j}$ together, and we obtain $v_{j+1}$ and $V_{j+1}$ which satisfy the conditions of the lemma. Hence we obtain $V_{k}$ and $v_{k}$ ($k=3,4,\ldots$) inductively. ∎

Now we prove that the natural map

[TABLE]

is an isomorphism and

[TABLE]

for $0<q<n-2$.

Proof.

Let $U$ be an open neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$ in $X$. Let $u\in\Omega^{(0,q-1)}(U,F)$ ($1\leq q\leq n-2$) such that $\overline{\partial}u=0$. Take $\{s_{j}\}$ and $\{v_{j}\}$ as in Lemma 5. We can define $v\in\Omega^{(0,q-1)}(X,F)$ by $v(z)=v_{j+2}(z)$ when $z\in D_{\varphi}(s_{j})$. Then $\overline{\partial}v=0$ and $v$ is equal to $u$ on a neighborhood of $\mathrm{supp}\,i\partial\overline{\partial}\varphi$. Hence the natural map $H^{q-1}(X,F)\to\underset{\mathrm{supp}\,i\partial\overline{\partial}\varphi\subset V}{\varinjlim}H^{q-1}(V,F)$ is surjective. Since $H^{q-1}(X,F)=0$ for $q\geq 2$, this completes the proof. ∎

Let $X$ be a projective manifold and let $T$ be a closed positive current of type $(1,1)$ on $X$ such that the cohomology class $\{T\}$ belongs to $\mathcal{K}_{NS}$. There exist very ample line bundles $L_{1},\ldots,L_{p}$ and positive numbers $a_{1},\ldots,a_{p}$ such that $\{T\}=a_{1}c_{1}(L_{1})+\cdots+a_{p}c_{1}(L_{p})$ where $c_{1}(L_{j})$ is the first Chern class of $L_{j}$. Let $\omega_{j}$ be a smooth closed positive form such that $\omega_{j}\in c_{1}(L_{j})$. Put $\omega=a_{1}\omega_{1}+\cdots a_{p}\omega_{p}$. Then there exists an almost plurisubharmonic function $\varphi$ on $X$ such that $T=\omega+i\partial\overline{\partial}\varphi$ (see Section 14 of [4]). Here we say that a function $\varphi$ is almost plurisubharmonic if, for any $x\in X$, there exists a smooth function $\psi$ on a neighborhood of $x$ such that $\varphi+\psi$ is a plurisubharmonic function. Then we have that points where $\varphi$ is not continuous belong to $\mathrm{supp}\,T$. (We note that this claim needs not hold if $\varphi$ is not almost plurisubharmonic, even when $\varphi$ is a locally integrable upper-semicontinuous function.)

First we assume that $\varphi$ is bounded on $X$. Take non-zero holomorphic sections $s_{1}\in\Gamma(X,L_{1}),\ldots,s_{p}\in\Gamma(X,L_{p})$. Define $s=s_{1}\otimes\cdots\otimes s_{p}\in\Gamma(X,L_{1}\otimes\cdots\otimes L_{p})$.

Lemma 10**.**

Let $U\subset X$ be an open neighborhood of $\mathrm{supp}\,T=\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. Here $\varphi$ is a bounded almost plurisubharmonic function. Let $u\in\Omega^{(0,q-1)}(U,F)$ ($1\leq q\leq n-2$) such that $\overline{\partial}u=0$. Let $K\subset X\setminus\{z\in X\,|\,s(z)=0\}$ be a compact set. Then there exist an open neighborhood $U_{1}\subset X$ of $K\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$, a $\overline{\partial}$-closed $F$-valued form $u_{1}\in\Omega^{(0,q-1)}(U_{1},F)$ and a bounded almost plurisubharmonic function $\varphi_{1}$ on $X$ which satisfy the following conditions:

• (a)

$u=u_{1}$* on an open neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$,*

• (b)

$\omega+i\partial\overline{\partial}\varphi_{1}\geq 0$,

• (c)

$K\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi_{1})\subset U_{1}$.

Proof.

