Invariant Volume forms in Complex
Geometry and their plurisubharmonic
variations
Hajime TSUJI
December 1, 2018
Oka Symposium
Abstract
In this report, I would like to explain several invariant volume forms inseveral complex variables and the pseudoconvex variation of them.

Contents
1 Convexity in Several Complex Variables

2

2 Decomposition of geometry of compact K´’ahler manifolds
2.1 K´’ahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 One dimensional case . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Calabi’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 M.M.P (minimal model program) . . . . . . . . . . . . . . . . . .
2.5 Iitaka fibration and Abundance conjecture . . . . . . . . . . . . .

3
3
4
5
6
7

3 K´’ahler Ricci flow
3.1 K´’ahler Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Singular K´’ahler-Ricci flows . . . . . . . . . . . . . . . . . . . . .
3.3 Rough correspondence between KRF and MMP . . . . . . . . . .

8
8
8
9

4 Basic Problems
10
4.1 Case of projective families . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Variation of supercanonical measures . . . . . . . . . . . . . . . . 12
4.3 Basic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
125 Bergman kernel
5.1 Generalization of Bergman Kernels . . . . . . . . . . . . . . . . . 13
5.2 Extremal Property of Bergman Kernels . . . . . . . . . . . . . . 13
5.3 Plurisubharmonic Variation of Bergman Kernels . . . . . . . . . 14
6 Variation of K´’ahler-Einstein metrics and Canonical measures 14
6.1 Polynomial Approximation of K´’ahler-Einstein volume form . . . 15

1

7 Semipositivity of the direct image of pluricanonical systems
16
7.1 Viehweg’s weak semipositivity . . . . . . . . . . . . . . . . . . . . 16
7.2 KLT version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.3 Canonical measure . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7.4 Relative canonical measure . . . . . . . . . . . . . . . . . . . . . 20
7.5 Dynamical construction of the canonical measure . . . . . . . . . 20
8 Global generation
21
8.1 Scheme of the proof of Theorem 8.1 . . . . . . . . . . . . . . . . 22
8.2 Relative Iitaka fibration . . . . . . . . . . . . . . . . . . . . . . . 22
8.3 Regularity of relative canonical measure . . . . . . . . . . . . . . 22
8.4 Monge-Ampère foliation . . . . . . . . . . . . . . . . . . . . . . . 23
8.5 Weak stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.6 Metrized canonical models . . . . . . . . . . . . . . . . . . . . . . 24

1

Convexity in Several Complex Variables

In several complex variables, the pseudoconvexity plays a central role. I would
like to explain what is the pseudoconvexity and why it is important.
K. Oka proved the following fundamental theorem in 1953.
Theorem 1.1 ([O]) Let Ω be a domain in Cn and let dΩ : Ω −→ R be thedistance function , then the followings are equivalent.
(1) Ω is a domain of holomorphy, i.e., there exists a holomorphic functionf ∈ O(Ω) which does not have an analytic continuation to a stictly larger
domain.
(2) Ω is pseudoconvex, i.e,, − log dΩ is plurisubharmonic near the boundary,i.e,, −i∂ ∂¯ log dΩ is semipositive in the sense of current.
This theorem asserts that the natural existence domain of holomorphic functions has some convexity, i.e. “pseudoconvexity”. The pseudoconvexity is thecomplex analytic counterpart of the geometric convexity.
For a domain in C with the smooth boundary, the diﬀerence of the geometricconvexity and the pseudoconvexity is as follows.
Definition 1.1 Let Ω be a domain in Cn with smooth boundary ∂Ω.
Ω is said to be geometrically convex, if there exists a local defining functionof partialΩ such that the Hessian of the defininig function is positive definite
on the boundary.
Ω is said to be pseudoconvex, if the local defining function of ∂Ω such thatthe complex Hessian of the defining fuction is positive definite on the boundary.
To generalize the notion of convexity for complex manifolds, we need to
inroduce the notion of plurisubharmonic functions.
Definition 1.2 Let Ω be a domain in C and let ϕ : Ω −→ [−∞, ; ∞) be an
uppersemicontinuous function. ϕ is said to be subharmonic, if for every z ∈ Ωand suﬃciently small r > 0,
∫ 2π
1ϕ(z + reiθ )dθ
ϕ(z) ≦
2π 0
2

holds.
It is well known ϕ ∈ C 2 (Ω) this ϕ is subharmonic, if and only if ∆ϕ ≧ 0
holds on Ω.
Now we shall introduce the notion of plurisubharmonic functions.
Definition 1.3 Let Ω be a domain in Cn , φ : Ω −→ [−∞, ∞) is said to beplurisubharmonic , if for every z ∈ Ω and w ∈ Cn , φ(z + tw) is subharmonicin t for every suﬃcitnly small |t|.
¯ is
φ ∈ C 2 (Ω) is plurisubharmonic. if and only if the complex Hessian i∂ ∂φpositive semidefinite on Ω.
We denote the set of plurisubharmonic function on Ω by P SH(Ω). For
φ ∈ P SH(Ω), e−φ is considered to be a hermitian metric on the trivial bundle¯ in the sense of current.
Ω × C and its curvature is ∂ ∂φ
We define the intrinsic pseudoconvexity of a complex manifold as follows.
Definition 1.4 Let X be a complex manifold. X is said to be pseudoconvex, if
there exists a plurisubharmonic exhaustion function ϕ : X −→ R.
Following the fundamental result of K. Oka, the basic philosophy of several
complex variables is that every natural geometric object in several complex
variables is pseudoconvex, in other words, is of (semi)positive curvature.
This is similar to the following fundamental theorem in algebraic geometry.
Theorem 1.2 ([Kod]) Let X be a compact complex manifold and let (L, hL ) bea hermitian line bundle with strictly positive curvature. Then
H q (X, KX + L) = 0
holds for every q ≧ 1.
In fact K. Oka’s fundamental theorem has proved in terms of the extension of
Kodaira’s work by Andreotti and Visentini([A-V] ) and L. H´’ormander ([H]).

2

Decomposition of geometry of compact K´’ahler
manifolds

In this section, we shall discuss the decomposition of geometry of compact
K´’ahler manifolds. This is called the minimal model program (MMP) in algebraic geometry.

2.1

K´’ahler manifolds

X : complex manifold∂, ∂ )g : Hermitian metric on X. gij̄ = g( ∂zi ∂zjWe define the fundametal 2-form ω by
√
−1 ∑
ω=
gij̄ dzi ∧ dz̄j2

3

X; K´’ahler ⇔ dω = 0
Hereafter we consider K´’ahler manifolds.
The volume form of the n-dimensional K´’ahler manifold (X, ω) is given bydV (ω) =ωn= det(gij̄ )|dz1 ∧ · · · ∧ dzn |2n!

