GENERALIZED BERGMANN METRICS AND INVARIANCE OF PLURIGENERA

HAJIME TSUJI

February 14, 1996

Abstract. An invariant kernel for the pluricanonical system of a projective manifold of general type is introduced. Using this kernel we prove that the Yau volume form on a smooth projective variety has seminegative Ricci curvature. As a biproduct we prove the invariance of plurigenera for smooth projective deformations of manifolds of general type.

Contents

1. Introduction 2

Part I. Generalized Bergmann Kernels 4

2. Generalized Bergmann metrics 4 2.1. Yau pseudovolume form 2.2. Construction of the generalized Bergmann kernels

3. Limit of the Generalized Bergmann kernels 9 3.1. First properties of Generalized Bergmann kernels 3.2. Poinwise upper estimate of dμ 3.3. Lower estimate of the limit of Bergmann kernel

4. Invariance of Plurigenera 18 4.1. Analytic Zariski decomposition 4.2. Growth estimate for the sections 4.3. Analytic Zariski Decomposition of the family 4.4. Comparison with the global growth 4.5. Use of L^{2}-extention theorem

Part II. Analytic construction of AZD 25 5. Deformation of Kähler form I 25

Research at MSRI is supported in part by NSF grant DMS-9022140.

5.1. Hamilton’s equation

5.2. Reduction to the parabolic Monge-Ampère equation 5.3. C^{0}-estimate 5.4. C^{2}-estimate

6. Deformation of Kähler form II 30 6.1. The current ω_{E} 6.2. Kodaira’s Lemma 6.3. Construction of a suitable ample divisor 6.4. The Dirichlet problem 6.5. C^{0}-estimate 6.6. C^{1}-estimate on ∂K 6.7. C^{1}-estimate on K 6.8. C^{2}-estimate on ∂K 6.9. C^{2}-estimate on K 6.10. Costruction of ω_{E}

References 54

1. Introduction

Let Ω be a bounded domain in C^{n} and let {ϕ_{i}}_{i=1}^{∞} be a complete orthonormal basis of of L^{2} holomorphi n-forms on Ω. Then the Bergmann kernel of Ω is defined by

K(z, w) = ∑_{i=1}^{∞} ϕ_{i}(z)ϕ̄_{i}(w)

and the Bergmann Kähler form is defined by

ω = \sqrt{−1}∂ ¯∂log K(z, z).

The Kähler metric associated with ω is called the Bergmann metric of Ω. One of the most important property of the Bergmann metric is the invaiance underthe action of the holomorphic automorphisms of Ω.

In the same way, one can introduce the Bergmann kernel and the Bergmann metric on a complex manifold X such that the space of L^{2} canonical forms H_{(2)}^{0}(X, O_{X}(K_{X})) is very ample. In the case of compact complex manifolds, the class of such manifolds corresponds to the class of complex manifolds whose canonical bundles are very ample. Hence it is a relatively small class.

In this paper, we generalize the Bergmann metric to wider class of manifolds and discuss some applications. Roughly speaking, we introduce a Bergmann kernel function for pluricanonical systems.

Theorem 1. Let X be a projective manifold of general type. Then for every sufficiently large integer m, there exists a singular Kähler form ω_{m} on X and a reproducing kernel K_{m}(z, w) of holomorphic m-ple canonical forms on X with respect to the inner product induced from the −m-th powe of the Yau intrisic pseudovolume form on X such that

(1) ω_{m} is invariant under Aut(X),

(2) there exists a singular (degenerate) m-ple volume form K_{m}(z, z) on X such that

ω_{m} = \frac{\sqrt{−1}}{m}∂ ¯∂log K_{m}(z, z)

holds.

Remark 1. To construct ω_{m} one can use the inner product defined by the −m-th power of the Kobayashi pseudovolume form. Then we have another singular Kähler form. The reason why we use Yau pseudovolume form is mainly the bimeromophic invariance of it. This enables us to use birational geometry instead of biholomorphic geometry. But I do not have any specific example which tells us that these pseudovolume forms are actually different.

Theorem 2. Let X be a projective manifold of general type. Let dμ be the Yau volume form on X. Then the Ricci curvature of dμ is negative in the sense of current. In fact

dμ(z) = lim sup_{m→∞} K_{m}(z, z)^{\frac{1}{m}} (z ∈ X)

and

−Ric dμ = lim_{m→∞} ω_{m}

hold.

As an application of this theorem, we prove the following theorem.

Theorem 3. Let π : X ⟶ S be a smooth projective family of projective varieties. Assume that every fibre of π is of general type. Then the plurigenera of the fibres are locally constant on S.

Theorem 3 is a partial answer to the following conjecture.

Conjecture 1. The plurigenera is invariant under smooth projective deformations.

There have been several works on this conjecture.

The conjecture is trivial for the family of curves. As for the deformation of surfaces, S. Iitaka([10]) proved the conjecture using classification of compact complex surfaces. In general dimensions an interesting approach was proposed by M. Levine ([18]). He computed the obstruction for the extention of the member of pluricanonical system on a fibre and proved that it always vanishes by using Hodge theory in the case that the member is smooth. On the other hand, N. Nakayama pointed out that if the minimal model program has been completed (also in the relative case), the conjecture is a direct consequence ([20]). By the completion of minimal model program for the case of 3-folds, J. Kollar and S. Mori proved Conjecture 1 in the case of 3-folds ([13, p.535, Theorem 1.3]). These approaches require some smoothness for a general member of the pluricanonical systems on the fibres.

Our approach is completely different from the above ones. We use the Yau pseudovolume form on the fibre to controle the growth of plurigenera and use the continuity of the pseudovolume form under the deformation. To get the disired result we use the theory of analytic Zariski decompositions and L^{2}-extention theorem in [21].

The author would like to express his hearty thanks to Professor S. Kobayashi who pointed out the reference [27] and to Professor T. Ohsawa for his frank criticism. He also would like to express his hearty thanks to Professor S. Lu who explained Yau pseudovolume form to him. Finally he would like to express his hearty thanks to MSRI for their hospitality and the stimulating atomosphere.

Remark 2. Recently T. Mabuchi and I have proved the conjecture in full generality by a different method (in preparation). Hence Theorem 3 has beed generalized. But the proof of Theorem 3 would be of independent interest.

Part I. Generalized Bergmann Kernels

2. Generalized Bergmann metrics

2.1. Yau pseudovolume form. In [28], S.-T. Yau introduced an intrinsic pseudovolume form on a complex manifold which is very similar to Kobayashi volume form.

Let M be a n-dimensional connected complex manifold. Let Δ^{n} denote the unit open polydisk in C^{n}. Let us take a point x on M. Let f : Δ^{n} ⟶ M be a meromorphic mapping which satisfies the following three conditions:

(1) f is holomorphic on a neighbourhood of O,

(2) f(O) = x,

(3) f is nondegenerate at O. Then we define a pseudovolume form dλ(x) at x by

dμ(x) = inf(f^{−1})^{∗}dμ_{Δ^{n}}

where the infimum is taken with respect to all the f which satisfies the above three conditions and dμ_{Δ^{n}} is the Poincaré volume form on Δ^{n} defined by

dμ_{Δ^{n}} = (∏_{i=1}^{n} \frac{4}{(1− |z_{i} |^{2})^{2}} )(\sqrt{−1})^{n}dz_{1} ∧ dz̄_{1} ∧ ⋯ ∧ dz_{n} ∧ dz̄_{n}

Let f : X − ⋯ → Y be a meromorphic mapping between complex manifolds. Then we have the inequality:

f^{∗}dμ_{Y} ≤ dμ_{X}

by the definition of the pseudovolume form. We call this property the volume decreasing property of Yau pseudovolume form. By this property it is clear that Yau pseudovolume form is invariant under bimeromorphic mappings.

It is well known that dμ is an uppersemicontinuous pseudovolume form on X. It is clear that Yau pseudovolume form is smaller than Kobayashi pseudovolume form (I do not know any example which shows these volume forms are actually different). We call a complex manifold X to be meromorphically measure hyperbolic if for any nonempty open subset of X the measure defined by dμ is nonzero. It is easy to see that every Kobayashi hyperbolic manifolds are meromorphicall y measure hyperbolic.

Theorem 4. ( [28]) Let X be a projective manifold of general type. Then dμ is nondegenerate on a nonempty Zariski open subset of X. In particular X is meromorphically measure hyperbolic.

Lemma 1. Let X be a projective manifold of general type. Then there exists a proper subvariety V of X such that dμ is continuous on X − V.

Proof. This fact is essentially in [28] and was used in [27] implicitly (see the proof of [27, p. 367, Theorem 8]). The proof goes as follows.

For the first by the definiciton we see that dμ is uppersemicontinuous. Hence we only need to prove that dμ is lowersemicontinuous. If we take m sufficiently large, we may assume that | mK_{X} | gives a birational rational mapping into a projective space. Let σ_{0}, …, σ_{N} be global holomorphic sections of mK_{X}(m >> 1) which give a birational rational mapping Φ into P^{N} and the pullback of the Fubini-Study form

\sqrt{−1}∂ ¯∂log ∑_{i=0}^{N} | σ_{i} |^{2}

is a strictly positive on X. The existence of σ_{0}, …, σ_{N} follows from Kodaira’s lemma (cf.[Appendix][14]). Let V denote the support of the base locus of σ_{0}, …, σ_{N}. Let f_{i} : Δ^{n} ⟶ X be a sequece of meromorphic mappings such that

(1) f_{i} is holomorphic near the origin and {f_{i}(O)} converges to a point x on X −V,

(2) the Jacobian determinant Jac df_{i,O} are not zero and converges to a nonzero element.

We note that f_{i}^{∗}σ_{j}(0 ≤ j ≤ N) are holomorphic pluricanonica l forms on Δ^{n} by Hartogs extention theorem. Then by the maximum principle, there exists a constant C independent of i such that

(\frac{(\sqrt{−1})^{m \frac{n(n−1)}{2}} f_{i}^{∗}(∑_{j=0}^{N} σ_{j} ∧ σ̄_{j})}{dμ_{Δ^{n}}^{m}}) ≤ C

holds. Here as usual we apply the maximum principle for

(\frac{(\sqrt{−1})^{m \frac{n(n−1)}{2}} f_{i,ε}^{∗}(∑_{j=0}^{N} σ_{j} ∧ σ̄_{j})}{dμ_{Δ^{n}}^{m}}),

where

f_{i,ε}(z) = f_{i}((1 − ε)z) ((1 − ε)z = ((1 − ε)z_{1}, …, (1 − ε)z_{n})) (0 < ε << 1)

and letting ε tend to 0.