Put $Y=\{z\in X\,|\,s(z)=0\}$. Then $X\setminus Y$ is a Stein manifold. Let $\|\cdot\|_{j}$ be a smooth hermitian metric of $L_{j}$ whose Chern curvature is $\omega_{j}$. Let $\tilde{\varphi}=\varphi-\sum_{j=1}^{p}\frac{1}{2\pi}\log\|s_{j}\|^{2a_{j}}$ on $X\setminus Y$. Then $\tilde{\varphi}$ is an exhaustive plurisubharmonic function and $\mathrm{supp}\,i\partial\overline{\partial}\tilde{\varphi}=(X\setminus Y)\cap\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. Let $D_{\tilde{\varphi}}(r)=\{z\in X\setminus Y\,|\,\tilde{\varphi}(z)<r\}$. Take $s<t$ such that $K\subset D_{\tilde{\varphi}}(s)\subset\subset D_{\tilde{\varphi}}(t)$. There exists $v\in\Omega^{(0,q-1)}(D_{\tilde{\varphi}}(t),F)$ such that $\overline{\partial}v=0$ and that $u=v$ on an open neighborhood of $D_{\tilde{\varphi}}(s)\cap\mathrm{supp}\,i\partial\overline{\partial}\tilde{\varphi}$ by Lemma 6. As in the proof of Lemma 9, we can glue $v$ and $u$ together, and we obtain an open neighborhood $U_{1}\subset X$ of $K\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$ and $u_{1}\in\Omega^{(0,q-1)}(U_{1},F)$ which satisfy (a). By lemma 8, there exists bounded from below, exhaustive plurisubharmonic function $\tilde{\varphi}_{1}$ such that $K\cup\mathrm{supp}\,i\partial\overline{\partial}\tilde{\varphi}\subset\mathrm{supp}\,i\partial\overline{\partial}\tilde{\varphi}_{1}\subset U_{1}$ and that $\tilde{\varphi}_{1}=\tilde{\varphi}$ on $X\setminus(Y\cup D_{\tilde{\varphi}}(t))$. Put $\varphi_{1}=\tilde{\varphi}_{1}+\sum_{j=1}^{p}\frac{1}{2\pi}\log\|s\|^{2a_{j}}$. Since $\varphi_{1}=\varphi$ on a neighborhood of $Y$, we can consider $\varphi_{1}$ as a bounded almost plurisubharmonic function on $X$. Then $\varphi_{1}$ is a function we are looking for. ∎

Now we prove that the natural map

[TABLE]

is an isomorphism for $0\leq q<n-2$ and is injective for $q=n-2$.

proof of the case where $\varphi$ is bounded.

We first show that $H^{q}(X,F)\to\underset{\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset V}{\varinjlim}H^{q}(V,F)$ is surjective for $0\leq q\leq n-3$. Let $u\in\Omega^{(0,q)}(U,F)$ be a $\overline{\partial}$-closed $F$-valued form where $U$ is an open neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. We can take compact sets $K_{l}\subset X$ and $s_{l,j}\in\Gamma(X,L_{j})$ ($1\leq l\leq N$, $1\leq j\leq p$) such that $\bigcup_{l=1}^{N}K_{l}=X$ and that $K_{l}\cap\bigcup_{j=1}^{p}\{z\in X\,|\,s_{l,j}(z)=0\}=\emptyset$ for any $l$. Put $\varphi_{0}=\varphi$, $u_{0}=u$. By using Lemma 10 repeatedly, there exist $U_{l}\subset X$, $u_{l}\in\Omega^{(0,q)}(U_{l},F)$ ($1\leq l\leq N$) and $\varphi_{l}$ ($1\leq l\leq N-1$) such that $\bigcup_{k=1}^{l}K_{k}\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi_{l})\subset U_{l}$ and that $u_{l}=u_{l-1}$ on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi_{l-1})$. Hence $u_{N}\in\Omega^{(0,q)}(X,F)$ is $\overline{\partial}$-closed $F$-valued form such that $u_{N}=u$ on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. This proves the surjectivity.