We define the Ricci form Ric(ω) by√Ric(ω) = − −1∂ ∂¯ log det(gij̄ )Let KX denote the canonical bundle of X (the line bundle of (n, 0)-forms on X.
−1
Then deg(gijdefines a hermitian metric on KX ,i.e.,
( √
) 12
2√
( −1)n −1η ∧ η̄dV (ω)
is a hermitian norm of η.
Hence −Ric(ω) represents the de Rham cohomology class of the 1-st Chernclass c1 (KX ).

2.2

One dimensional caseIn the case of one dimenisonal compact complex manifolds, it is classically known
that every compact Riemann surface admits a K´’ahler metric with constantcurvature.
Let C be a Riemann surface, i.e., a connected complex manifold of dimension
1. Let π : C̃ −→ C be the universal covering. Then we have the following
theorem:
Theorem 2.1 (Koebe) C̃ is biholomorphic to P1 , C or ∆, where ∆ denotes theunit open disk in C (with) center O.
This implies that every one dimensional complex manifold admits a complete
K´’ahler metric with constant curvature√−1dz ∧ dz̄
P1 :(1 + |z|2 )2√−1
C:dz ∧ dz̄
2√−1dz ∧ dz̄
∆:(1 − |z|2 )2In particular we have the following theorem.
Theorem 2.2 Let C be a compact Riemann surface of genus g. Then there
exists a metric with constant (Ricci = sectional) curvature on C.
This theorem follows from K´’oebe’s uniformization and the invariance of the
Poincaré metric√−1dz ∧ dz̄
(1 − |z|2 )2on ∆ under the action of the automorphism group Aut(∆) ≃ SL(2, R).
4

2.3

Calabi’s conjecture

By Koebe’s uniformization theorem (Theorem 2.1), every one dimensional complex manifold admits a metric with constant curvature.
But in the case of higher dimensional compact K´’ahler manifolds. we cannot
expect the existence of K´’ahler metric with constant sectional curvature. In thecase of higher dimensional compact K´’ahler manifolds, the natural object tocontrole is the Ricci curvature.
Definition 2.1 Let (X, ω) be a K´’ahler manifold. X is said to be K´’ahlerEinstein, if there exists a real number c such that
Ric(ω) = c · ωholds.
We say that c1 (X) is positive, 0 or negative, if the de Rham chomology classc1 (X) contains a K´’ahler form, 0 or −c1 (X) contains a K´’ahler form
For the higher dimensional analogue of Koebe’s uniformization theorem, thefollowing conjecture is well known.
Conjecture 2.1 (Calabi’s conjecture) Let (X, ω) be a compact K”ahler manifolds,(1) If c1 (X) is negative, then, there exists a unique K´’ahler-Einstein form ωsuch that−Ric(ω) = ω.
(2) For every volume form dµ on X, there exists a K´’ahler form ω such that
dµ =

1 n
ω
n!

holds.
This conjecture has been proven by T, Aubin and S.T. Yau in (1978).[Au,Y1].
Theorem 2.3 ([Au], [Y1]) Let X be a compact K´’ahler manifold with ample
KX . Then there exists a unique K´’ahler form ω on X such that
−Ric(ω) = ω
holds.
Theorem 2.4 ([Y1]) Let X be a compact K´’ahler manifold of dimension n andlet ω be a K´’ahler form on X and let dµ be a C ∞ -volume form on X such that∫∫ω n = dµXholds. Then there exists a C ∞ -function φ on X such that¯ n = dµ(ω + i∂ ∂φ)holds.
In the case of c1 (X) > 0, there is a necessary and suﬃcient condition for the
existence of K´’ahler-Einstein form (Chen-Donaldson-Sung). But this result israther complicated.
5

2.4

M.M.P (minimal model program)

The minimal model program (MMP) is a program to describe the geometry of
projective algebraic varieties in terms of the following 3-classes of varieties:(1) projective varieties with c1 > 0 (Fano varieties),(2) projective varieties with c1 = 0 (Calabi-Yau varieties or Holomorphic symplective varieties),(3) projective varieties with c1 < 0 (projective varieties of general type).
These are the direct generalization of the case of compact Riemann surfaces.
Definition 2.2 (Fano manifold(Ricci positive part) Let X be a smoothprojective variety. X is said to be Fano manifold, if −KX is ample, i.e., c1 (X)is positive.
By Theorem2.4, every smooth Fano manifold admits a C ∞ -K´’a]hler metric withpositive Ricci curvature.
Definition 2.3 (Ricci flat part)Let X be a smooth projective variety. X is
said to be varieties with trivial canonical classs, if KX is numerically trivial,
i.e., c1 (X) = 0.
Definition 2.4 (Ricci negative part) Let X be a smooth projective variety.
X is said to be canonical model, if KX is ample, i.e., KX is positive.
By Theorem2.4, every smooth Fano manifold admits a C ∞ Ricci flat K´’a]hler
metric.
This definition can be generalized to the case of varieties with mild singularities (i.e. with canonical singularities).
Definition 2.5 Let X be a normal projective variety and let Xreg be the regular locus of X. Let i : Xreg −→ X be the natural inclusion. Let KX :=i∗ OX (KXreg ).
In the case of higher dimensional projective varieties, the geometry would be
decomposed into 3-parts, in terms of minimal model program. This is a straightforward generalization of geometry of Riemann surfaces.
Definition 2.6 A line bundle L on a compact K´’ahler manifold is said tobe pseudoeﬀective (p.e.), if c1 (L) sits on the closure of eﬀetive cone (thecone of the classes of closed positive currents). A line bundle L is said to benef(numerically eﬀetive), if c1 (L) sits on the closure of the K´’ahler cone of
X.
There is a conjecture in algebraic geometry similar to Calabi’s conjecture(Conjecture ??) in diﬀerential geomety,
Conjecture 2.2 (Minimal Model Conjecture) Let X be a smooth projective
variety. Then one of the followings holds.
(1) There exists a fibration f : X −→ Y such that a general fiber is a Fano
variety (with positive dimension).
6

(2) There exists a projective variety Xmin birationally equivalent to X suchthat X is of canonical singularities and KX ∈ Div(X) ⊗ Q is nef.
Definition 2.7 Let X be a projective variety with only canonical singularities.
X is said to be minimal, if KX is nef.
The minimal model conjecture asserts that we may single out the ‘’Fanopart” of a projective variety can be single out as a fiber of the fibration or the
excepsional sets of the modification. The minimal model conjecture has beensolved in the case of projective varieties of dimension ≦ 4. But in general it isstill open.
The next step is to single out the geometry of varieties with trivial canonicalclass, i.e., varieties with c1 = 0.

