数理解析研究所講究録
第 819 巻 1993 年 203-217

203

Analytic Zariski Decomposition
Hajime Tsuji

Introduction

1

Let $X$ be a projective variety and let $D$ be a Cartier divisor on $X$ . The
following problem is fundamental in algebraic geometry.

Problem 1 Study the linear system $|\nu D|$ for$\nu\geq 1$.

To this problem, there is a rather well developped theory in the case of$\dim X=1$ . In the case of $\dim X=2$ , in early 60-th, $0$ . Zariski reduced thisproblem to the case thatis $nef$ ( $=numerically$ semipositive) by using his
famous Zariski decomposition ([12]).
Recently Fujita, Kawamata etc. generalized the concept of Zariski decompositions to the case of $\dim X\geq 3([2,4])$ . The definition is as follows.
$D$

Definition 1 Let $X$ be a projective va reety and let
on X. The expression$D$

be a R-Cartier divisor$D=P+N(P, N\in Div(X)\otimes R)$

is called a Zariski decomposition $ofD$ , if the following conditions are satisfied.

1.

$P$

is

$nef_{J}$2. $N$ is effective,
3.

holds for all$[]s$ denote the integral parts of divisors.

$H^{0}(X, \mathcal{O}_{X}([\nu P]))\simeq H^{0}(X, \mathcal{O}_{X}([\nu D]))$$\nu\in Z_{\geq 0_{f}}$

where

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In the case of $\dim X=2$ , for any pseudoeffective divisor $D$ on $X$ , a Zariski
decomposition of $D$ exists ([12]). But in the case of $\dim X\geq 3$ , althoughmany useful applications of this decomposition have been known ([2, 4, 7]),
as for the existence, very little has been known. There is the following (rather
optimistic) conjecture.

Conjecture 1 Let $X$ be a normal projective variety and let $D$ be a pseudoeffective R-Cartier divisor on X. Then there exists a modification : $Yarrow X$
such that $f^{*}D$ admits a Zariski decomposition.
$f$

The purpose of this paper show how to construct an analytic counterpartof Zariski decomposition. Please see $[9, 10]$ for detail and further applications.
In this paper, all algebraic varieties are defined over C.

2Statement of the results

The main idea in this paper is to use d-closed positive $(1, 1)$ -currents, insteadof divisors. d-closed positive currents is far more general object than effective
algebraic cycles. The advantage of using d-closed positive currents is in theflexibility and completeness of them.

Definition 2 Let $X$ be a normal projective variety and let $D$ be a R-Cartierdivisor on X.
is called big if$D$$\kappa(D)$$:= \lim_{\nuarrow+}\sup_{\infty}\frac{\log\dim H^{0}(X,\mathcal{O}_{X}([\nu D]))}{\log\nu}=\dim X$.

holds. $D$ is called pseudoeffective , iffor any ample divisor $H_{f}D+\epsilon H$ is big.
for every$\epsilon>0$

Definition 3 Let $M$ be a complex manifold of dimension and let
denote the space offorms of compact support on $M$ with usual
Frechet space structure. The dual space $D^{p,q}(M)$ $:=A_{c}^{n-p,n-q}(M)^{*}$ is called$n$$C^{\infty}(p, q)$$A_{c}^{p,q}(M)$205
the space of $(p, q)$ -currents on M. The linear operators$D^{p+1,q}(M)$ and $\overline{\partial}:D^{p,q}(M)arrow D^{p,q+1}(M)$ isdefined by$\partial T(\varphi)=(-1)^{p+q+1}T(\partial\varphi),$ $T\in$\partial$: $D^{p,q}(M)arrow$D^{p,q}(M),$ $\varphi\in A_{c}^{n-p-1,n-q}(M)$and$\overline{\partial}T(\varphi)=(-1)^{p+q+1}T(\overline{\partial}\varphi),T\in D^{p,q}(M),$ $\varphi\in A_{c}^{n-p,n-q-1}(M)$.