Next, we show that $H^{q}(X,F)\to\underset{\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset V}{\varinjlim}H^{q}(V,F)$ is injective for $0\leq q\leq n-2$. Let $\alpha\in\Omega^{(0,q)}(X,F)$ such that $\overline{\partial}\alpha=0$. Suppose that there exists an open neighborhood $U\subset X$ of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$ and $v\in\Omega^{(0,q-1)}(U,F)$ such that $\alpha=\overline{\partial}v$ on $U$. Let $\chi\in C^{\infty}(X)$ be a function such that $\mathrm{supp}\,\chi\subset U$ and that $\chi=1$ on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. Then $\alpha-\overline{\partial}(\chi v)$ is $\overline{\partial}$-closed $F$-valued form which vanishes on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. Take $K_{1},\ldots,K_{N}$ as in the proof of the surjectivity. As in the proof of Lemma 6, there exists $F$-valued $(0,q-1)$-form $v^{\prime}_{1}$ which is defined in an open neighborhood of $K_{1}$ such that $\overline{\partial}v^{\prime}_{1}=\alpha-\overline{\partial}(\chi v)$ and that $v^{\prime}_{1}=0$ on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. By the trivial extension, we may assume that $v^{\prime}_{1}$ is defined on a neighborhood $U_{1}$ of $K_{1}\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. Define $v_{1}=\chi v+v^{\prime}_{1}$ on $U_{1}$. We have that $\overline{\partial}v_{1}=\alpha$ and that $v_{1}=v$ on a neighborhood of $\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)$. As in Lemma 10, there exists a bounded function $\varphi_{1}$ such that $\omega+i\partial\overline{\partial}\varphi_{1}\geq 0$ and that $K_{1}\cup\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi_{1})\subset U_{1}$. If we replace $K_{1},v,U,\varphi$ by $K_{2},v_{1},U_{1},\varphi_{1}$ respectively, we obtain $v_{2},U_{2},\varphi_{2}$ which satisfy the suitable conditions. By repeating this process, we obtain $v_{N}\in\Omega^{(0,q-1)}(X,F)$ such that $\overline{\partial}v_{N}=\alpha$. ∎

proof of the case where $\varphi$ is unbounded.

Define $\varphi_{c}=\max\{\varphi,c\}$ for $c\in\mathbb{R}$ and $T_{c}=\omega+i\partial\overline{\partial}\varphi_{c}$. Then $\varphi_{c}$ is bounded and $T_{c}\geq 0$ since $\omega>0$. It is easy to see that $\mathrm{supp}\,T=\mathrm{supp}\,(\omega+i\partial\overline{\partial}\varphi)\subset\mathrm{supp}\,T_{c}$. Let $U$ be any open neighborhood of $\mathrm{supp}\,T$, We show that $\mathrm{supp}\,T_{c}\subset U$ for sufficiently small $c$. Assume that there exist points $x_{k}\subset X\setminus U$ such that $x_{k}\in\mathrm{supp}\,T_{-k}$ for any $k\in\mathbb{N}$. Let $\{x_{k(j)}\}_{j\in\mathbb{N}}$ be a convergent subsequence and $x=\lim_{j\to\infty}x_{k(j)}\in X\setminus U$. Then almost plurisubharmonic function $\varphi$ is unbounded on a neighborhood of $x$, and is not continuous at $x$. We have that $x\in\mathrm{supp}\,T$, which gives a contradiction. Now we can reduce the proof to the case where $\varphi$ is bounded. ∎

6. Proof of Main results

The proof of Theorem 1 is similar to that of Theorem 2. Hence we only prove Theorem 2 by induction on $m$. The case $m=1$ holds by the arguments in Section 5. Assume now that $m\geq 2$ and that the case $m-1$ has already been proved. We first note that the cohomology class of $T_{j}+T_{k}$ is in $\mathcal{K}_{NS}$ and

[TABLE]

for any $1\leq j,k\leq m$. Put $A=\bigcap_{j=1}^{m-1}\mathrm{supp}\,T_{j}$ and $B=\mathrm{supp}\,T_{m}$. Then Mayer-Vietoris exact sequence yields the commutative diagram

[TABLE]

where the rows are exact sequences. We have that

[TABLE]

and

[TABLE]

Then we complete the proof by the induction hypothesis and a diagram-chasing argument. ∎

Let $X={\bf P}^{n}$ be the complex projective space of dimension $n\geq 3$. Then any non-zero closed positive current of type $(1,1)$ belongs to $\mathcal{K}_{NS}$ and we can apply Theorem 2 to this case.