2.5

Iitaka fibration and Abundance conjecture

To single out the “Ricci flat part” of a compact K´’ahler manifold, it isDefinition 2.8 : Let X be a smooth projective variety and let KX be the canonical line bundle on X. We define the Kodaira dimension κ(X) of X by
κ(X) := lim supm−→∞log h0 (X, mKX ).
log mIt is known that κ(X) is one of −∞, 0, 1, · · · , dim X.
If κ(X) ≧ 0, for m >> 1,, we have a rational fibration
Φ|m!KX | : X − · · · → Y = Φ|m!KX | (X)with dim Y = κ(X). This is called the Iitaka fibration. The rational map
Φ|m!KX | is defined by
Φ|m!KX | (x) = [σ0 (x) : · · · : σN (x)] ∈ PN
where {σ0 , · · · , σN } is a set of basis of H 0 (X, m!KX ). The rational map Φ|m!KX |is not defined on the base locus: Bs|m!KX | = {x ∈ X|σj (x) = 0, 0 ≦ j ≦ N }.
Theorem 2.5 (Iitaka) For a suﬃciently large m, Y does not depend on m.
And a general fiber F of f : X − · · · → Y , κ(F ) = 0 holds.
The Iitaka fibration single out the κ = 0 part of X.
Definition 2.9 Let X be a smooth projective variety, We define the numerical
Kodaira dimension ν(X) of X bylog h0 (X, A + mKX )
log m
m→∞

ν(X) = sup lim sup
A

where A runs all the ample line bundle on X.
Conjecture 2.3 (abundance conjecture) Let X be a smooth projective variety.
Then κ(X) = ν(X) holds.

7

If the minimal model conjecture holds, then the abundance conjecture isequivalent to the following conjecture:
Conjecture 2.4 Let X be a minimal projective algebraic variety. Then KX issemiample, i.e., there exists a positive integer N such that for every m ≧ N ,|m!KX | is base point free and gives a fibration
Φ|m!KX | : X −→Ysuch that a general fiber F is a minimal algebraic variety with m!KF is trivial.
The minimal model conjecture and the abundance conjecture are still open. But
the basic philosophy of these conjectures is we may single out the Ricci positive
part and the Ricci flat part of the projective variety by some algebro-geometric
operations.
X

3

K´’ahler Ricci flow

The K´’ahler-Ricci flow is the diﬀerential geometric counterpart of the minimalmodel conjecture and the abundance conjecture.

3.1

K´’ahler Ricci flow

Let (X, ω0 ) be a compact K´’ahler manifold. We consider the following evolution
equation.
∂ω = −Ric(ω) on X × [0, T ),(1)∂tω(0) = ω0 on X × {0},where T is the maximal existence time for the C ∞ solution.
Then we see that the de Rham cohomology class [ω] of ω satisfies[ω] = [ω0 ] + 2πt · c1 (KX )holds. Let K denote the K´’ahler cone, i,e, the set of the K´’ahler classes on X.
Then the following theorem holds.
Theorem 3.1 T is given by
T = sup{t|[ω0 ] + 2πt · c1 (KX ) ∈ K}.

3.2

Singular K´’ahler-Ricci flows

By Theorem 3.1, if KX is not nef, then the maximal existence time of the smooth
solution for (1) is finite. But if we allow some singularities on the solution ω,
we may construct a long time singluar solution for the K´’ahler-Ricci flows.
Theorem 3.2 Suppose that KX is pseudoeﬀective. Then there exsist closed
positive currenet ω(t) and a discrete sequence 0 < T0 < T1 · · · < Tm < · · · anda sequence of nonempty Zariski open subsets {Um }X = U0 ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Um ⊃ · · ·such that8

(1) ω(t) is C ∞ on Um × [Tm−1 , Tm ),(2) ω(t) satisfies the evolution equation :∂ω = −Ric(ω)∂ton Um × [Tm−1 , Tm ),(3) ω(t) is a closed positive current of minmal singularities in the class [ω0 ] +2πt · c1 (KX ).
Here the current of minimal singularities is in the following sense.
Definition 3.1 Let T be a closed positive(1, 1) current on a compact complex
manifold X. Let η be a closed C ∞ − (1, 1) form in T ]¯
T = η + i∂ ∂φfor some φ ∈ L1 (X). We say that T is of minimal singularity if φ has minimal
singularity among the closed positive current (1, 1)-current in the same class, inthe sense that for any φ̃ ∈ L1 (X) such that η + i∂ ∂¯φ̃ is closed positive, there
exists a positive constant C such that φ̃ ≦ φ + C holds on X.
The following theorem follows from the result of [L, p.26].
Theorem 3.3 Let X be a compact K´’ahler manifold and let c ∈ H 1,1 (X, R) be
a pseudoeﬀective class. Then there exists a closed positive (1, 1)-current T in cwith minimal singularities.
Theorem 3.2 can be considered as a diﬀerential geometric version of minimalmodel program with scaling.

3.3

Rough correspondence between KRF and MMP

Construction of minimal model corresponds the KRF and the abundance conjecture corresponds the study of the limit of KRF.
Theorem 3.4 Let X be a smooth pojective variety with pseudoeﬀective KX .
Let ω0 be a smooth K´’ahler form on X. Let us consider the KRF:∂
ω(t) = −Ric(ω(t))∂tstarting from ω0 . Suppose that KX is abundant, i.e., κ(X) = ν(X) holds. Thenthe limit
1
ω∞ := lim ω(t)t−→∞ texists a generically smooth semi-K´’ahler form on X which is the generalized
K´’ahler-Einstein form on X.