Then by the normal family argument (Montel’s theorem), we see that there exists a subsequence {f_{k}} of {f_{i}} such that {f_{k}^{∗}σ_{j}}(j = 0, …, N) converges to a holomorphic m-ple canonical forms on Δ^{n} compact uniformly. Then {Φ ∘ f_{k}} converges to a meromorphic mapping from Δ^{n} into P^{N}. By the biratonality of Φ {f_{k}} of {f_{i}} converges to a meromorphic mapping f_{∞} : Δ^{n} ⟶ X such that

df_{∞,O} = lim_{k→∞} df_{k,O}

holds. We note that by the assumption x ∈ X − V and the nondegeneracy condition of the Jacobian, f_{∞} is holomorphic at O. Then by this fact it is easy to see that dμ is continuous on X − V. Q.E.D.

The following corollary is an easy consequence of the above theorem.

Corollary 1. Let X be a projective manifold of general type. Then there exits a proper subvariety V in X such that for every x ∈ X − V, there exists a meromorphic mapping f : Δ^{n} ⟶ X which satisfies the following three conditions:

(1) f is holomorphic on a neighbourhood of O,

(2) f(O) = x holds,

(3) (f^{−1})^{∗}(dμ_{Δ^{n}})(x) = dμ_{X}(x) holds.

2.2. Construction of the generalized Bergmann kernels. In this subsection we shall prove Theorem 1. Because in some applications, it is more convinient to construct the generalized Bergmann kernel on the universal covering space, we shall consider not only the projective manifolds of general type but also its universal covering spaces. The proof we present here is for the universal coverings. If we restrict ourselves to the case of the manifold itself, the proof is much more straightforward. But it seems to be natural to consider the universal covering in connection withapplications. The following proposition follows from the definition of Yau pseudovolume form.

Proposition 1. Let M be a complex manifold and let π : M̃ ⟶ M be an unramified covering. Then dμM̃ = π^{∗}dμ_{M} holds. In particular M is meromorphically measure hyperbolic if and only if M̃ is meromorphically measure hyperbolic.

Let X be a meromorphically measure hyperbolic manifold. We define the subspace H_{m} = H_{m}(X) of H^{0}(X, O_{X}(mK_{X})) by

H_{m} = \{η ∈ H^{0}(X, O_{X}(mK_{X})) | (\sqrt{−1})^{m \frac{n(n−1)}{2}} ∫_{X} \frac{η ∧ η̄}{dμ_{X}^{m}}dμ_{X} < ∞\}.

We introduce an inner product on H_{m} by

(η, τ) = (\sqrt{−1})^{m \frac{n(n−1)}{2}} ∫_{X} \frac{η ∧ τ̄}{dμ_{X}^{m}}dμ_{X}.

The inner product exists by Schwarz inequality.

Lemma 2. H_{m} is a Hilbert space.

Proof. Let {η_{k}} be a Cauchy sequence in H_{m}. Let f : Δ^{n} ⟶ X be an embedding into a relatively compact subset of X. We shall prove {f^{∗}η_{k}} converges. By the volume decreasing property of Yau pseudovolume form, we see that {f^{∗}η_{k}} is a Cauchy seqence with respect to the inner product

(η, τ) = (\sqrt{−1})^{m \frac{n(n−1)}{2}} ∫_{X} \frac{η ∧ τ̄}{dμ_{Δ^{n}}^{m}}dμ_{Δ^{n}}.

Hence {f^{∗}η_{k}} converges to a m-ple canonical form. Since f is arbitrary, {η_{k}} converges. Q.E.D.

Let {ϕ_{i}} be a complete orthonormal basis for H_{m}. We set

K_{m}(z, w) = ∑_{i} ϕ_{i}(z)ϕ̄_{i}(w)

and

ω_{m} = \frac{\sqrt{−1}}{m}∂ ¯∂log K_{m}(z, z).

K_{m}(z, w) exists since for every x inf X, ev_{x} : H_{m} ⟶ C ≃ O_{X}(mK_{X})/M_{x} defined by

ev_{x}(f) = f(x)

is a bounded functional. If H_{m} is very ample, ω_{m} is a C^{∞}-Kähler metric on X. In this case we call ω_{m} the m-ple Bergmann metric of X.

Let X be a projective manifold of general type and let π : X̃ ⟶ X be an universal covering of X. Let ω_{X} be a Kähler form on X. Let H_{(2)}^{0}(X̃, OX̃(mKX̃)) be the space of L^{2} holomorphic sections of mKX̃ with respect to π^{∗}ω_{X}. Then by the standard L^{2}- estimates for ¯∂ operator, we see that for every large m, H_{(2)}^{0}(X̃, OX̃(mKX̃)) defines a bimeromorphic mapping on to its image in some infinite dimensional projective space.

Lemma 3. H_{m}(X̃) = H_{(2)}^{0}(X̃, OX̃(mKX̃)) holds.

Proof. For the first we shall prove that

H^{0}(X, L^{2}(mK_{X}, dμ_{X}^{−m})) ≃ H^{0}(X, O_{X}(mK_{X}))

holds for every positive integer m, where L^{2}(mK_{X}, dμ_{X}^{−m})is the sheaf defined by

L^{2}(mK_{X}, dμ_{X}^{−m})(U) = \{σ ∈ Γ(U, O_{X}(mK_{X})) | \frac{| σ |^{2}}{dμ^{m}} ∈ L_{loc}^{1}(U)\}.

Let m be a sufficiently large positive integer such that | mK_{X} | gives a birational rational mapping from X into a projective space. Let

π_{m} : X_{m} ⟶ X

be a resolution of Bs |mK_{X} | and let

π_{m}^{∗} |mK_{X} |=|P_{m} | +F_{m}

be the decomposition into the free part and the fixed part. Then by the assumption P_{m} is nef and big. We identify O_{X_{m}}(P_{m}) as a subsheaf of O_{X_{m}}(π_{m}^{∗}(mK_{X})) Then by Kodaira’s lemma, we see that there exists an effective Q-divisor E such that P_{m} −εE is an ample Q-divisor for every sufficiently small positive rational number ε. Let r = r(ε) be a positive integer such that r(P_{m} − εE) is a very ample Cartier divisor. Let σ_{0}, …, σ_{N} be a basis of H^{0}(X_{m}, O_{X_{m}}(r(P_{m}−εE))). Then for every meromorphic mapping f : Δ^{n} ⟶ X we have the inequality

\frac{(\sqrt{−1})^{rm \frac{n(n−1)}{2}} ∑_{j=0}^{N} f^{∗}σ_{j} ∧ σ̄_{j}}{dμ_{Δ^{n}}^{rm}} ≤ C

holds, where C is a positive constant independent of f. Hence we have that

dμ_{X} ≥ \frac{1}{\sqrt[rm]{C}} (\sqrt{−1})^{\frac{n(n−1)}{2}} (∑_{j=0}^{N} σ_{j} ∧ σ̄_{j})^{\frac{1}{rm}}

holds on X. This implies that

π_{m}^{∗}L^{2}(mrK_{X}, dμ_{X}^{−mr})) ⊇ O_{X_{m}}(r(P_{m} − εE))

holds.

Letting ε tend to 0, by using the above lower estimate for dμ_{X} we see that

π_{m}^{∗}L^{2}(mK_{X}, dμ_{X}^{−m}) ⊇ O_{X_{m}}(P_{m})

holds. Hence for a sufficiently large m we have the desired isomorphism:

H^{0}(X, L^{2}(mK_{X}, dμ_{X}^{−m})) ≃ H^{0}(X, O_{X}(mK_{X})).

By using the ring structure of the canonical ring R(X, K_{X}), we conclude that the above isomorphim holds for every positive m.

Suppose that there exists an element ϕ in H_{(2)}^{0}(X̃, OX̃(mKX̃)) which is not contained in ϕ ∋ H_{m}. We set for a positive integer k > 1,

av(ϕ^{k}) = ∑_{γ∈π_{1}(X)} γ^{∗}ϕ^{k}.

Then as in [26], av(ϕ^{k}) exists and defines an element of H^{0}(X, O_{X}(mkK_{X})). Then we have for every x ∈ X

mult_{x}(av(ϕ^{k})) ≥ mult_{x}Bs |mkK_{X} |

holds. Let us note that the following simple fact that if a sequence of complex numbers {a_{γ}} such that ∑_{γ} a_{γ}^{k} converges for every integer k ≥ 1 and

∑_{γ} a_{γ}^{k} = 0

holds for every k ≥ 1, then a_{γ} = 0 for every γ.

Hence we have for every x ∈ X̃

mult_{x}(ϕ) ≥ lim_{k→∞} k^{−1}mult_{π(x)}Bs |mkK_{X} |

holds. This contradicts the assumption that ϕ is not contained in H_{m} by Proposition 1 and the first half of the proof of this lemma. Q.E.D.

This lemma implies that the kernel K_{m} exists on X̃. And ω_{m} = \sqrt{−1}∂ ¯∂log K_{m}(z, z) defines a singular Kähler form on X̃. We see that by the definition that ω_{m} is invariant under Aut(X̃). Hence ω_{m} definies a singular Kähler form on X.

Corollary 2. ( [26]) Let X_{1},X_{2} be compact unramified quotients of a complex manifold X̃. Then X_{1} is of general type if and only if X_{2} is of general type.

Proof. Suppose that X_{2} is of general type. Then by Theorem 1, X_{1} carries a singular Kähler form ω_{m} which is the curvature form of a singular hermitian metric of the canonical bundle of X_{1}. Then by the standard L^{2}-estimates for ¯∂ operator, we see that X_{1} is also of general type. Q.E.D.

3. Limit of the Generalized Bergmann kernels

In this section we shall denote K_{m}(x, x) by K_{m}(x) for simplicity.