. $T\in D^{p,q}(M)$ is called closed if $dT=0$ . $T\in D^{p,p}(M)$We setis called real if
holds for all $\varphi\in A_{c}^{n-p,n-p}(M).$ A real current$(p,p)$ -current
is called positive ifholds for all$d=\partial+\overline{\partial}$$T(\varphi)=T(\overline{\varphi})$$T$$\eta\in A_{c}^{p,0}(M)$$(\sqrt{-1})^{p(n-p)}T(\eta\wedge\overline{\eta})\geq 0$.

Since codimension subvarieties are considered to be closed positive $(p,p)-$currents, closed positive $(p,p)$ -currents are considered as a completion of thespace of codimension subvarieties with respect to the topology of currents.
For a $R$ divisor on a smooth projective variety $X$ . We denote the class ofin $H^{2}(X, R)$ by $c_{1}(D)$ .
$p$

$p$

$D$

$D$

Definition 4 Let be a closed positive $(p,p)$ -current on the open unit ball$B(1)$ inwith centre O. The Lelong $number\Theta(T, O)$ ofT at is defined$T$$C^{n}$$O$$by$$\Theta(T, O)=\lim_{r\downarrow 0}\frac{1}{\pi^{n-p}r^{2(n-p)}}T(\chi(r)\omega^{n-p})$where
open ball of radiusandwith centre in$\omega=\frac{\sqrt{-1}}{2}\sum_{i=1}^{n}dz_{i}\wedge d\overline{z}_{i}$$r$$\chi(r)$$O$$C^{n}$,

be the charcterristic function of the.

It is well known that the Lelong number is invariant under coordinatechanges. Hence we can define the Lelong number for a closed positive $(p,p)-$current on a complex manifold. It is well known that if a closed positive
current is defined by a codimension p-subvariety the Lelong number $\Theta(T,x)$
coincides the multiplicities of the subvariety at . In this sense the Lelong
number is considered as the multiplicity of a closed positive current.
We note that thanks to Hironaka resolution of singularities, to solve theconjecture, we can restrict ourselves to the case that $X$ is smooth. Our
theorem is stated as follows.
$T$

$x$

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Theorem 1 Let $X$ be a smooth projective variety and let be a line bundle
on X. Then there exists a closed positive $(1, 1)$ -current such that
$L$

$T$

1.

$T$

represents$c_{1}(L)$in $H^{2}(X, R)$ ,2. For every modification$f$: $Yarrow X\nu\in Z_{\geq 0}$ and $y\in Y$ ,$mult_{y}Bs|f^{*}(\nu L)|\geq\nu\Theta(f^{*}T,y)$holds.
We call$T$

an Analytic Zariski decomposition(AZD) of

$L$

. Let

$T=T_{abc}+T_{sing}$

be the Lebesgue decomposition of , where
denote the absolutely
continuous part and the singular part of respectively. As you see below,
this decomposition corresponds to Zariski decomposition.
The relation between Zariski decomposition and AZD is described by thefollowing corollary and proposition.
$T$$T_{abc},$ $T_{sing}$$T$

Corollary 1 Let $X$ be a smooth projective variety and let $D$ be a $nef$ andbig $R$ divisor on X. Then $c_{1}(D)$ can be represented by a closed positive$(1, 1)$ -current
.
with
$T$

$\Theta(T)\equiv 0$

Proposition 1 Let $X$ be a smooth projective variety and let $D$ be a $R$ divisor
on $X$ such that $2\pi c_{1}(D)$ can be represented by a closed positive $(1, 1)$ current
with. Then $D$ is $nef$.

$T$$\Theta(T)\equiv 0$

Let $X,L$ be as in Theorem 1. Suppose that there exists a modification$f$ : $Yarrow X$ such that there exists a Zariski decomposition $f^{*}L=P+N$ of
on Y. Then by Cororally 1 there exists a closed positive $(1, 1)$ current. Then the push-forward $T=f_{*}(S+N)$ issuch that $c_{1}(P)=[S]$ anda AZD of L. The main advantage of AZD is that we can consider the existence
without changing the space by modifications. One may ask whether AZD
substitutes ZD(Zariski decomposition). In some case the answer is “Yes”. In
this paper, I would like to show some applications, too.
$S$$f^{*}L$$\Theta(S)\equiv 0$207

3Outline of the proof of Theorem 1
Now I would like to show the outline of the proof of Theorem 1. Let $X,L$ be
as in Theorem 1. Let be a-hermitian metric on and let
be theK\"ahler form on $X$ such thatcurvature form of . Let
be a
$h$

$h$

$L$

$C^{\infty}$$\omega_{0}$$\omega_{\infty}$$C^{\infty}$$\omega_{0}-\omega_{\infty}>0$

holds on $X$ . We set$\omega_{t}=(1-e^{-t})\omega_{\infty}+e^{-t}\omega_{0}$

.