9

4

Basic Problems

Conjecture 4.1 (Invariance of plurigenera) Let f : X −→ S be a smooth
K´’ahler family of compact K´’ahler manifolds (with connected base S). We set
Pm (s) = dim H 0 (Xs , mKXs ), Then Pm (s) is constant on S for every m ≧ 1.
This conjecture is still open.
Conjecture 4.2 Let f : X −→ S be a smooth K´’ahler family of compact K´’ahler
manifolds. Suppose that a fiber has a pseudoeﬀective canonical bundle. Thenthe relative canonical bundle KX/S := KX ⊗ f ∗ KS−1 is also pseudoeﬀective

4.1

Case of projective families

In the case of projective family, we can solve the conjecture as follows. Let
f :X−→ S be a smooth projective family.
Definition 4.1 Let X be a smooth projective manifold with pseudoeﬀective
canonical bundle. Let A be a suﬃcitnly ample line bundle on X and let hA
be a C ∞ hermitian metric on A. We set
∫
1
1
2
2
0
m
m
hAm · |σ| m = 1}
dµm = sup{hA |σ| |σ ∈ H (X, A + mKX ),X

where sup denotes the pointwise supremum. We set
dµcan := (lim sup dµm )∗m→∞where ( )∗ denotes the uppersemicontinuous envelope. We call dµcan the supercanonical measure.
Definition 4.2 Let L be a holomorphic line bundle on a complex manifold X
and h0 is a C ∞ -hermitian metric on L. h is called a singular hermitian
metric on L, if there exists φ ∈ L1loc (X)| such thath = e−φ · h.
We call φ the weight function of h with respect to h0 and we define the ideal
sheaf J (h) by
J (h)(U ) = {f ∈ O(U )||f |2 e−φ ∈ L1loc (U )}.
We call J (h) the multiplier ideal sheaf of h. We define the curvature current
Θh by¯Θh = Θh0 + ∂ ∂φ.
Theorem 4.1 (Nadel’s vanishing theorem[N]) Let (X, ω) be a compact K´’ahler
manifold and let (L, h) be a singular hermitian line bundle on X such that thecurvature current satisfies the inequality iΘh ≧ εω for some positive number ε.
Then the multiplier ideal sheaf J (h) is coherent on X and we have the vanishing:H q (X, OX (KX + L) ⊗ J (h)) = 0
holds for every q ≧ 1.
10

AZD (Analytic Zariki Decomposition)
Definition 4.3 Let X be a compact complex manifold and let L be a holomorphic line bundle on X. A singular hermitian metric h on L is said to be an
analytic Zariski decomposition(AZD), if the followings hold.
1. Θh is a closed positive current,
2. for every m ≥ 0, the natural inclusion
H 0 (X, OX (mL) ⊗ J (hm )) → H 0 (X, OX (mL))
is an isomorphim.
Lemma 4.1 dµ−1
can is an AZD of KX , i.e., −Ric(dµcan ) is a closed (semi)positive
current and0
H 0 (X, OX (mKX ) ⊗ J (dµ−mcan )) ≃ H (X, OX (mKX ))holds for every m ≧ 1
Lemma 4.2 Let f : X −→ S be a smooth projective family such that a fiber X0
has pseudo eﬀective KX0 . Then KXs is pseudoeﬀective for every s ∈ S.
Let A be a suﬃcietly ample line bundle on X . Then shrinking S if necessary,
we may prove that for every m ≧ 1. We see that the restriction map
H 0 (X, OX (mKX/S + A)) −→ H 0 (X0 , OX0 (mKX0 + A))is surjective (by induction on m and the L2 -extension theorem below).
Theorem 4.2 (L2 -extension theorem) Let Ω be a bounded pseudoconvex domain in Cn contained in {|zn | < 1}. Let φ ∈ P SH(Ω) and let Ω0 = Ω ∩ {zn =0}. We set∫|F |2 e−φ < ∞}A2 (Ω, φ) = {F ∈ O(Ω)|
Ω

∫

and
A2 (Ω0 , φ) = {f ∈ O(Ω)||f |2 e−φ < ∞}.

Ω0Then for every f ∈ A2 (Ω0 , φ), there exists F ∈ A2 (Ω, φ) such that F |Ω0 = f
and
∫
∫
2 −φ
|F | e ≦ π|f |2 e−φΩΩ0This L2 -extension theorem can be generalized to the case of extension of twisted
canonical forms.
Theorem 4.3 Let f : X −→ ∆ be a smooth projective family over a unit open
disk ∆ in C. And let (L, h) be a singular hermitian line bundle on X withsemipositive curvature current.
Then the restriction map :H 0 (X, OX (KX + L) ⊗ J (h)) −→ H 0 (X0 , OX0 (KX0 + L|X0 ) ⊗ J (h0 ))is surjective.
11

4.2

Variation of supercanonical measures

In the projective case invariance of plurigenera can be proven as follows.
Theorem 4.4 (Invariance of plurigenera[T5]) Let f : X −→ S be a smoothprojective family. Suppose that a fiber X0 has pseudoeﬀective canonical bundle.
Then Xs has pseudoeﬀective canonical bundle for every s ∈ S. We set
dµX/S,can |Xs = dµcan,s (s ∈ S)
Then dµ−1
X/S,can is a singular hermitian metric on KX/S with semipositive curvature current. And Pm (s) = h0 (Xs , mKXs ) is constant on S.

4.3

Basic strategy

To solve these conjecture, we shall use the following procedure:
Let f : X −→ S be a smooth K´’ahler family of compact K´’ahler manifolds.
(1) Construct a family of invariant volume forms dµs for every s ∈ S suchthat dµ−1
s is an AZD of KXs .
(2) Define the relative volume form dµX/S by dµX/S |Xs = dµs (s ∈ S) andprove that −Ric(dµX/S ) is semipositive (in the sense of current).
(3) Apply the L2 -extension and prove the conjectures.

5

Bergman kernel

G: a domain in Cn
O(G): the space of holomorphic functions on G
O(G) ∩ L2 (G) is a Hilbert space by the inner product∫(f, g) :=f (z) · g(z) dµGwhere dµ denotes the standard Lebesgue measure on Cn . But it is more convenient to consider the space of holomorphic n-forms instead of holomorphic
functions. Instead of f, g ∈ A2 (G), we consider the holomorphic n-form:
σ := f (z) · dz1 ∧ · · · ∧ dzn , τ := g(z) · dz1 ∧ · · · ∧ dzn
Then we have√2( −1)n∫σ ∧ τ̄ = 2n (f, g)Gholds. For a complex n-fold X, we denote KX the holomorphic line bundle of(n, 0)-forms on X. KX is said to be the canonical bundle of X.
G:a boudned domain in Cn
}
{
∫
√
2
η ∧ η̄ < ∞ .
A2 (G) := η ∈ Γ(G, OG (KG )) ( −1)n√(η, η ′ ) := ( −1)n2G∫η ∧ η′G12(η, η ′ ∈ A2 (G)).

with norm1

∥ η ∥= (η, η) 2
Then A2 (G) is a Hilbert space, i,e., it is complete space. This follows from theMontel’s theorem and the (pluri)subharmonicity of the |f |2 (f ∈ O(G)).
2{ϕi }∞
i=1 be a complete orthonormal basis of A (G)KG (z) =∞∑|ϕi (z)|2 (Bergman volume form)

i=1

√
2
where |ϕ(z)|2 = ( −1)n ϕ(z) ∧ ϕ(z).
√ωG := −1∂ ∂¯ log KG (Bergman Kähler form)
The advantage of the Bergman volume form is that it is invaeriant under theaction of holomorphic automorphism group Aut(G). This follows from the invariance of the inner product under that action of Aut(G).
ωG is nothing but the pull-back of the Fubini-Study K´’ahler form on P∞ by
the projective embedding
Φ:G−
→ P∞
defined by
Φ(z) = [ϕ1 (z) : ϕ2 (z) : · · · : ϕm (z) : · · · ]By the invariance of KG (z) under the action of Aut(G), we see that ωG is also
invariant under the action of Aut(G).