10 HAJIME TSUJI

3.1. First properties of Generalized Bergmann kernels. Lemma 4. For every x ∈ X

K_{m}(x) = max_{∥ϕ∥=1}(ϕ(z)ϕ̄(x))

holds.

Proof It is clear that

K_{m}(x) ≥ max_{∥ϕ∥=1}(ϕϕ̄)(x)

holds by the definition of K_{m}. On the other hand,

K_{m} ≤ max_{∥ϕ∥=1}(ϕϕ̄)

follows from Riesz’s theorem. Q.E.D.

Lemma 5.

lim sup_{m→∞} ∫_{X} Km^{\frac{1}{m}} ≤ ∫_{X} dμ

holds.

Proof. By Hölder inequality we see that

∫_{X} Km^{\frac{1}{m}} ≤ (∫_{X} \frac{K_{m}}{dμ^{m}} dμ)^{\frac{1}{m}} · (∫_{X} dμ)^{\frac{m−1}{m}}

holds. We note that

∫_{X} \frac{K_{m}}{dμ^{m}} dμ = dim H^{0}(X, O_{X}(mK_{X}))

holds.

By Riemann-Roch theorem, we see that there exists a constant c such that

dim H^{0}(X, O_{X}(mK_{X})) = cm^{n} + o(m^{n})

holds. Hence we have that

lim_{m→∞}(∫_{X} \frac{K_{m}(z, z)}{dμ^{m}}dμ)^{\frac{1}{m}} = 1

holds. This implies that

lim sup_{m→∞} ∫_{X} K_{m}(z, z)^{\frac{1}{m}} ≤ ∫_{X} dμ

holds. Q.E.D.

3.2. Poinwise upper estimate of dμ. Let us consider the function

F_{m} = \frac{dμ^{m}}{K_{m}}

on X.

For the first we shall assume that K_{X} is ample. In this case there exists a point x_{0} where F_{m} takes its minimum since F_{m} is continuous. We set

h_{m} = (\frac{1}{F_{m}(z_{0})K_{m}})^{1/m}.

Then h_{m} is a singular hermitian metric on K_{X}.

Lemma 6. ( [22, p.103, Lemma 1.2]) Let X be a smooth projective manifold and let (L, h) be a positive line bundle on X. Let x_{0} be a point on X. Let g be the Kähler metric associated with the curvature of h. Choose a normal coordinate (z_{1}, …, z_{n}) at x_{0} such that x_{0} = (0, …, 0). We choose a local holomorphic frame e_{L} of L at x_{0} such that the local representation function a is the hermitian metric h has the property

a(x_{0}) = 1, da(x_{0}) = 0.

For an n-tuple of integers (p_{1}, …, p_{n}) ∈ Z_{≥0}^{n} and an integer p′ > p = p_{1}+⋯+p_{n} be a positive integer there exists m_{0} > 0 such that for m > m_{0}, there exists a holomorphic section S in H^{0}(X, O_{X}(mL)), satisfying

(1)

∫_{X} ∥ S ∥_{h^{m}} dV_{g} = 1,

(2)

S(z) = λ_{(p_{1},··· ,p_{n})}(z_{1}^{p_{1}} ⋯ z_{n}^{p_{n}} + O(| z |^{2p′})e_{L}^{m}(1 + O(\frac{1}{m^{2p′}})),

where

λ_{(p_{1},... ,p_{n})}^{−2} = ∫_{X \{|z|≤ \frac{log m}{\sqrt{m}} \}} | z_{1}^{p_{1}} ⋯ z_{n}^{p_{n}} |^{2} a^{m}dV_{g}.

where dV_{g} be the volume form with respect to g and |z |= \sqrt{∑ |z_{i} |^{2}}.

We call the above S’s the peak sections of (L, h) at x_{0}. Let g be the Kähler metric associated with curv h_{m}. For a positive integer ℓ we define the number p(ℓ) by

p(ℓ) = max{∥ σ ∥_{h_{m}^{ℓ}} (x_{0}) | σ ∈ H^{0}(X, O_{X}(ℓK_{X})), ∫_{X} ∥ σ ∥_{h_{m}^{ℓ}}^{2} dV_{g} = 1},

and let σ_{ℓ} denote the section which attains the maximum. Then by Lemma 6, we see that

lim_{ℓ→∞} p(ℓ)^{1/ℓ} = 1.

Since h_{m} ≥ dμ^{−m} holds on X by the definition of h_{m}, by Lemma 6, we see that

∫_{X−\{|z|≤ \frac{log ℓ}{\sqrt{ℓ}} \}} ∥ σ_{ℓ} ∥_{dμ^{−mℓ}}^{2} dV_{g} = O(\frac{1}{ℓ^{2}} ).

Since dμ is continuous and h_{m}(x_{0}) = dμ^{−1}(x_{0}) holds , we see that

lim_{ℓ→∞}(∫_{\{|z|≤ \frac{log ℓ}{\sqrt{ℓ}} \}} ∥ σ_{ℓ} ∥_{dμ^{−mℓ}}^{2} dV_{g})^{1/ℓ} = 1

holds. By Lemma 4 this implies that

lim sup_{ℓ→∞} K_{ℓ}(x_{0})^{\frac{1}{ℓ}} ≥ dμ(x_{0})

holds.

On the other hand, for a positive integer ℓ we define the number q(ℓ) by

q(ℓ) = max{∥ τ ∥_{dμ^{−ℓ}} (x_{0}) | τ ∈ H^{0}(X, O_{X}(ℓK_{X})), ∫_{X} ∥ σ ∥_{dμ^{−ℓ}}^{2} dμ = 1}

and let τ_{ℓ} denote the section which attains the maximum. By Lemma 4, we have

q(ℓ) = \frac{K_{ℓ}(x_{0})}{dμ^{ℓ}(x_{0})}

holds. By the inequality

dμ^{−1} ≤ h_{m},

as above we see that L^{2} norm of τ_{ℓ} with respect to dμ^{−ℓ} concentrates around x_{0} as ℓ tends to infinity. In fact let (z_{1}, …, z_{n}) be a local coordinate around x_{0} as in Lemma 6 (with respect to h_{m}). Let ε be a sufficiently small positive number. Let ρ be a nonnegative C^{∞} function of | z |^{2}= ∑_{i=1}^{n} | z_{i} |^{2} such that

(1) ρ ≡ 1 on B(O, ε/2)(= {z ∈ X | | z |≤ ε/2}),

(2) | dρ |≤ 3/ε,

(3) ρ ≡ 0 on X − B(0, ε). If ε sufficiently small, ρ is well defined. Let a be a positive function on B(x, ε) defined as in Lemma 6. Then there exists a posive constant c such that

a ≤ (1 − c ∑_{i=1}^{n} | z_{i} |^{2})

holds on B(O, ε). This follows from the fact that curv h_{m} is positive at x_{0}. On the other hand since F_{m} takes its minimum at x_{0}, we see that the Taylor expansion of τ_{ℓ} around x_{0} is of the form

τ_{ℓ} = (c_{ℓ} + O(| z |^{2})σ_{m}^{ℓ/m}

holds if m | ℓ, where c_{ℓ} is a constant depending on ℓ. For the case that m does not divide ℓ we have the similar expansion of the hermitian norm of τ_{ℓ} with respect to h_{m}^{ℓ}.

We set

G := ¯∂(ρσ_{ℓ}).

Let ψ be a function on X defined by

ψ = nρ log(∑_{i=1}^{n} | z_{i} |^{2})

Then by the Taylor expansions of τ_{ℓ} and a, we see that there exist a positive constant C independent of ℓ and a positive number 0 < β < 1 such that

∫_{X} e^{−ψ} ∥ F ∥_{h_{m}^{ℓ}}^{2} dv ≤ C · c_{ℓ} · β^{ℓ}.

holds, where dv is the pseudovolume form defined by

dv = Km^{\frac{1}{m}} .

Then by the standard Hörmander’s L^{2} estimates for ¯∂ opearator , we have that there exists u ∈ C^{∞}(X, νK_{X}) such that

¯∂u = G

and

∫_{X} e^{−ψ} ∥ u ∥_{h_{m}^{ℓ}}^{2} dv ≤ C_{ℓ} ∫_{X} e^{−ψ} ∥ F ∥_{h_{m}^{ℓ}}^{2} dv ≤ C · C_{ℓ} · c_{ℓ} · β^{ℓ}

holds, where C_{ℓ} is a positive constant satisfying

C_{ℓ} = O(\frac{1}{ℓ}).

Moreover since dμ^{−1} ≤ h_{m} we see that

∫_{X} e^{−ψ} ∥ u ∥_{dμ^{−ℓ}}^{2} dv ≤ C · C_{ℓ} · c_{ℓ} · β^{ℓ}

holds. By the construction

τ̃_{ℓ} := ρτ_{ℓ} − u

is a holomorphic ℓ-ple canonical form on X and

τ̃_{ℓ}(x_{0}) = τ_{ℓ}(x_{0})

holds. Since β < 1, we see that

lim_{ℓ→∞} \frac{∥ τ_{ℓ} ∥}{∥ τ̃_{ℓ} ∥} = 1

holds.

Then it is clear that

lim sup_{ℓ→∞} q(ℓ)^{1/ℓ} ≤ 1

holds. Hence we have that

lim sup_{ℓ→∞} K_{ℓ}(x_{0})^{\frac{1}{ℓ}} = dμ(x_{0})

holds. Since m can be arbitrary large we have that

dμ ≥ lim sup_{ℓ→∞} Kℓ^{\frac{1}{ℓ}}

holds. This implies that the L^{2}-norm of τ_{ℓ} concentrates around x_{0} as ℓ tends to infinity. This means that as for the asymptotics, we can work only on a small neighbourhood of x_{0}. By using this fact, we see that

lim sup_{ℓ→∞} q(ℓ) ≤ 1

holds.

Combining the above arguments we see that

dμ(x_{0}) = lim sup_{ℓ→∞} K_{ℓ}(x_{0})^{1/ℓ}

holds. Letting m tend to infinity, we see that

dμ ≥ lim sup_{ℓ→∞} K_{ℓ}^{1/ℓ}

holds on X.