Let
be a
volume form on $X$ . Now we consider the following initialvalue problem.
$C^{\infty}$$\Omega$$=$$\frac{\partial u}{\partial t}$$\log\frac{(\omega_{t}+\sqrt{-1}\partial\overline{\partial}u)^{n}}{\Omega}-u$

$=0$

$u$

on $X\cross[0,t_{0}$ )(1)

on $X\cross\{0\}$ ,(2)

where $n=\dim X$ and is the maximal existence time for the
By the standard implicit function theorem is positive. Sinceby direct calculation we have the partial differential inequality$C^{\infty}$$t_{0}$$T$$\frac{\partial}{\partial t}(\frac{\partial u}{\partial t})\leq\tilde{\Delta}\frac{\partial u}{\partial t}-\frac{\partial u}{\partial t}$

solution .
,$u$$\omega_{0}-\omega_{\infty}>0$,

where dnotes the Laplacian with respect to the K\"ahler form
Hence by maximum principle, there exists a positive constant$\tilde{\Delta}$$\omega_{t}+\sqrt{-1}\partial\overline{\partial}u$$C_{0}^{+}$.

such that$\frac{\partial u}{\partial t}\leq C_{0}^{+}e^{-t}$holds onfor the solution). But unfortunately, we do not have uniform lower bound. Actually we cannot expect the uniform lower bound for$X\cross[0, t_{0}$

$u$

$u$

.
The above equation corresponds to the following Hamilton type equation:$\frac{\partial\omega}{\partial t}$$=$$-Ric_{\omega}-\omega+(Ric\Omega+curvh)$$\omega=\omega_{0}$om $X\cross\{0\}$ .

on$X\cross[0, t_{0}$)208This equation preserves the K\"ahlerity of . Hence it is meaningful to take
. By a calculation, we see thatthe de Rham cohomology class$\omega$$[\omega]$$[\omega]=(1-e^{-t})2\pi c_{1}(L)+e^{-t}[\omega_{0}]$holds. Let $A(X)$ denote the K\"ahler cone of $X$ . By the above equation, we
see that $[\omega]\in A(X)$ , if $t\in[0,t_{0}$ ). Conversly we have:
Lemma 1 $T= \sup$ {$t$I$[\omega]\in A(X)$}.

sits on the closure of $A(X)$ , we cannot
But this means that unless
expect $T=\infty$ .
Hence we should consider a current solution$2\pi c_{1}(L)$$\omega_{t}+\sqrt{-1}\partial\overline{\partial}u$solution, where $u:Xarrow[-\infty, \infty$ ) To construct a current
instead of asolution we need to find the place where the estimate of the solution breaks.
We set$C^{\infty}$$u$$S= \bigcap_{\nu>0}${ $x\in X|H^{0}(X,$$\mathcal{O}_{X}(\nu L))$does not separate$TX_{x}$}and we expect that the solution is
on $X-S$ .
The natural way to construct such a singular solution is to construct
the solution by as a limit of the solution of Dirichlet problems on relatively
compact subdomains in $X-S$ which exhaust $X-S$ . So we would like toapply the theory of Dirichlet problem for complex Monge-Amp\‘ere equations
developped recently ([1]).
But in fact, we need to subtract a little bit larger set because $X-S$ is notstrongly pseudoconvex. Otherwise the theory does not work (this phenomena
is caused by the lack of good barriers for the estimates, if the domain is not: $X_{\nu}arrow X$ be a resolution of Bs $|\nu L|$ and let
pseudoconvex). Let$u$$C^{\infty}$$f_{\nu}$$|f_{\nu^{*}}(\nu L)|=|P_{\nu}|+N_{\nu}$be the decomposition into the free part and the fixed part. The following
lemma is well known and very useful.