5.1

Generalization of Bergman Kernels

Let Ω be a bounded domain in Cn and let φ ∈ P SH(Ω) be a plurisubharmonic
function on Ω. We set∫
20
A (Ω, φ) = {η ∈ H (Ω, O(KΩ ))||η|2 · e−φ < ∞}.
ΩThen A2 (Ω, φ) is a Hilbert space as before and we may define the weighted
Bergman kernel K(Ω, φ) similarly as above.
Another generalization is the compact case. Let X be a compact complex
manifold and let (L, hL ) be a (possibly singular) hermitian line bundle on X.
Then we define the finite dimensional Hilbert space∫
A2 (X, KX + L, hL ) := {σ ∈ H 0 (X, OX (KX + L))||σ|2 · hL < ∞}.
X

Then we may define the Bergman kernel K(X, KX + L, hL )(z, w) entirely as
above.

5.2

Extremal Property of Bergman Kernels

Here we consider the diagonal part K(Ω, φ) and denote it as K(Ω, φ)(z)(z ∈ Ω).
The following property is called the extremal property of the Bergman kernel
and it is very fundamental.

13

Proposition 5.1
K(Ω, φ)(z) = sup{|σ|2 (z)| ∥ σ ∥2 = 1, σ ∈ A2 (Ω, φ)}holds.
Using this property J.P. Demailly proved the following important approximation
thoeorem.
Theorem 5.1 ([Dem]) Let φ ∈ P SH(Ω). Then1
φ = limlog K(Ω, mφ)m−→∞ m
holds.
This theorem implies that every psh functions can be approximated by thelogarithm of Bergman kernels.

5.3

Plurisubharmonic Variation of Bergman Kernels

Theorem 5.2 (Maitani-Yamaguchi, Berndtsson) Let Ω be a bounded pseudoconvex domain in Cn × C. Let φ ∈ P SH(Ω). Let f : Ω −→ C be the second
projection. Let K(Ωt , φt ) be the Bergman kernel of Ωt = f −1 (t) with weightφt := φ|Ωt . Then ther relative voulme form K defined by
K|Ωt := K(Ωt , φt )is PSH in the sense that i∂ ∂¯ log K ≧ 0. holds on Ω.
The following theorem the projective version.
Theorem 5.3 ([B, B-P]) Let f : X −→ Y be an algebraic√fiber space and let(L, hL ) be a singular hermitian line bundle on X such that −1ΘhL ≧ 0. Thenthe singular hermitian metric h on KX/Y + L defined by
h|Xy := K(Xy , KXy + L, hL |Xy )−1 (y ∈ Y ◦ )has semipositive curvature on X, where Y ◦ denotes the complement of the discriminant locus of f .

6

Variation of K´’ahler-Einstein metrics and Canonical measures

In this section, we shall investigate the plurisubharmonic variation propertiesof K´’ahler-Einstein volume forms and canonical measures,

14

6.1

Polynomial Approximation of K´’ahler-Einstein volume
form

Let X be a smooth projective variety with ample canonical bundle of dimension
n. Let A be a suﬃciently ample line bundle on X and let hA be a C ∞ -hermitian
metric on A. We set
K1 := K(X, KX + A, hA ).
Here every Bergman kernel is the diagonal part. For m ≥ 2, inductively wedefine−1
Km = K(X, mKX + A, Km−1)
Then we have the following theorem.
Theorem 6.1

1 √dV∞ = lim hAm m (m!)−n Kmm−→∞is nothing but the K´’ahler-Einstein volume form on X, i.e.,dV∞ = (2π)n

1 n
ω
n! E

holds, where ωE is the unique K-E form such that −Ric(ωE ) = ωE .
The above theorem asserts that we can approximate the K´’ahler he following
theorem is very important.
Theorem 6.2 ([T1]) Let f : X −→ S be a smooth projective family with relatively ample canonical bundle. Let dVs denote the unique K-E volume form on
Xs . Then the relative volume form
dVX/S |Xs = dVs
is semipositive in the sense that −Ric dVX/S ≧ 0 holds.
This strategy can be generalized to the case of K´’ahler-Ricci flow and gives thefollowing theorem.
Theorem 6.3 Let f : X −→ S be a smooth projective family. Let A be an
ample line bundle on X and let hA be a C ∞ -hermitian metric on A with positive
curvature. Let ω0 = iΘ(hA ). We consider the family of K´’ahler-Ricci flow∂ωs (t) = −Ric(ωs (t))∂tωs (0) = ω0 |Xs .
Suppose that for some X0 , KX0 is pseudoeﬀective. Then the family of KRF
gives a fiberwise volume form dVs (t) such thatωs (t) = −t · RicdVs (t) + ω0 |Xs .
And the relative volume form
dVX/S (t)|Xs = dVs (t)satisfies−tRic dVX/S + ω0
is semipositive current for every t.
15

Besides Bergman volume form, there are several invariant volume forms incomplex geometry. Since there does not exist enough line bundle on a compact
K”ahler manifold, in the case of K´’ahler manifolds we need to consider theinvariant volume forms other than Bergman volume forms. Here is a candidate:Definition 6.1 (Extremal measure) Let Ω be a bounded domain in Cn . We set∫
dµext (Ω) = (sup{dV | − Ric dV ≧ 0,
dV = 0})∗Ω

where dV runs uppersemicontinuous volume form on Ω. For a closed positive
current T we define the twisted version:∫dµext (Ω, T ) = (sup{dV | − Ric dV + T ≧ 0,
dV = 0})∗ .
ΩWe also construct the extremal measure for an adjoint class on compact
K´’ahler manifolds.
f :X−→ Y : algebraic fiber space, i.e.,• X, Y are smooth projective varieties.
• f is projective surjective morphism with connected fibers.
• KX/Y := KX ⊗ f ∗ KY−1 : the relative canonical bundle.

7

Semipositivity of the direct image of pluricanonical systems

The following theorem is fundamental in algebraic geometry.
⊗m
Theorem 7.1 (Kawamata, 1982) If dim Y = 1, then for every m > 0, f∗ KX/Y⊗m
is semipositive in the sense that every quotient Q of f∗ KX/Y
, deg Q ≧ 0 holds.