If K_{X} is not ample, we need to modify the argument as follows. By Kodaira’s lemma (cf. [14, Appendix]) there exists an effective Q-divisor E such that K_{X} − E is ample. We may assume that the support of E is V in Lemma 1. Let r be a positive intger such that rE is a Z-divisor. For every m ≥ r we define the modified generalized Bergmann kernel K̃_{m}(z, w) by

K̃_{m}(z, w) = ∑_{i} ϕ̃_{i}(z)¯ϕ̃_{i}(w),

where {ϕ̃_{i}} is a orthonormal basis of H^{0}(X, O_{X}(mK_{X} −rE)) with respect to the inner product introduced in 2.2. By Lemma 4 one can show easily that for x ∈ X − V

lim_{m→∞}(\frac{K̃_{m}(x)}{K_{m}(x)})^{1/m} = 1

holds by looking at the product:

H^{0}(X, O_{X}(m_{0}K_{X} − rE)) × H^{0}(X, O_{X}(mK_{X})) → H^{0}(X, O_{X}((m + m_{0})K_{X} − rE)),

where m_{0} is a sufficiently large positive integer. It is clear that for a sufficiently large m, dμ^{m}/K̃_{m} is infinity on V and the Ricci form of K̃_{m} is strictly negative. Using K̃_{m} instead of K_{m}, we can argue exactly the same way. Letting m tend to infinity, we obtain the following lemma.

Lemma 7.

dμ ≥ lim sup_{ℓ→∞} Kℓ^{\frac{1}{ℓ}}

holds on X.

3.3. Lower estimate of the limit of Bergmann kernel. If we have the pointwise inequality

lim sup_{m→∞} K_{m}(x, x)^{\frac{1}{m}} ≥ dμ(x) (1)

for every x ∈ X, by Lemma 5 , we have the desired equality

lim sup_{m→∞} K_{m}(x)^{\frac{1}{m}} = dμ(x)

for every x ∈ X. Hence we shall prove (1). Let m be a sufficiently large positive integer. Let V be a proper subvariety of X as in Lemma 1 and let x be a point on X − V. Let ∥ ∥_{m} denote the hermitian norm on mK_{X} with respect to the hermitian metric K_{m}^{−1} and let ∥ ∥ denote the hermitian norm with respect to dμ^{−m}.

Let σ ∈ H^{0}(X, O_{X}(mK_{X})) be a section of mK_{X} such that ∥ σ ∥_{m}= 1 and (σ, σ) = 1 where ( , ) denotes the inner product with respect to the hermitian metric dμ^{−m} as in the last section. Then by Lemma 4, x is the point where ∥ σ ∥ takes its maximum. Hence we have that

d ∥ σ ∥ (x) = 0

holds. Let f : Δ^{n} ⟶ X be a meromorphic mapping such that (1) f is holomorphic on a neighbourhood of O,

(2) f(O) = x,

(3) (f^{−1})^{∗}dμ_{Δ^{n}} = dμ(x). Such f exists by Corollary 1. If we restrict f^{−1} on a small neighbourhood of x, we may consider f^{−1} as a local coordinate around x. We shall denote the coordinate by (z_{1}, …, z_{n}). We note that the inequality

dμ ≤ ∏_{i=1}^{n} \frac{4\sqrt{−1}dz_{i} ∧ dz̄_{i}}{(1− | z_{i} |^{2})^{2}}

holds on a neighbourhood of x by the definition of dμ and the equality holds at x. Let g_{m} be the hermitian metric on mK_{X} on the neighbourhood defined by

g_{m} = (∏_{i=1}^{n} \frac{4\sqrt{−1}dz_{i} ∧ dz̄_{i}}{(1− | z_{i} |^{2})^{2}} )^{−m}.

Then it is clear that

d ∥ σ ∥_{g_{m}} (x) = 0

holds. This implies that

d(\frac{∏_{i=1}^{n}(1− | z_{i} |^{2})^{2m}σ ∧ σ̄}{∏_{i=1}^{n}(dz_{i} ∧ dz̄_{i})^{m}})(x) = 0

holds. Hence if we write (dz_{1} ∧ ⋯ ∧ dz_{n})^{m} as

(dz_{1} ∧ ⋯ ∧ dz_{n})^{m} = b(z)σ (f ∈ O_{X,x})

on a neighbourhood of x, we have that

b(z_{1}, …, z_{n}) = c + ∑_{i,j} c_{ij}z_{i}z_{j} + O(| z |^{3})

holds, where c, c_{ij} are constants. Let ε be a sufficiently small positive number. Let ρ be a nonnegative C^{∞} function of | z |^{2}= ∑_{i=1}^{n} | z_{i} |^{2} such that

(1) ρ ≡ 1 on B(x, ε/2)(= {z ∈ X | | z |≤ ε/2}),

(2) | dρ |≤ 3/ε,

(3) ρ ≡ 0 on X − B(0, ε). If ε sufficiently small, ρ is well defined.

Let a be a positive function on B(x, ε) defined by

a :=∥ σ ∥_{m}^{2} .

We note that lim sup_{m→∞} K_{m}^{−1/m} has strictly positive curvature on X − V. This fact follows from for a sufficiently large ℓ K̃_{ℓ}^{1/ℓ} defined in the last subsection has a strictly positive curvature on X and satisfies the inequality

K̃_{ℓ}^{1/ℓ} ≤ C_{ℓ}dμ

on X for some positive constant C_{ℓ} by the result in the last subsection. Then applying Lemma 6 to the cases (p_{1}, …, p_{n}) = (1, 0, …, 0), …, (0, …, 0, 1) we see that there exists a posive constant c independent of m such that

a ≤ (1 − c ∑_{i=1}^{n} | z_{i} |^{2})^{m}

holds for every sufficiently large m. We set for a positive integer ν

F = ¯∂(ρ\frac{(dz_{1} ∧ … ∧ dz_{n})^{⊗ν}}{c_{ν}}),

where c_{ν} is a positive number defined by

c_{ν} = 4^{−nν} ∫_{Δ^{n}} ∏_{i=1}^{n}(1− | z_{i} |^{2})^{ν−1}dλ

where dλ is the usual Lebesgue measure.

Let ψ be a function on X defined by

ψ = nρ log(∑_{i=1}^{n} | z_{i} |^{2})

Then by the Taylor expansions of b and a, we see that there exist a positive constant C independent of ν and a positive number 0 < α < 1 such that

∫_{X} e^{−ψ} ∥ F ∥_{m,ν}^{2} dv ≤ C(\frac{dμ^{m}(x)}{K_{m}(x)})^{ν/m}α^{ν}.

holds, where ∥ ∥_{m,ν} denotes the hermitian norm induced by K_{m}^{−ν/m} and dv is the pseudovolume form defined by

dv = Km^{\frac{1}{m}} .

Then by the standard Hörmander’s L^{2} estimates for ¯∂ opearator , we have that there exists u ∈ C^{∞}(X, νK_{X}) such that

¯∂u = F

and

∫_{X} e^{−ψ} ∥ u ∥_{m,ν}^{2} dv ≤ C_{ν} ∫_{X} e^{−ψ} ∥ F ∥_{m,ν}^{2} dv ≤ C · C_{ν}(\frac{dμ^{m}(x)}{K_{m}(x)})^{ν/m}α^{ν}

holds, where C_{ν} is a positive constant satisfying

C_{ν} = O(\frac{1}{ν}).

By the construction

S_{m,ν} := ρ\frac{(dz_{1} ∧ ⋯ ∧ dz_{n})^{⊗ν}}{c_{ν}} − u

is a holomorphic ν-ple canonical form on X and

S_{m,ν}(x) = \frac{(dz_{1} ∧ ⋯ ∧ dz_{n})^{⊗ν}}{c_{ν}} (x)

holds. We note that

lim_{ν→∞}(\frac{(dz_{1} ∧ ⋯ ∧ dz_{n})^{⊗ν}}{c_{ν}})^{\frac{1}{ν}} (O) = dμ_{Δ^{n}}(O)

holds. Hence if

\frac{dμ(x)}{\sqrt[m]{K_{m}(x)}}α < 1

holds, by the above estimate and Lemma 7 we see that

dμ(x) ≤ lim sup_{m→∞} K_{m}(x)^{\frac{1}{m}}

Let U be the subset of X − V defined by

U := \{x ∈ X − V | (lim sup_{m→∞} Km^{\frac{1}{m}})(x) = dμ(x)\}.

Then by the proof of Lemma 7, we see that U is nonempty. We note that since \sqrt{−1}∂ ¯∂log(lim sup_{m→∞} Km^{\frac{1}{m}}) is a closed positive current, lim sup_{m→∞} Km^{\frac{1}{m}} is uppersemicontinuous on X. Then by Lemma 1 we see that U is closed. On the other hand by the above consideration we see that U is open. Hence we conclude that U is equal to X − V. This implies that the equality

lim sup_{m→∞} Km^{\frac{1}{m}} = dμ

holds on X − V. Since the curvature of both lim sup_{m→∞} Km^{\frac{1}{m}} and dμ express the first Chern class of X. We see that

lim sup_{m→∞} Km^{\frac{1}{m}} = dμ

holds on X.

4. Invariance of Plurigenera

In this section we shall prove Theorem 3.

4.1. Analytic Zariski decomposition. The notion of analytic Zariski decompositon (AZD) was introduced by the author to study big line bundles on an projective variety. It is known such a decomposition exists for any big line bundles on projective manifolds ([25]). For the first we shall recall the definition of singular hermitian metrics.

Definition 1. Let L be a holomorphic line bundle over a complex manifold M. h is said to be a singular hermitian metric on L, if there exists a C^{∞}-hermitian metric h_{0} on L and a locally L^{1}-function φ such that

h = e^{−φ}h_{0}

holds.

For a singular hermitian line bundle (L, h) as above , we define the curvature curv h of (L, h) by

curv h = curv h_{0} + \sqrt{−1}∂ ¯∂φ.