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Lemma 2 (Kodaira’s lemma) Let $X$ be a smooth projective variety and let$D$
be a big divisor on X. Then there exists an effective Q-divisor $E$ suchthat $D-E$ is an ample Q-divisor.

Then by Kodaira’s lemma, we can find an effective divisoronsuchthat for every sufficiently small positive rational number
is anample Q-divisor.
Let us take sufficiently large so that the free divisor
is nef and big.
Let : $X_{\nu}arrow P^{N}$ be an embedding of
into a projective space and let$R_{\nu}$$\epsilon,$$X_{\nu}$$P_{\nu}-\epsilon R_{\nu}$$P_{\nu}$$\nu$$\Phi$$X_{\nu}$: $X_{\nu}arrow P^{n}(\alpha=1, \ldots,m)$$\pi_{\alpha}$

be generic projections and we set$W_{\alpha}$: the ramification divisor of$H_{\alpha}$$:=\pi_{\alpha}^{*}(z_{0}=0)$$\pi_{\alpha}$

,

where
:.. . :
be the homogeneous coodinate of. For simplicity
we shall denote the support of a divisor by the same notation as the one, if
without fear of confusion. If we take $m$ sufficiently large, we may assume thefollowing conditions:$[z_{0}$$P^{n}$$z_{n}]$,

1.

$\bigcap_{\alpha=1}^{m}(W_{\alpha}+H_{\alpha})=\phi$2.

$D$ $:=(F_{\nu}+\Sigma_{\alpha=1}^{m}(W_{\alpha}+H_{\alpha}))_{red}$is an ample divisor with normal cross-

ings,

3.

$D$

constains S U$R_{\nu}$,

4. $K_{X_{\nu}}+D$ is ample.
Then $U=X_{\nu}-D_{\nu}$ is strongly pseudoconvex and is identified with a Zariski
open subset of $X$ . Let $K$ be a relatively compact strongly pseudoconvex
boundary. Thanks to the condition 1 above, forsubdomain of with
$K$ , we can apply the theory in [1] developped on strongly pseudoconvex
boundary in a complex Euclidean space, although $K$ isdomains with
not inside.
$U$$C^{\infty}$$C^{n}$$C^{\infty}$

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Hence we can solve a Dirichlet problem for a complex Monge-Amp\’ereequation on $K$ . In our case the equation is parabolic, so we need to modify
the theory. To get the-estimate for the solution, we shall change theand let be aunknown. Let be a section ofwith divisor. Then there exists a hermitian metrics
section ofwith divisor
respectively such thaton$C^{0}$$R_{\nu}$$\mathcal{O}_{X_{\nu}}(R_{\nu})$$h_{F},h_{R}$$\mathcal{O}_{X_{\nu}}(F_{\nu}),$$\mathcal{O}_{X_{\nu}}(R_{\nu})$$f_{\nu^{*}} \omega_{\infty}-\frac{1}{\nu}$is a K\"ahler form onchange by$\lambda$$F_{\nu}$$\mathcal{O}_{X_{\nu}}(F_{\nu})$$\tau$

curv$h_{F}-\epsilon$. curv$h_{R}$

for every sufficiently small postive number$X_{\nu}$$\epsilon$. Let us$u$.

$v=u-(1-e^{-t})( \frac{1}{\nu}\log h_{N}(\tau,\tau)-\epsilon h_{R}(\lambda, \lambda))$Then since$\omega_{t}’=(1-e^{-t})$($f_{\nu^{*}} \omega_{\infty}-\frac{1}{\nu}$curv$h_{F}-\epsilon$. curv$h_{R}$)$+e^{-t}f_{\nu^{*}}\omega_{0}$is uniformly positive on , we can solve the O-Dirichlet boundary value problem for on) for any relatively compact strongly pseudoconvex
boundary.
subdomain $K$ with
$U$

$v$

$K\cross[0, \infty$$C^{\infty}$

Remark 1 Here we need to worry about the Gibb’s phenomena for the parabolic
equation. But this is rather thechnical and not essential. Hence we shall omit
$it$.