The proof depends on the variation of Hodge structure (VHS) due to
Griﬃths and Schmid. The reason why we do not have the semipositive curvature⊗mproperty of f∗ KX/Y
is that the proof depends on the Finslar metric :(∫∥ σ ∥:=

) m2
|σ|

2
m

X/Y
⊗m
on f∗ KX/Y.

7.1

Viehweg’s weak semipositivity

E, Viehweg studied the direct image of pluricanonical systems by the fiber product method and find the weak semistability theorem..
Definition 7.1 Let Y be a quasi-projective reduced scheme, Y0 ⊆ Y an open
dense subscheme and let G be locally free sheaf on Y , of finite constant rank.
Then G is weakly positive over Y0 , if for an ample invertible sheaf H on Y
and for a given number α > 0 there exists some β > 0 such that S α·β (G) ⊗ Hβ
is globally generated over Y0 .
16Definition 7.2 Let F be a locally free sheaf and let A be an invertible sheaf,
both on a quasi-projective reduced scheme Y . We denote
F⪰

b
· A,
a

if S a (F) ⊗ A−b is weakly positive over Y , where a, b are positive integers.
Theorem 7.2 ([V1]) f : X −→ Y : an algebric fiber space such that KX/Y is
f -semiample over the complement of the discriminant locus Y ◦ .
m
1. (Weak positivitiy) f∗ KX/Y
(m > 0) is weakly positive over Y ◦ .

2. (Weak semistability) There exists e > 0 such that⊗mf∗ KX/Y⪰1⊗m· det(f∗ KX/Y
) on Y ◦ .
e · r(m)• Y ◦ : complement of the discriminant locus of f . X ◦ := f −1 (Y ◦ ).
• r = rank f∗ OX (mKX/Y ), X r := X ×Y X ×Y · · · ×Y X be the r-times fiber
product over Y• f r : Xr −→ Y be the natural morphism.
det f∗ OX (mKX/Y ) → ⊗r f∗ OX (mKX/Y ) = f∗r OX r (mKX r /Y ).
Hence we have the canonical global section()γ ∈ Γ X, f r∗ (det f∗ OX (mKX/Y ))−1 ⊗ OX r (mKX/Y ) .

(2)

(3)

Let Γ denote the zero divisor of γ. It is clear the Γ does not contain any
fiber over Y ◦ . Now we setδ0 := sup{δ > 0|(Xyr , δ · Γy ) is KLT for every y ∈ Y ◦ }.

(4)

Theorem 7.3 Let f : X −→ C be an eﬀectively parametrized family of canonically polarized varieties over a smooth quasiprojective curve C. Let m be a⊗mpositive integer such that f∗ KX/C
̸= 0. Then⊗mdeg det f∗ KX/C≦

1
deg KCrε⊗m⊗mholds, where ε is the threshold of f∗ KX/C
as in Theorem 8.1 and r := rank f∗ KX/C.

In the case of the higher dimensional base, we have the following result.
Theorem 7.4 Let f : X −→ Y be an eﬀectively parametrized family of canonically polarized varieties over a smooth quasiprojective curve C. Let m be a⊗mpositive integer such that f∗ KX/Y
̸= 0. Then⊗m
det f∗ KX/Y−

1
deg KYrε⊗m
is not pseudoeﬀective, where ε is the threshold of f∗ KX/C
as in Theorem 8.1⊗m
and r := rank f∗ KX/Y .

17

Theorem 7.5 (Weak Semistabitlity Theorem [V2]) f : X −→ Y : algebraic fiber space and let Y ◦ be the complement of the discriminant locus.
⊗m⊗m
1. (Semipositivity) For m > 0 such that f∗ KX/Y
̸= 0, then f∗ KX/Y
is
Griﬃth semipositive with respect to the relative canonical measure.

2. (Weak semistability) There exist e > 0 and a singular hermitian metricHm,e on
⊗m
⊗m −e
KX/Y
⊗ (f ∗ det f∗ KX/Y)
with semipositive curvature current such that for every y ∈ Y ◦ Hm,e |Xy⊗m⊗m −eis an AZD of KX/Y
⊗ (f ∗ det f∗ KX/Y
) |Xy .

7.2

KLT version

Theorem 7.6 Let f : X −→ Y be an algebraic fiber space and let D be aneﬀective Q divisor on X such that (X, D) is KLT. Let Y ◦ denote the complementof the discriminant locus of f . We set
Y0 := {y ∈ Y |y ∈ Y ◦ , (Xy , Dy ) is a KLT pair}• Let a be a minimal positive integer such that mD is Cartier. Then there
exist a positive integers b and m0 such that for every m ≧ m0 , b|m,m(KX/Y + D) is Cartier and f∗ OX (m(KX/Y + D)) is globally generated
over Y0 .
• Let r denote rank f∗ OX (⌊m(KX/Y + D)⌋) and let X r := X ×Y X ×Y
· · · ×Y X be the r-times fiber product over Y and let f r : X r −→ Y berr
the naturalmorphism.
And
letD
denotethedivioron
X
defined
by
∑r
∗
r
r
Dr =
π
D,
where
π
:
X
−→
X
denotestheprojection:
X
∋
i
i=1 i
(x1 , · · · , xn ) 7→ xi ∈ X.
There exists a canonically defined eﬀective divisor Γ (depending on m) on
X r which does not conatin any fiber Xyr (y ∈ Y ◦ ) such that if we we define
the number δ0 by
δ0 := sup{δ | (Xyr , Dyr + δΓy ) is KLT for all y ∈ Y ◦ },then for every ε < δ0f∗ OX (⌊m(KX/Y + D)⌋) ⪰mε· det f∗ OX (⌊m(KX/Y + D)⌋)
(1 + mε)rholds over Y0 .
• There exists a singular hermitian metric Hm,ε on (1+mε)(KX r /Y +Dr )−ε · f ∗ det f∗ OX (⌊m(KX/Y + D)⌋)∗∗ such that√
1. −1 ΘHm,ε ≧ 0 holds on X in the sense of current.
2. For every y ∈ Y0 , Hm,ε |Xyr is well defined and is an AZD of(1 + mε)(KX r /Y + Dr ) − ε · (f r )∗ det f∗ OX (⌊m(KX/Y + D)⌋)∗∗ |Xy

18

7.3

Canonical measure(Generalized Kähler-Einstein metrics)⊗m! ∗∗
Let f : X −→ Y be an Iitaka fibration such that (f∗ KX/Y
) is locally free
on Y for some m (hence for every suﬃciently large m), where ∗∗ denotes the
double dual. We define the Q-line bundleL :=1⊗m! ∗∗(f∗ KX/Y)m!

on Y . We note that L is independent of a suﬃciently large m. L carries thenatural singular hermitian metric hL defined by
(∫
)m!
2

|σ| m!

hm!
L (σ, σ) =.