We define the sheaf L^{2}(O_{X}(L), h) by

L^{2}(O_{X}(L), h)(U) = {σ ∈ Γ(U, O_{X}(L)) | h(σ, σ) ∈ L_{loc}^{1}(U)},

where U runs open subset of M. We call L^{2}(O_{X}(L), h) the sheaf of germs of L^{2} holomorphic sections of (L, h). By a theorem of Nadel [19], we see that L^{2}(O_{X}(L), h) is coherent if the curv h is bounded from below by a minus of some Kähler form locally.

We shall define an analogy of Zariski decompositions for line bundles over a compact complex manifold.

Definition 2. Let (L, h) be a singular hermitian lins bunndle over a compact complex manfold M. We call (L, h) an analytic Zariski decomposition, if it satisfies the following conditions.

(1) curv h is a closed positive (1, 1)-current on M,

(2) the natural morphism H^{0}(X, O_{X}(mL)) → H^{0}(X, L^{2}(O_{X}(mL), h^{m})) is an isomorphim for every positive integer m.

Theorem 5. Let X be a smooth projective variety of general type and let dμ be the Yau pseudovolume form on X. Then (K_{X}, dμ^{−1}) is an analytic Zariski decomoposition of K_{X}.

Proof. By Theorem 2, curv dμ^{−1} is a closed positive current. By Lemma 3 (and its proof), we have the isomorphism:

H^{0}(X, L^{2}(O_{X}(mK_{X}), dμ^{−m})) ≃ H^{0}(X, O_{X}(mK_{X}))

for every positive m. Q.E.D.

4.2. Growth estimate for the sections.

Definition 3. Let L be a holomorphic line bundle over a compact complex n-manifold M. Let gw(L) be the number defined by

gw(L) = lim sup_{m→∞} m^{−n} dim H^{0}(M, O_{M}(mL)).

We call gw(L) the growth of L.

Let X be a smooth projective variety of general type and let dμ be the Yau pseudovolume form on X. Let Θ_{μ} denote the curvature of dμ^{−1}. Then we have the Lebesgue decomposition:

Θ_{μ} = Θ_{μ,abc} + Θ_{μ,sing},

where Θ_{μ,abc} denotes the absolutely continuous part of Θ_{μ} and Θ_{μ,sing} denotes the singular part of Θ_{μ}. We shall study the meaning of this decomposition.

Theorem 6.

∫_{X}(Θ_{μ,abc})^{n} = (2π)^{n}n!gw(K_{X})

Proof. Let π_{m} : X_{m} ⟶ X be a resolution of the base locus of the complete linear system |mK_{X} |. Let

|π_{m}^{∗}mK_{X} |=|P_{m} | +F_{m}

be the decomposition of | π_{m}^{∗}mK_{X} | into the free part and the fixed part. We shall take {π_{m!}} such that π_{(m+1)!} factors throgh π_{m!}. We shall write π_{(m+1)!} as

π_{(m+1)!} = ξ_{m+1} ∘ π_{m!}

Then it is clear that

P_{(m+1)!} − ξ_{m+1}^{∗}(m + 1)P_{m!}

is effective. Since P_{(m+1)!} and P_{m!} are nef, we see that

P_{(m+1)!}^{n} − (ξ_{m+1}^{∗}(m + 1)P_{m!})^{n} ≥ 0

holds for every positive integer m. This implies that

lim_{m→∞} \frac{P_{m!}^{n}}{(m!)^{n}}

exists. It is clear that

gw(K_{X}) ≥ \frac{1}{(2π)^{n}n!} lim_{m→∞} \frac{P_{m!}^{n}}{(m!)^{n}}

holds. But by the work of Fujita ([4, p.1,Theorem]) we see that the equality

gw(K_{X}) = \frac{1}{(2π)^{n}n!} lim_{m→∞} \frac{P_{m!}^{n}}{(m!)^{n}}

holds. On the other hand by the definition of ω_{m} , we have that

∫_{X} ω_{m,abc}^{n} = m^{−n}P_{m}^{n}

holds. Since

Θ_{μ} = lim_{m→∞} ω_{m}

holds, we have that

\frac{1}{(2π)^{n}n!} ∫_{X}(Θ_{μ,abc})^{n} ≤ gw(K_{X})

holds. On the other hand, the Lelong number n(Θ_{μ}), n(ω_{m}) of Θ_{μ}, ω_{m} respectively satisfy the inequality

n(Θ_{μ}, x) ≤ n(ω_{m}, x) (2)

for every x ∈ X (see the proof of Lemma 3). And we note that the de Rham cohomology class of Θ_{μ} and ω_{m} are 2πc_{1}(K_{X}). Let B_{m} denote the base locus of mK_{X}. By the construction the support of n(Θ_{μ}, x), n(ω_{m}, x) are contained in B_{m}. We shall compute the integral

I_{m} = ∫_{X−B}(Θ_{μ}^{n} − ω_{m}^{n}).

Then we have

I_{m} = ∫_{X−B}(\sqrt{−1}∂ ¯∂log \frac{dμ}{Km^{\frac{1}{m}}})(Θ_{μ}^{n−1} + Θ_{μ}^{n−2}ω_{m} + ⋯ + ω_{m}^{n−1})

holds. We note that

Θ_{μ}^{n−1} + Θ_{μ}^{n−2}ω_{m} + ⋯ + ω_{m}^{n−1}

is an absolutely continuous positive form on X − B_{m}. Then by Stokes theorem and (2), we conclued that I_{m} is nonnegative. Hence we have

\frac{1}{(2π)^{n}n!} ∫_{X} Θ_{μ,abc}^{n} ≥ gw_{(}K_{X})

holds. This completes the proof of Theorem 6. Q.E.D.

Remark 3. The other proof will be given (implicitly), in the proof of Theorem 7 below (see also the proof of Lemma 11).

4.3. Analytic Zariski Decomposition of the family. Let π : X ⟶ S be a smooth projective family of general type variety. To prove Theorem 3, we may assume that S is the unit open disk Δ in C. For every t ∈ S, we denote the fibre π^{−1}(t) by X_{t}. Let dμ_{t} denote the Yau pseudovolume form on X_{t} and let Θ_{t} denote the curvature of (K_{X_{t}}, dμ_{t}^{−1}).

Lemma 8. There exists a proper subvariety V which does not contain any fibre of π such that dμ_{t} is a continuous relative pseudovolume form on X − V.

Proof. Since K_{X/S} is relativiely big by Kodaira’s lemma [14, Appendix] , we have a finite number of global sections σ_{0}, … σ_{N} of R^{0}π_{∗}O_{X}(mK_{X/S}) which generates an ample subbundle of mK_{X_{t}} for every t ∈ Δ. Then we can apply the same proof as that of Lemma 1. This proves the lowersemicontinuity of dμ_{t}.

The uppersemicontinuity of dμ_{t} follows from the projectivity of π. Since π is projective, we may assume that the family X is embedded in a projective space P^{N} by an very ample subsehaf of the relative m-ple canonical bundle for some large m as above. Let f_{0} : Δ^{n} − ⋯ → X_{0} be a meromorphic mapping which is holomorphic at O and nondegenerate at O. We want to construct a meromorphic extention f : Δ^{n} × Δ(ρ) − ⋯ → X such that f(z, 0) = f_{0}(z)(z ∈ Δ^{n}) and f(z, t) ∈ X_{t}(z ∈ Δ^{n}), where ρ is a sufficiently small positive number and Δ(ε) is the open polydisk of radius ρ. We claim that such an extension exists if we replace f(z) by f((1−ε)z) for every positive number ε < 1. Let F_{1}, …, F_{r} ∈ O_{Δ}(Δ)[X_{0}, …, X_{N}] be a homogeneous polynomials with coefficients in O_{Δ}(Δ) such that for every t ∈ Δ F_{i,t} := F_{i}(t)(1 ≤ i ≤ r) is a set of defining equations of X_{t} in P^{N}, where we have considered F_{i} as a holomorphic mapping from Δ to C[X_{0}, …, X_{N}]. We set F := (F_{1}, …, F_{r}) : C^{N+1}×Δ ⟶ C^{r}. To construct f is equivalent to find set of holomorphic functions g(z, t) = (g_{0}, …, g_{N}) : Δ^{n} × Δ(ε) ⟶ C^{N} such that the equation

(∗) F (g(z, t), t) = 0

holds with the initial condition

g(z, 0) = g_{0}

where g_{0} is a lifting of f_{0} to a holomorphic mapping to C^{N+1}. We shall expand the solution g(z, t) as

g(z, t) = ∑_{ν=0}^{∞} g_{|ν}(z)t^{ν}.

Then g_{|1}(z) satisfies the equation:

\frac{∂(F_{1}, …, F_{r})}{∂(X_{0}, …, X_{N})}(g_{0})^{t}g_{|1}(z) + \frac{∂F}{∂t} (g_{0}) = 0.

This equation for g_{|1}(z) has a holomorphic solution on Δ^{n} since the polar set of g_{0} is of codimension ≥ 2 on Δ^{n} and the equation (∗) is equivalent to a equation with r = 1 by a generic projection to P^{n+1}. The equation for g_{|ν} is of the form:

\frac{∂(F_{1}, …, F_{r})}{∂(X_{0}, …, X_{N})}(g_{0})^{t}g_{|ν}+terms with holomorphic coefficients in g_{|i}(0 ≤ i ≤ ν − 1) = 0.

Inductively we can determine holomorphic functions g_{|ν}(ν ≥ 1). To assure the convergence of the formal solution, we need to construct g_{|ν}(ν ≥ 1) with some estimate. Let ε be a positive number less than 1. By using the standard argument (see for example [17]), if we replace f(z) by f((1 − ε)z) we can construct a solution g_{|ν} with the estimate

| g_{|ν}(z) |≤ C · M^{ν} (z ∈ Δ^{n})

for some positive constants C and M independent of ν. This assures the convergence of the solution g(z, t) for | t |< 1/M.

The uppersemicontinuity of dμ_{t} follows immediately from the existence of this extention. Q.E.D.

Lemma 9. ∫_{X_{t}} Θ_{t,abc}^{n} is a constant function on t ∈ S.

We set

I(t) := ∫_{X_{t}} Θ_{t,abc}^{n}.