of and consider
Let us take an strongly pseudoconvex exhaustiona family of Dirichlet problems of parabolic complex Monge-Amp\’ere equation(1).
The next difficulty is the convergence of the solutions of this familyof Dirichlet problems. Here we note that there exists a complete K\"ahleron thanks to the conditions 3, 4 above and [6]. Then if
Einstein form
we choose the boundary values properly, we can dominate the volume forms
by maximum
associated with the solutions from above by a constant times
principle. This ensures the convergence.
Let $u\in C^{\infty}(U)$ be the solution of (1) on . Then by the -estimate of, we see that extends to a -function on $X$ for every .
$\{K_{\mu}\}$$\omega_{D}$

$U$

$U$

$\omega_{D}^{n}$$C^{0}$

$U$

$u$

$u$

$L^{1}$$t$

211
Now we set$T=_{tarrow}m_{\text{科}}(\omega_{t}+\sqrt{-1}\partial\overline{\partial}u)$

where$\partial\overline{\partial}$,

is taken in the sense of current.

Remark 2 On the first look,seems to depend on the choice of . But
actually,
is indpendent of . This follows from the uniqueness property ofthe equation (1).
$T$

$T$

$\nu$

$\nu$

Then we can verify that$T$

is an AZD of$D$

by using the$C^{0}$-estimate of$u$.

Basic properties of AZD

4

As a direct consequence of the construction, an AZD has following properties.

Proposition 2 Let $X$ be a smooth projective variety and let be a big line
bundle on X. Let be an $AZD$ of then has the following properties.
$L$

$T$

$L_{f}$$T$

1. Let $T=T_{abc}+T_{\epsilon ing}$ denote the Lebesgue decomposition of T. Then
there exists a reduced very ample divisor on $X$ such thatison $X-D$ .
$D$2.

$T_{a^{n}bc}$$T_{abc}$

is of Poincare‘ growth along D. In particular$T_{a^{n}bc}$$C^{\infty}$

is integrable on

X.
3.

$T$

is of finite order along $D,$ $i.e.$ , only polynomial growth along $D$ .

need not be of normal crossings. Hence the word “Poincar\’eRemark 3
growth “ means a little bit generalized sense, $i.e$ . if we take any modificationsuch that the total transform of $D$ becomes of normal crossings, the pull-backis of Poincar\’e growth along the total transform.
of$D$$T_{a^{n}bc}$

Remark 4 I think the third property of $AZD$ should be the key to solve theconjecture in the introduction.
By using Kodaira’s lemma and H\"ormander’s
we can easily get.

$L^{2}$-estimate for -operator,$\overline{\partial}$

212
Proposition 3 Let $X,L,T$ be as in Proposition 1. Then for every modification : $Yarrow X$ and any $y\in Y$ ,$f$$\Theta(f^{*}T, y)=\lim_{\nuarrow+}\inf_{\infty}\nu^{-1}mult_{y}Bs|\nu L|$

holds.
Proposition 2 means that although an AZD is not unique, but the singularpart is in some sense unique and the AZD controlles the asymptotic behaviorof the base shemes of the multilinear systems.
Instead of using AZD itself, sometimes it is more useful to use the “potential” of AZD.

Definition 5 Let be a line bundle on a complex manifold X. is called asingular hermitian metric on
if there exist a -hermitian metric onand locally -fuction such that
$L$

$h$

$C^{\infty}$$L_{f}$$L$$h_{0}$$L^{1}$$\varphi$$h=e^{-\varphi}h_{0}$holds.
We note that for a singular hermitian meric it is meaningful to take curvatureof it in the sense of current.
One of the most useful property of AZD is the following vanishing theorem.

Theorem 2 Let $X,L$ be as in Theorem 1 and let be an $AZD$ of $D$ constructed as above. Let
be a singular he $7mitian$ metmc onsuch that$T=curvh$ . For a positive integer $m$ we set
$T$

$L$

$h$

$\mathcal{F}_{m}$$:=sheaf$ of germs of local

Then$\mathcal{F}_{m}$$L^{2}$-holomorphic sections ofis coherent sheaf on $X$ and$H^{p}(X, K_{X}\otimes \mathcal{F}_{m})=0$

holds for $p\geq 1$ .