X/Y(L, hL ) : Hodge Q-line bundle
Theorem 7.7 (Existence of canonical measures (Song-Tian, T-)) In
the above notations, there exists a unique singular hermitian metric on hK on
KY + L and a nonempty Zariski open subset U in Y such that
1. hK is an AZD of KY + L.
2. hK is real analytic on U .
√3. ωY = −1 ΘhK is a Kähler form on U .
√4. −RicωY + −1 ΘhL = ωY holds on U .
The above equation:−RicωY +√−1 ΘhL = ωY(5)

is similar to the Kähler-Einstein equation :The correction term√

−RicωY = ωY .
−1 ΘhL represents the isomorphism :R(X, KX )(a) = R(Y, KY + L)(a)for some positive integer a, where for a graded ring R := ⊕∞i=0 Ri , where for agraded ring R := ⊕∞i=0 Ri and a positive integer b, we set
R(b) := ⊕∞i=0 Rbi .
We setdµGKE := f ∗

(

ωYn
· h−1
L
n!

)

is called the canonical measure on X. dµcan has the following properties.
• dµGKE is a bounded volume form on X which degenerates along subvarieties on X.
• dµ−1
GKE is an AZD of KX .
• dµGKE is unique and birationally invariant.
19

7.4

Relative canonical measure

Theorem 7.8 ([T9]) Let f : X −→ S be a projective family such that X, S aresmooth and f has connected fibers. And(let D be an eﬀective) divisor on X suchthat (X, D) is KLT. Suppose that f∗ OS ⌊m(KX/S + D)⌋ ̸= 0 for some m > 0.
Then there exists a singular hermitian metric hK on KX/Y + D such that√
1. Let us define ωX/S := −1 ΘhK . Then ωX/S ≧ 0 holds on X.
2. For a general smooth fiber Xs := f −1 (s) such that (Xs , Ds ) is KLT, hK |Xs
is dµ−1can,(Xs ,Ds ) , where dµcan,(Xs ,Ds ) denotes the canonical measure on(Xs , Ds ). In particular ωX/S |Xs is the canonical semipositive current on(Xs , Ds ) constructed as in Theorem 8.1.

7.5

Dynamical construction of the canonical measure

To prove the plurisubharmonic variation of canonical measure, we need to use
the dynamical systems of Bergman kernels.
• X: a complex manifold,• (L, hL ) : a singular hermitian line bundle on X.
• Hilbert space:√2
A2 (X, KX + L) := {σ ∈ Γ(X, OX (KX + L))|( −1)n∫hL σ ∧ σ < +∞}X• inner product:√
2(σ, σ ′ ) := ( −1)n∫hL · σ ∧ σ̄ ′X

• {σi } : a complete orthonormal basis of A2 (X, KX + L)∑
• K(X, KX + L, hL ) = i |σi |2 : Bergman kernel of KX + L with respect
to hL .
•
K(X, KX + L, hL )(x) = sup{|σ|2 (x); σ ∈ A2 (X, KX + L, hL ), ∥ σ ∥= 1}
We shall consider the following setting.
• f :X−→ Y : Iitaka fibration• (L, hL ): Hodge Q-line bundle• A: suﬃciently ample line bundle on Y
• hA : C ∞ hermitian metric on A
• a : least positive integer such that aL ∈ Div(Y )

20



 K(Y, KY + A, hA ),

K1 :=





if a > 1

K(Y, KY + A + L), hA · hL ), if a = 1

and h1 := 1/K1 .
Inductively we define {Km } and {hm } bym
 K(Y, mKY + ⌊ a ⌋aL + A, hm−1 ), if a ̸ |m

Km :=



K(Y, m(KY + L) + A, hm−1 · haL ), if a|m
Theorem 7.9 (Dynamical construction) Let X be a smooth projective variety of nonnegative Kodaira dimension and let f : X −→ Y be the Iitaka fibration
as above. Let m0 and {hm }m≥m0 be the sequence of hermitian metrics as above
and let n denote dim Y . Then√h∞ := lim inf m (m!)n · hmm→∞is a singular hermitian metric on KY + L such that√ωY = −1 Θh∞holds, where ωY is the canonical Kähler current on Y as in Theorem 8.1 andn = dim Y .
√In particular ωY = −1Θh∞ (in fact h∞ ) is unique and is independent of
the choice of A and hA .
By Theorem 7.9, the theorem follows from the following theorem.
Theorem 7.10 ([B]) Let f : X −→ Y be an algebraic √fiber space and let (L, hL )be a singular hermitian line bundle on X such that −1ΘhL ≧ 0. Then the
singular hermitian metric h on KX/Y + L defined by
h|Xy := K(Xy , KXy + L, hL |Xy )−1 (y ∈ Y ◦ )has semipositive curvature on X, where Y ◦ denotes the complement of the discriminant locus of f .

8

Global generation

The following theorem is an important application of Theorem 7.9,
Theorem 8.1 ([T9]) Let f : X −→ S be an algebraic fiber space. Then for every
suﬃciently large m, f∗ OX (m!KX/Y )) is globally generated on a fixed nonempty
Zariski open subset S0 of S.

21

8.1

Scheme of the proof of Theorem 8.1

The scheme of the proof of Theorem 8.1 is summerized as follows:• Plurisubharmonic variation of Canonical measures.
• Two Monge-Ampère foliations on the relative Iitaka fibrations and thebase spaces induced by the −Ric of the relative canonical measure.
• Comparison of the two Monge-Ampère foliation in terms of the weaksemistability• Metrized canonical models are locally trivial along the leaves on the base.
• Leaves are closed and are the fibers of the moduli map to the moduli of
metrized canonical models.
• The family of canonical measures defines a positive Q-line bundle on themoduli space of the metrized canonical models.

8.2

Relative Iitaka fibration

f :X−→ Y be an algebraic fiber space such that Kod(X/Y ) ≧ 0. Let Z be theimage of the relative pluricanonical map⊗m!
Φ : X − ··· −→ P(f∗ KX/Y)

for m >> 1.

g

X
f@

@
R

-Z
h

Y
For a suﬃciently large m we see that a general fiber F of g : X − · · · → Z
is connected and Kod(F ) = 0. We call g : X − · · · → Z the relative Iitaka
fibration. By taking a suitable modification of X, we may assume that g is amorphism.
Let f : X −→ Y be an algebraic fiber space and let g : X −→ Z be a relative⊗m!
Iitaka fibration associated with f∗ KX/Y
. Taking a suitable modification we may
and do assume the followings :• g is a morphism,• Z is smooth.
⊗m! ∗∗• (g∗ KX/Z
) is a line bundle on Z.