By Theorem 6, we have that

I(t) = gw(K_{X_{t}})

holds. By the well known uppersemicontinuity theorem of the cohomology (cf. [16, p.351, Theorem 7.8]), we have that I(t) is constant outside of a countably many points on S and strictly larger on these points.

On the other hand by Lebesgue-Fatous’ lemma , for every convergent sequence {t_{j}} in S, we have that

I( lim_{ȷ→∞} t_{j}) ≤ lim inf_{ȷ→∞} I(t_{j})

holds. This implies that I(t) is constant on S. Q.E.D.

4.4. Comparison with the global growth. Let π : X ⟶ S be a smooth projective family of varieties of general type over the unit open polydisc in C. To prove Theorem 3 we may assume that this family is embedded into an irrational pencil

φ : Y ⟶ C

such that (1) Y is smooth,

(2) Y is of general type,

(3) the ristriction morphism:

H^{0}(Y, O_{Y} (mK_{Y})) → R_{0}ϖ_{∗}O_{Y} (mK_{Y})_{t}

is surjective for every t ∈ C. We may assume the third condition by the semipositivity of the direct image of the relative pluricanonical sheaf ([11, Thorem 1][5]). Let dμ be the ristriction of the Yau volume form on Y to X and let Θ be the curvature of (K_{X}, dμ^{−1}). By Theorem 2, we see that Θ is a closed positive current on X. By the construction we see that the restriction Θ_{|t} : Θ |_{X_{t}} is well defined for every t ∈ S.

Lemma 10. For every t, we have the equality

I(t) = ∫_{X} Θ_{t,abc}^{n} = ∫_{X_{t}}(Θ_{|t})_{abc}^{n}

Proof By Lemma 6, I(t) is a constant function on S. On the other hand outside a countably many points on S, the equality

I(t) = ∫_{X_{t}}(Θ_{|t})_{abc}^{n}

holds by the construction of Θ_{|t} and the upperseimicontinuity theorem. Then by the trivial inequality,

I(t) ≥ ∫_{X_{t}}(Θ_{|t})_{abc}^{n}

we completes the proof of Lamma 7. Q.E.D.

We set

F_{|t}(m) := L^{2}(mK_{X_{t}}, dμ_{|X_{t}}^{−m}).

Lemma 11.

F_{|t}(m) = L^{2}(X_{t}, dμ_{t}^{−m})

holds.

Proof. To prove this result we shall use the construction of analytic Zariski decompositions via parabolic Monge-Ampère equation in Part 2. We note that we are considering the case of canonical bundles. Then by the construction there exists a Kähler-Einstein form ω_{1},ω_{2} on a nonempty Zariski open subset U of X_{t} such that

L^{2}(X_{t}, dμ_{t}^{−m}) = L^{2}(X_{t}, (ω_{1}^{n})^{−m})

and

F_{|t}(m) = L^{2}(X_{t}, (ω_{1}^{n})^{−m})

holds for every m ≥ 1. Since ω_{1}, ω_{2} are Kähler-Einstein on U by the construction of these currents (see Lemma 34,35 in Part 2) we have the pointwise inequality,

ω_{1}^{n} ≥ ω_{2}^{n}

holds on U. On the other hand by the proof of Theorem 6, we see that

∫_{U} ω_{1}^{n} = ∫_{U} Θ_{t,abc}^{n}

and

∫_{U} ω_{2}^{n} = ∫_{U} Θ_{|t,abc}^{n}

hold. Hence by Lemma 10, we have the equality,

∫_{U} ω_{1}^{n} = ∫_{U} ω_{2}^{n}

holds. Combining the pointwise inequality between ω_{1}^{n} and ω_{2}^{n}, we have that

ω_{1} = ω_{2}

holds. Since (K_{X}, (ω_{i}^{n})^{−1})(i = 1, 2) are AZD, we complete the proof of Lemma 11. Q.E.D.

Remark 4. Lemma 11 is the only part which requires the use of singular Kähler-Einstein metrics. If one succeed in avoiding this very analytic tool, the proof of Theorem 3 will become much easier.

4.5. Use of L^{2}-extention theorem.

Lemma 12. The restriction morphism

H^{0}(X, O_{X}(K_{X} ⊗ L^{2}(O_{X}(mK_{X}), dμ^{−m})) → H^{0}(X_{t}, O_{X}(K_{X_{t}} ⊗ F_{|t}(m)))

is surjective for every t ∈ S.

Proof. This directly follows from the L^{2}-extention theorem of [21]. By Kodaira’s lemma, we see that there exists a effective Q-divisor on X, such that K_{X} − E is relatively ample on X. Then there exists an integer r such that r(K_{X} − E) is a very ample Cartier divisor on X. Let H be a smooth member of | r(K_{X} − E) |. Then X −H −Supp E is Stein and moreover K_{X} is trivial on X −H −Supp E. Now we can apply the extension theorem of [21] and completes the proof of Lemma 12. Q.E.D.

Now we shall completes the proof of Theorem 3. By Theorem 5 and Lemma 11, we see that

H^{0}(X_{t}, O_{X}(K_{X_{t}} ⊗ F_{|t}(m)) ≃ H^{0}(X_{t}, O_{X_{t}}((m + 1)K_{X_{t}}))

holds. By Lemma 12, we completes the proof of Theorem 3.

Part II. Analytic construction of AZD

This part is more or less independent of the last part. We shall prove the following theorem. More general results are in [25].

Theorem 7. Let X be a smooth projective manifold of general type. Then there exists a nonempty Zariski open subset U of X and a closed positive (1, 1)-current ω_{E} such that

(1) ω_{E} is a Kähler-Einstein form on U with constant negative Ricci curvature,

(2) (K_{X}, (ω_{E}^{n})^{−1}) is an analytic Zariski decomposition.

5. Deformation of Kähler form I

In this section we shall consider Hamilton’s equation on a smooth projective variety of general type and determine the maximal existence time for the smooth solution. This section is more or less independent from the other sections. Hence a reader who is familiar with Monge-Ampère equation may skip this section. But the basic method of the estimate of parabolic Monge-Ampère equations is introduced in this section.

5.1. Hamilton’s equation. Let X be a smooth projective n-fold of general type. We consider the initial value problem:

(3)  \frac{∂ω}{∂t} = −Ric_{ω} − ω on X × [0, T)
(4)  ω = ω_{0} on X × {0},

where ω_{0} : a C^{∞}-Kähler form on X, T : the maximal existence time for C^{∞}-solution.

Since

\frac{∂}{∂t}(dω) = −dω on X × [0, T)

dω_{0} = 0 on X × {0},

we have that dω = 0 on X × [0, T), i.e., the equation preserves the Kähler condition.

5.2. Reduction to the parabolic Monge-Ampère equation. Let ω denote the de Rham cohomology class of ω in H_{DR}^{2}(X, R). Since −(2π)^{−1}Ric_{ω} is a first Chern form of K_{X}, we have

[ω] = (1 − exp(−t))2πc_{1}(L) + exp(−t)[ω_{0}]. (5)

Let Ω be a C^{∞}-volume form on X and let

ω_{∞} = curv Ω.

We set

ω_{t} = (1 − exp(−t))ω_{∞} + exp(−t)ω_{0}. (6)

Since [ω] = [ω_{t}] on X × {t} for every t ∈ [0, T), there exists a C^{∞}-function u on X × [0, T) such that

ω = ω_{t} + \sqrt{−1}∂ ¯∂u. (7)

By (3), we have

\frac{∂}{∂t}(ω_{t} + \sqrt{−1}∂ ¯∂u) = \sqrt{−1}∂ ¯∂log(ω_{t} + \sqrt{−1}∂ ¯∂u)^{n} − (ω_{t} + \sqrt{−1}∂ ¯∂u)

Hence

exp(−t)(ω_{∞} − ω_{0}) + \sqrt{−1}∂ ¯∂(\frac{∂u}{∂t})

= \sqrt{−1}∂ ¯∂log(ω_{t} + \sqrt{−1}∂ ¯∂u)^{n} − \sqrt{−1}∂ ¯∂log Ω + exp(−t)(ω_{∞} − ω_{0}).

Then (3) is equivalent to the initial value problem:

\frac{∂u}{∂t} = log \frac{(ω_{t} + \sqrt{−1}∂ ¯∂u)^{n}}{Ω} − u on X × [0, T), (8)

GENERALIZED BERGMANN METRICS AND INVARIANCE OF PLURIGENERA 27

where

u = 0 on X × {0}.

Let

A(X) = {[η] | η : Kähler form on X} ⊂ H_{DR}^{2}(X, R)

be the Kähler cone of X. Since [ω] moves on the segument connecting [ω_{0}] and [ω_{∞}] = 2πc_{1}(K_{X}), we cannot expect T to be ∞, unless 2πc_{1}(K_{X}) is on the closure of A(X) in H_{DR}^{2}(X, R). We shall determine T. It is standard to see that T > 0 ([8]). Theorem 8. If ω_{0} − ω_{∞} is a Kähler form, then T is equal to

T_{0} = sup{t > 0 | [ω_{t}] ∈ A(X)}.

The proof of Theorem 8 is almost parallel to that of [26, p.126, Theorem 3]. 5.3. C^{0}-estimate. Lemma 13. If ω_{0} − ω_{∞} is a Kähler form, then there exists a constant C_{0} such that

\frac{∂u}{∂t} ≤ C_{0} exp(−t).

Proof.

\frac{∂}{∂t}(\frac{∂u}{∂t}) = Δ_{ω} \frac{∂u}{∂t} − \frac{∂u}{∂t} − exp(−t)tr_{ω}(ω_{0} − ω_{∞})

holds by defferentiating (5) by t. By the maximum principle, we have

\frac{∂u}{∂t} ≤ (max log \frac{ω_{0}^{n}}{Ω} ) exp(−t).

Q.E.D.