$(\mathcal{O}_{X}(mD), h^{\otimes m})$.

213
By Corollary 1, we get the following well known vanishing theroem.

Corollary 2 ([5]) Let $X$ be a smooth projective manifold and let
and big line budnle. Thenbe a $nef$$L$$H^{p}(X, K_{X}\otimes L)=0$

holds for $p\geq 1$ .

Some direct applications of AZD

5

In this section, we shall see that we can controle the asumptotic behavior of
the multilinear systems associated with big line bundle in terms of its AZD.

Definition 6 Let$L$

be a line bundle over a projective n-fold X. We set$vol(X, L)= \lim_{\nuarrow+\infty}\nu^{-n}\dim H^{0}(X, \mathcal{O}_{X}(\nu L))$and call it the L-volume of $X$ or the volume of $X$ with respect to$L$.

We can express the volume in terms of AZD.
Theorem 3 Let be a big line bundle over a smooth projective n-fold $X$
and let $T=T_{abc}+T_{sing}$ be an $AZD$ of constructed as in Section 3. Then
we have$L$$L$$vol(X, L)= \frac{1}{(2\pi)^{n}n!}\int_{X}T_{abc}^{n}$holds.
The following therem follows from the existence of AZD and LebesgueFatou’s lemma.
Theorem 4 Let : $Xarrow S$ be a smooth projective family \‘of projective
varieties over a connected complex manifold and let be a relatively bigline bundle on X. For $s\in S$ , we set $X_{s}=\pi^{-1}(s)$ and $L_{s}=L|X_{s}$ . Then$vol(X_{s}, L_{s})$ is an uppersemicontinuousfunction on .
$\pi$

$S$

$L$

$S$

The following theorem follows from Theorem 2.

214
Theorem 5 Let : $Xarrow S$ be a smooth projective family of projective
varieties over a connected complex manifold and let be a line bundle on
X. Suppose that $aL-K_{X}$ is relatively big for some $a>0$ . Then $vol(X_{s}, L_{8})$is a constant function on .
$\pi$

$L$

$S$

$S$

Proof of Theorem 2. Let

be as in Theorem 3. Let be as in
Proposition 1. By taking a modification of X-, we may assume thatis adivisor with simple normal crossings. By Kodaira’s lemma there exists aneffective divisor such that $L-E$ is an ample Q-line bundle. Let be aK\"ahler form on $X$ which represents $c_{1}(L-E)$ . Let be a section of. Then forsuch that $(\sigma)=H$ and let be a-hermitian metric ona sufficiently small positive number ,$D$$X,$ $L,$ $T$

$D$

$E$

$Q$

$\overline{\omega}$$\mathcal{O}_{X}(D)$$\sigma$$h$$C^{\infty}$$\mathcal{O}_{X}(D)$$c$$\omega=\overline{\omega}+c\sqrt{-1}\partial\overline{\partial}\log(-\log h(\sigma, \sigma))$

is a complete K\"ahler form on $X-D$ . We note that there exist positive
constants
such that$C_{1},$ $C_{2}$$-C_{1}\omega<Ric_{\omega}<C_{2}\omega$

on $X-D$ by direct computation (actually has bounded geometry). Thenby the -Riemann-Roch inequality ([8]), we have that for every
wehave the inequality$\omega$$\epsilon>0$$L^{2}$$\frac{1}{(2\pi)^{n}n!}\int_{X-D}(T_{abc}+\epsilon\omega)^{n}\leq vol((1+\epsilon)L-\epsilon E)\leq(1+\epsilon)^{n}vol(X, L)$

Letting$\epsilon$.

tend to , we have the inequality$0$$\frac{1}{(2\pi)^{n}n!}\int_{X-D}T_{abc}^{n}\leq vol(X,L)$.