Let h : Z −→ Y be the natural morphism.

8.3

Regularity of relative canonical measure

Let f : X −→ Y be an algebraic fiber space such that
Kod(X/Y ) ≧ 0and let g : X −→ Z be the relative Iitaka fibration as above.
g
-Z
X
f@
h
@
R
Y
22

By the dynamical construction and the generalized Kähler-Einstein equation,wehave the following lemma.
Lemma 8.1 Let dµX/Y,can is C ω on a Zariski open subset of X. Also therelative canonical Kähler current ωZ/Y is C ω on a Zariski open subset of Z.
This can be proven by using the Dirichlet problem for complex Monge-Ampèreequations and implicit function thoerems([C-K-N-S, Bl]).

8.4

Monge-Ampère foliation

The following theorem is very important and classical.
Theorem 8.2 ([B-K]) Let Ω: domain in Cn and let f ∈ C 3 (Ω) be a plurisubharmonic function such that ddc f has constant rank say r on Ω. Then¯ = 0}
F := {ξ ∈ T Ω|ddc f (ξ, ξ)defines a foliation on Ω such that the leaves are complex submanifolds of dimension n − r.
This foliation F is said to be a Monge-Ampère foliation on Ω associated with
ddc f .
Now we shall compare the two Monge-Ampere foliations.
ωZ/Y defines a Monge-Ampère foliation FZ on the generic point of Z. Let⊗m!
us consider the singular hermitian line bundle (det f∗ KX/Y
, det hm ), where′∫−(m!−1)hm (σ, σ ) :=X/Yσ · σ ′ · dµX/Y,can .

Θdet hm defines a Monge-Ampère foliation FY on Y on the generic point of Y .
The following is the key observation.
Lemma 8.2 h∗ FZ = FY holds.

8.5

Weak stability

Now we shall use the fiber product technique of [?].
• f :X−→ Y : algebraic fiber space with Kod(X/Y ) ≧ 0.
• r := rank f∗ OX (mKX/Y ).
• X r := X ×Y X ×Y · · · ×Y X(r-times),• f r : Xr −→ Y :the natural morphism.
⊗m⊗mr• f∗r KX
r /Y ≃ ⊗ f∗ KX/Y
⊗m
⊗m −1
r∗
• Γ ∈ |KX
det f∗ KX/Y
) |: corresponding to the inclusion :r /Y ⊗ (f(f r )∗ (det f∗ OX (mKX/Y )) ,→ (f r )∗ f∗r OX r (mKX r /Y ) ,→ OX r (mKX r /Y ).
•δ0 := sup{δ | (Xyr , δ · Γy ) is KLT for all y ∈ Y ◦ },23

• For every ε < δ0f∗ OX (mKX/Y ) ⪰mε· det f∗ OX (mKX/Y )(1 + mε)rholds over Y ◦ .
• There exists a singular hermitian metric Hm,ε on(1 + mε)KX r /Y − ε · (f r )∗ det f∗ OX (mKX/Y )∗∗ such that√
−1 ΘHm,ε ≧ 0 holds on X r in the sense of current.
• For every y ∈ Y ◦ , Hm,ε |Xyr is well defined and is an AZD of(1 + mε)KX r /Y − ε · (f r )∗ det f∗ OX (mKX/Y )∗∗ |Xy .
• Weak semistability ⇒ Θh∗ det hm |FZ ≡ 0nSince hK := (ωZ/Y)−1 · hL is an AZD of KZ/Y + L,n(ωZ/Y)−1 · hL = O(Hm,ε ⊗ (h∗ det hm )ε )holds. This implies that hK is more positive than (h∗ det hm )ε .
This implies the assertion.
• Along the→ Y is locally trivial, This is√ leaves of FY , h : (Z, (L, hL )) −because −1 Θhm ≧ 0 and trace Θhm ≡ 0 on FY . HenceΘhm ≡ 0 along FY⊗mThen the parallel transport on f∗ KX/Y
trivialize (Z, (L, hL )) along FY .

Hence we have thatf∗ FZ = FY

8.6

Metrized canonical models• (X, D): KLT pair with Kod(X, D) ≧ 0.
• R(X, KX + D) := ⊕∞m=0 Γ(X, OX (⌊m(KX + D)⌋)) : the log canonical ringof (X, D) : finitely generated.
• Y := Proj R(X, KX + D) : the canonical model of (X, D).
()∗∗• L := m10 ! f∗ OX (m0 !(KX/Y + D)) (m0 >> 1) : the Hodge Q-line bundle.
• hL : the Hodge metric on L.
• ωY : the canonical Kähler current.
• hK := n!(ωYn )−1 · hD (n = dim Y ) : canonical metric on KY + L.
Definition 8.1 The pair (Y, (L, hL )) is called the metrized canonical model
associated with the KLT pair (X, D).

24

The moduli space of metrized canonical models
Let (Zy , (L, hL )|Zy ) be the canonical model Zy of Xy and the metrized Hodge
bundle. The Hodge metric comes from a variation of Hodge structure on the
canonical cyclic cover Wy◦ −→ Zy◦ .
M = {(Zy , (L, hL )|Zy )}/ ∼where the equivalece ∼ is defined by
φ : Zy −→ Zy ′
covered by the biholomorphism φ̃ : Wy◦ −→ Wy◦′ which induces an isomorphism
between flat bundles preserving the Hodge line bundles.
Theorem 8.3 M has a structure of separable complex space and for m >> 1
(some multiple of ) det f∗ OX (m!KX/Y ) decends to a polarization of M. In
particular M is quasiprojective.
This theorem implies that the leaves of FY is the fiber of the classifying map:

Φ:Y◦ −→ M.

Then some symmetric power S r (f∗ OX (m!KX/Y )) decends to a vector bundle on
M. Then by the weak semistability, we see that for m >> 1 S r (f∗ OX (m!KX/Y ))decends to a very ample vector bundle on M. Then f∗ OX (rm!KX/Y ) is globally
generated on Y ◦ for m >> 1.
This theorem gives an alternative proof of the following theorem.
Theorem 8.4 (Viehweg) Let Mpol,min be the polarized minimal algebraic varieties with semiample canonical divisors, then Mpol,min is quasiprojective .

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Author’s address
Hajime Tsuji
Department of Mathematics
Sophia University
7-1 Kioicho, Chiyoda-ku 102-8554
Japane-mail address: tsuji@mm.sophia.ac.jp or h-tsuji@h03.itscom.net

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