To estimate u from below, we modify (10) as

\frac{∂u}{∂t} = log \frac{(ω_{t} + \sqrt{−1}∂ ¯∂u)^{n}}{ω_{t}^{n}} + f_{t} − u on X × [0, T_{1}), (9)

where

u = 0 on X × {0},

where

f_{t} = log \frac{ω_{t}^{n}}{Ω} (10)

and T_{1} = min{sup{t > 0 | ω_{t} > 0}, T }. (11)

If t ∈ [0, T_{1}), we have

log \frac{(ω_{t} + \sqrt{−1}∂ ¯∂u)^{n}}{ω_{t}^{n}} = ∫_{0}^{1} \frac{d}{ds} log \frac{(ω_{t} + \sqrt{−1}s∂ ¯∂u)^{n}}{ω_{t}^{n}} ds = ∫_{0}^{1} Δ_{s}uds,

where Δ_{s} is the Laplacian with respect to the Kähler form ω_{t} + \sqrt{−1}s∂ ¯∂u. Then by the minimum principle, (11) and Lemma 13, we have

Lemma 14.

u ≥ −C_{0} exp(−t) + min_{X} f_{t} on X × {t}, t ∈ [0, T_{1}).

We note that this estimate is depending on t and C_{0} is independent of the choice of Ω.

5.4. C^{2}-estimate. For the next we shall obtain a C^{2}-estimate of u.

Lemma 15. ( [29, p.351, (2.22)]) Let M be a compact Kähler manifold and let ω, ω̃ be Kähler forms on M. Assume that there exists a C^{∞}-function φ such that

ω̃ = ω + \sqrt{−1}∂ ¯∂φ.

We set

f = log \frac{ω̃^{n}}{ω^{n}} ,

Then for every positive constant such that

C + inf_{i≠j} R_{iījj̄} > 1,

exp(Cφ)Δ̃(exp(−Cφ)(n + Δφ)) ≥

(Δf − n^{2} inf_{i≠j} R_{iījj̄}) − Cn(n + Δφ)

+(C + inf_{i≠j} R_{iījj̄})(n + Δφ)^{\frac{n}{n−1}} exp(−\frac{f}{n − 1})

holds, where R_{iījj̄} : the bisectional curvature of ω, Δ : the Laplacian with respect to ω.

Applying this lemma to ω_{t} and ω = ω_{t} + \sqrt{−1}∂ ¯∂u, we have:

Lemma 16. For every C > 0 depending only on t ∈ [0, T_{1}) such that

C + inf_{i≠j} R_{iījj̄}(t) > 1 on X × {0},

exp(Cu)(Δ_{ω} − \frac{∂}{∂t})(exp(−Cu)tr_{ω_{t}}ω) ≥

−(Δ_{t} log \frac{ω_{t}^{n}}{Ω} + n^{2} inf_{i≠j} R_{iījj̄}(t) + n)

−C(n − \frac{1}{C} − \frac{∂u}{∂t})tr_{ω_{t}}ω − exp(−t)tr_{ω_{t}}((ω_{0} − ω_{∞}) · \sqrt{−1}∂ ¯∂u)

+(C + inf_{i≠j} R_{iījj̄}(t)) exp(\frac{1}{n − 1}(−\frac{∂u}{∂t} − u + log \frac{ω_{t}^{n}}{Ω} ))(tr_{ω_{t}}ω)^{\frac{n}{n−1}}

holds, on X × {t}) (t ∈ [0, T_{1}), where Δ_{t} : Laplacian with respect to ω_{t}, R_{iījj̄}(t) : the bisectional curvature of ω_{t} and tr_{ω_{t}}((ω_{0} − ω_{∞}) · \sqrt{−1}∂ ¯∂u) denotes the trace (with respect to ω_{t}) of the product of the endomorphisms A, B ∈ End(T X) defined by

\begin{aligned}ω_{t}(A(Z_{1}) ∧ Z̄_{2}) & = & (ω_{0} − ω_{∞})(Z_{1} ∧ Z̄_{2}) \\ ω_{t}(B(Z_{1}) ∧ Z̄_{2}) & = & (\sqrt{−1}∂ ¯∂u)(Z_{1} ∧ Z̄_{2}), \end{aligned}

where the pair (Z_{1}, Z_{2}) runs in T X ×_{X} T X.

Proof. Let

f = log \frac{ω^{n}}{ω_{t}^{n}} = \frac{∂u}{∂t} + u − log \frac{ω_{t}^{n}}{Ω} .

Then by Lemma 15, we have

exp(Cu)Δ_{ω}(exp(−Cu)tr_{ω_{t}}ω)

≥ (Δ_{t}f − n^{2} inf_{i≠j} R_{iījj̄}(t)) − Cn(n + Δ_{t}u)

+(C + inf_{i≠j} R_{iījj̄})(tr_{ω_{t}}ω)^{\frac{n}{n−1}} exp(−\frac{f}{n − 1}).

Since

Δ_{t}f = Δ_{t}(\frac{∂u}{∂t} + u − log \frac{ω_{t}^{n}}{Ω} )

= Δ_{t} \frac{∂u}{∂t} + tr_{ω_{t}}ω − n − Δ_{t} log \frac{ω_{t}^{n}}{Ω}

and

exp(Cu)\frac{∂}{∂t}(exp(−Cu)tr_{ω_{t}}ω)

= −C \frac{∂u}{∂t}tr_{ω_{t}}ω + tr_{ω_{t}} \frac{∂ω}{∂t} − tr_{ω_{t}} \frac{∂ω_{t}}{∂t} · ω

= −C \frac{∂u}{∂t}tr_{ω_{t}}ω + Δ_{t} \frac{∂u}{∂t} − exp(−t)tr_{ω_{t}}(ω_{0} − ω_{∞}) + exp(−t)tr_{ω_{t}}(ω_{0} − ω_{∞}) · ω,

we obtain the lemma. Q.E.D.

Let ε be an arbitrary small positive number. We set

T_{1}(ε) = min{sup{t > 0 | ω_{t} > 0} − ε, T }

and let C be a positive number such that

C + inf_{i≠j} R_{iījj̄}(t) > 1

for all t ∈ [0, T_{1}(ε)]. Then since the function x exp(−x) is bounded on [0, ∞), by the maximum principle and Lemma 16, we have that if exp(−Cu)tr_{ω_{t}}ω take its maximum at (x_{0}, t_{0}) ∈ X × [0, T_{0}(ε)], we have

tr_{ω_{t}}ω(x_{0}, t_{0}) < C_{ε}

for some C_{ε} > 0 depending only on ε. Then by the C^{0}-estimate of u , Lemma 13 and Lemma 14, by the maximum principle for parbolic equations we have that there exists a positive constant C′_{1,ε} such that

tr_{ω_{t}}ω < C′_{1,ε}.

Hence we obtain:

Lemma 17. There exists a positive constant C_{1,ε} depending only on T_{1}(ε) such that

∥ u ∥_{C^{2}(X)}≤ C_{2,ε}

for every t ∈ [0, T_{1}(ε)), where ∥ ∥_{C^{r}(X)} is the C^{2}-norm with repsect to ω_{0}.

Now by [23], for every r ≥ 2 there exists a positive constant C_{r,ε} depending only on T_{1}(ε) such that

∥ u ∥_{C^{r}(X)}≤ C_{r,ε}.

Letting ε tend to 0, we have that

T ≥ T_{1}

holds. Since [ω_{T_{0}}] is on the closure of the Kähler cone A(X), by changing Ω properly, we can make T_{0} − T_{1} > 0 arbitarary small. Hence we conclude that T = T_{0}. This completes the proof of Theorem 9. Q.E.D.

6. Deformation of Kähler form II

In this section we shall construct a Kähler form on a Zariski open subset of X by using a initial value problem similar to (3) in the last section. In this section we use the same notation as in the last section.

6.1. The current ω_{E}. To state our theorem we need the following definitions.

Definition 4. Let D be a R-Cartier divisor on a projective variety Y. Then the stable base locus of D is defined by

SBs(D) = ∩_{ν>0}SuppBs | [νD] | .

Definition 5. Let D be a Cartier divisor on a projective variety Y and let Φ_{|D|}; Y − ⋯ → P^{N(ν)} be the rational map associated with | νD |. Let μ_{ν} : Y_{ν} ⟶ Y be a resolution of the base locus of | νD | and let Φ̃_{|νD|}; X̃ ⟶ P^{N(ν)} be the associated morphism. We set

E(νD) = \overline{μ_{ν}(Ẽ(νD) ∩ (Y − Supp Bs | νD |)} (Zariski closure)

and call it the exceptional locus of | νD |. It is easy to see that E(νD) is independent of the choice of the resolution of the base locus μ_{ν}. We set

SE(D) = ∩_{ν>0}E(νD)

and call it the stable exceptional locus of D.

We set

S = SBs(L) ∪ SE(L).

The main result in this section is the following theorem.

Theorem 9. There exists a d-closed positive (1, 1)-current ω_{E} on X such that (1) ω_{E} is smooth on a nonempty Zariski open subset U of X,

(2) ω_{E} = −Ric_{ω_{E}} holds on U.

(3) [ω_{E}] = 2πc_{1}(L).

6.2. Kodaira’s Lemma. Let ν be a sufficiently large positive integer such that (1) | νL | gives a birational rational map from X into a projective space.

(2) Supp Bs| νL |= SBs(L). Let f_{ν} : X_{ν} → X be a resolution of the base locus of | νL | and let

F_{ν} = ∑_{i} b_{i}^{ν}F_{i}^{ν}

be the fixed part of | f_{ν}^{∗}(νL) |. We take f_{ν} so that F^{ν} is a divisor with normal crosssings. We set

b̃_{i}^{ν} = b_{i}^{ν}/ν.

Let σ_{i}^{ν} be a global holomorphic section of O_{X_{ν}}(F_{i}^{ν}) with divisor F_{i}^{ν}. Then there exist hermitian metrics ∥ ∥ on O_{X_{ν}}(F_{i}^{ν})’s such that

ω_{∞}^{ν} = f_{ν}^{∗}ω_{∞} + ∑ \sqrt{−1}b̃_{i}^{ν}∂ ¯∂log ∥ σ_{i}^{ν} ∥^{2}

is positive on f_{ν}^{−1}(X − S), if ν is sufficiently large. We may assume

log ∥ σ_{i}^{ν} ∥≤ 0

holds for every i. We set By Kodaira’s lemma ([14, Appendix] we have the following lemma

Lemma 18. There exists an effective Q-divisor

R_{ν} = ∑ r_{j}^{ν}R_{j}^{ν}

on X_{ν} such that

f_{ν}^{∗}(K_{X}) − ∑ b̃_{i}^{ν}F_{i}^{ν} − R^{ν}

is an ample Q-divisor on X_{ν}.