(3)

denote the free
Let $f_{\nu};X_{\nu}arrow X$ be a resolution of Bs $|\nu L|$ and let
part of. Assume that is sufficiently large so that is nef and. We set
denote a semipositive first Chern form ofbig. Let$|P_{\nu}|$$|f_{\nu^{*}}(\nu L)|$

$P_{\nu}$$\nu$$\mathcal{O}_{X_{\nu}}(f_{\nu}^{*}(\nu L))$$\omega_{\nu}$$T_{\nu}=\nu^{-1}(f_{\nu})_{*}(\omega_{\nu})$

Then sinceequalities:$|P_{\nu}|$.

is free, by Bertini’s theorem, we get the sequence of in$T_{abc}^{n}\geq T_{abc}^{n-1}T_{\nu}\geq\cdots\geq T_{\nu}^{n}$

215
on $X-D$ . Hence we see that$\int_{X_{\nu}}(T_{\nu})_{abc}^{n}=(2\pi)^{n}\nu^{-n}c_{1}^{n}(P_{\nu})\leq\int_{X}T_{abc}^{n}$holds. We need the following proposition.
Proposition 4 (Fujita) Let be a big line bundle on a projective manifold
X. Let : $X_{\nu}arrow X$ be a resolutiona of $Bs|\nu L|$ and let$L$$f_{\nu}$$|f_{\nu}^{*}(\nu L)|=|P_{\nu}|+F_{\nu}$be the decomposition into the free part and the fixed part. Then the equality$vol(X, L)= \frac{1}{n!}\lim_{\nuarrow+\infty}\nu^{-n}P_{\nu}^{n}$holds.
By Proposition 4, we have that(4)$vol(X, L) \leq\frac{1}{(2\pi)^{n}n!}\int_{X}T_{abc}^{n}$holds.
Combining (3) and (4), we complete the $pro$ of of Theorem 3.
Corollary 3 $LetX,$ $L,T$ be as in Theorem 1 and let
X. Thenis finite on $X$ .
A$\int_{X}T_{abc}^{k}$$\omega$

be a Kahler form on$\omega^{n-k}$

Proof of Theorem 4.

We may assume that $S=\{s\in C||s|<1\}$ .
Step 1. For the first we shall consider the case that is relatively big. Let$T_{s}=(T_{s})_{abc}+(T_{s})_{sing}(s\in S)$ be the family of AZD’s constructed by the flow
for the positive current ,$L$$\omega$$\partial\omega$$=$$-Ric_{\omega}-\omega+(Ric\Omega+curvh)$$\overline{\partial t}$$\omega=\omega_{0}$on$X_{s}\cross\{0\}$,on$X_{s}\cross[0, \infty$

)

216
where
: the Ricci current of-hermitian metric on ,: a (relative)-K\"ahler form on $X$ ,: a relative-volume form on $X$ .
a relative
Since is relatively big, as in Section 3, there exists a reduced divisor $D$ on$Ric_{\omega}$$:=-\sqrt{-1}\partial\overline{\partial}\log\omega^{n}$$h$$\omega$$L$$C^{\infty}$$C^{\infty}$$\omega_{0}$$C^{\infty}$$\Omega$$L$$X$such that

1. $D$ is equidimensional over ,$S$

2. $(X-D)$ admits a complete relative K\"ahler-Einstein metricstant Ricci curvature—l,3.

$\omega$is$C^{\infty}$$\omega_{D}$of con-on $X-D$ .

Then by Lebesgue’s bounded convergence theorem, we see that$vol(X_{s}, L_{s})= \int_{X_{s}}(T_{s})_{abc}^{n}$is an uppersemicontinuous function with respect toproof of Theorem 4.

$s$. This completes theReferences[1] Cafarelli, L-Kohn, J.J.-Nirenberg, L.-Spruck, J., The Dirichlet problems for nonlinear second-order elliptic equations II. Complex Monge$Ampere$ and uniformly elliptic equations, Comm. Pure and Appl. Math.

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[8] Nadel, A.- Tsuji, H., Compactification of complete K\"ahler manifolds
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[9] Tsuji, H., Analytic Zariski Decomposition, Part 1, Existence, preprint.
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Hajime Tsuji
Tokyo Institute.of Technology
2-12-1, Ohokayama, Megro, Tokyo 152Japan