We note that εR_{ν} has the same property as R_{ν} for ε ∈ [0, 1]. Let τ_{j}^{ν} be a global section of O_{X_{ν}}(R_{j}^{ν}) with divisor R_{j}^{ν}. Then there exists hermitian metrics ∥ ∥ on O_{X_{ν}}(R_{j}^{ν}) such that

ω_{∞}^{ν} + ∑_{j} \sqrt{−1}r_{j}^{ν}∂ ¯∂log ∥ τ_{j}^{ν} ∥^{2}

is a smooth Kähler form on X_{ν} and ∥ τ_{j}^{ν} ∥≤ 1 holds on X_{ν} for all j. We set

δ_{ν} = ∑_{j} \sqrt{−1}r_{j}^{ν} log ∥ τ_{j}^{ν} ∥^{2} .

Then for every ε ∈ [0, 1],

ω_{∞}^{ν} + ε\sqrt{−1}∂ ¯∂δ^{ν}

is a smooth Kähler form on X_{ν}.

We set

ξ_{ν} = ∑_{i} b̃_{i}^{ν} log ∥ σ_{i}^{ν} ∥^{2} .

6.3. Construction of a suitable ample divisor. To construct Kähler-Einstein current on X, we use the Dirichlet problem for parabolic Monge-Ampère equation. Hence we shall construct a strongly pseudoconvex convex exhaustion of a Zariski open subset of X_{ν} with certain properties. We fix sufficiently large ν hereafter. Let

Φ : X_{ν} ⟶ P^{N}

be a embedding of X_{ν} into a projective space. Let

π_{α}; X_{ν} ⟶ P^{n}(α = 1, … m)

be generic projections and we set

W_{α} : the ramification divisor of π_{α}, H_{α} := π_{α}^{∗}({z_{0} = 0}),

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

(1) ∩_{α=1}^{m}(W_{α} + H_{α}) = ϕ,

(2) D := (F_{ν} + ∑_{α=1}^{m}(W_{α} + H_{α}))_{red} is an ample divisor with normal crossings.

(3) D contains S ∪ R_{ν}.

(4) K_{X_{ν}} + D is ample. Then X_{ν} − D is strongly pseudoconvex and the following lemma is necessary for our purpose.

Lemma 19. There exists a positive strongly plurisubharmonic exhaustion function φ of X_{ν} − D such that ω_{φ} = \sqrt{−1}∂ ¯∂φ is a complete Kähler form on X_{ν} − D.

Proof. Let D = ∑_{k} D_{k} be the irreducible decomposition of D and let λ_{k} be a global holomorphic section of O_{X_{ν}}(D_{k}) with divisor D_{k}. Then there exist hermitian metrics ∥ ∥’s on O_{X_{ν}}(D_{k})’s such that

− ∑_{k} \sqrt{−1}∂ ¯∂log ∥ λ_{k} ∥^{2}

is a smooth Kähler form on X_{ν}. We set for a positive number ι

φ = − ∑_{k} log ∥ λ_{k} ∥ −ι log log \frac{1}{∥ λ_{k} ∥}.

Then if we choose ι sufficiently small, then

ω_{φ} = \sqrt{−1}∂ ¯∂φ

is a complete Kähler form on X_{ν} − D. Clearly by adding a sufficiently large positive number, we can make the exhuastion φ to be positive on X_{ν} − D. Q.E.D.

Remark 5. As one see in Lemma 24 below, ω_{φ} has a bounded Poincaré growth.

6.4. The Dirichlet problem. We set for c > 0,

K_{c} = {x ∈ X_{ν} : φ(x) ≤ c}.

It is easy to see that we may assume that there exists a positive constant c_{0} such that the boundary ∂K_{c} is smooth for every c ≥ c_{0}. We fix such c and set

K := K_{c}

for simplicity.

In the estimate of u, we try to make the estimate independent of c ≥ c_{0} for the later use.

Since X_{ν} − D is canonically biholomorphic to a Zariski open subset of X, we may consider K as a compact subset of X. We consider the following Dirichlet problem for a parabolic Monge-Ampère equation.

\frac{∂u}{∂t} = log \frac{(ω_{t}+\sqrt{−1}∂ ¯∂u)^{n}}{Ω} − u on K × [0, T)

u = (1 − e^{−t^{4}})ξ_{ν}  on ∂K × [0, T) (10)
u = 0  on K × {0},

where Ω is a smooth volume form on X,

ω_{t} = (1 − e^{−t^{4}})curv h + e^{−t^{4}}ω_{0}

and T is a maximal existence time for the smooth solution on K̄(the closure in the usual topology). We shall assume that

ω_{0}^{n} = Ω

holds. It is easy to find such ω_{0} and Ω by using the solution of Calabi’s conjecture ([29]). We set

ω_{∞} = curv h.

By multiplying a common sufficiently large positive number to ω_{0} and Ω, if neceessary, we may assume that

ω_{0} + Ric Ω = ω_{0} − ω_{∞} > 0

holds. Please do not confuse u with the one in the last section. We use the same notation for simplicity. For the first we shall show

Theorem 10. T is infinite and

u_{∞} = lim_{t→∞} u

exists in C^{∞}-topology on K̄.

6.5. C^{0}-estimate. We note that by the above choice of ω_{0},Ω and the Dirichlet condition, the Dirichlet problem (10) is compatible up to 3-rd order on the corner ∂K × {0}.

Hence by the standard implicit function theorem, we see that T is positive. Suppose T is finite. Then if lim_{t→T} u exist on K in C^{∞}-topology, then this is a contradiction. Because again by the implicit function theorem, we can continue the solution a little bit more. Hence to prove Theorem 10 it is sufficient to obtain an estimate of C^{k}-norm of u on K which is independent of t.

We begin with C^{0}-estimate.

Lemma 20. There exists a constant C_{0}^{+} such that

\frac{∂u}{∂t} ≤ C_{0}^{+}e^{−t} on K × [0, T)

holds.

Proof We set

ω = ω_{t} + \sqrt{−1}∂ ¯∂u

and

Δ̃ = tr_{ω}\sqrt{−1}∂ ¯∂.

As in the proof of Lemma 3.1, we have

\frac{∂}{∂t}(\frac{∂u}{∂t}) = Δ̃\frac{∂u}{∂t} − \frac{∂u}{∂t} − 4t^{3}e^{−t^{4}}tr_{ω}(ω_{0} − ω_{∞})

holds on K × [0, T). Since

u = (1 − e^{−t^{4}})ξ_{ν} on ∂K × [0, T)

and ω_{0} − ω_{∞} is a Kähler form on X, by maximal principle

\frac{∂u}{∂t} ≤ C_{0}^{+}e^{−t} on K × [0, T)

holds for

C_{0}^{+} = max\{max_{x∈K} log \frac{ω_{0}^{n}}{Ω} (x), max_{x∈∂K} ξ_{ν}(x)\}.

Q.E.D.

We set

v = u − (1 − e^{−t^{4}})ξ_{ν},
Ω_{ν} = exp(ξ_{ν})Ω,

and

ω̂_{t} = ω_{t} + (1 − e^{−t^{4}})\sqrt{−1}∂ ¯∂ξ_{ν}.

Then v satisfies the equation:

\frac{∂v}{∂t} = log \frac{(ω̂_{t}+\sqrt{−1}∂ ¯∂v)^{n}}{Ω_{ν}} − v on K × [0, T)

v = 0  on ∂K × [0, T) (11)
v = 0  on K × {0}

We note that ω̂_{t} is a Kähler form on (X_{ν} − D) × [0, ∞] and ω̂_{∞} = ω_{∞}^{ν}. Then since

log \frac{(ω̂_{t} + \sqrt{−1}∂ ¯∂v)^{n}}{ω̂_{t}^{n}} = ∫_{a=0}^{1} Δ̃_{a}v,

where Δ̃_{a} is the Laplacian with respect to the Kähler form

ω̂_{t} + a\sqrt{−1}∂ ¯∂v,

by maximum principle and Lemma 19, we have that

v ≥ min\{min_{x∈K} log \frac{ω̂_{t}^{n}}{Ω_{ν}} (x) − C_{0}^{+}e^{−t}, 0\} on K × [0, T)

holds. Hence we have

Lemma 21.

u ≥ C_{0}^{−} + (1 − e^{−t^{4}})ξ_{ν} on K × [0, T),

where

C_{0}^{−} = min\{ inf_{(x,t)∈K×[0,∞)} log \frac{ω̂_{t}^{n}}{Ω_{ν}} (x, t), 0\} − C_{0}

We note that C_{0}^{−} may depend on K because log(ω̂^{n}/Ω_{ν}) may not be bounded from below on X_{ν} − D. To obtain the C^{0}-estimate from below which is independent of K, we shall consider for ε ∈ (0, 1],

v_{ε} = u − (1 − e^{−t^{4}})(ξ_{ν} + εδ_{ν}).

Then by the same argument, we have

Lemma 22. Let ε ∈ (0, 1]. Then there exists a constant C_{0}^{−}(ε) which is independent of K such that

u ≥ C_{0}^{−}(ε) + (1 − e^{−t^{4}})(ξ_{ν} + εδ_{ν}) on K × [0, T)

The reason why C_{0}^{−}(ε) is independent of K is simply because

ω̂_{∞} + ε\sqrt{−1}∂ ¯∂δ_{ν}

on X_{ν} − D extends to a smooth Kähler form on X_{ν} and

exp(ξ_{ν} + εδ_{ν})f_{ν}^{∗}Ω

is a smooth semipositive (n, n) form on X_{ν}.

6.6. C^{1}-estimate on ∂K. Hereafter we estimate derivatives of v basically by using the method in [2]. But our estimates is a little bit more complicated because we are working on a quasi-projective variety which cannot admits a global flat Kähler metric. We set

ψ_{c} = b(φ − c),

where b is a positive constant. We note that

ψ_{c} = 0 on ∂K

and

ψ_{c} < 0 on K.

Then since ω_{φ} is a complete Kähler form of Poincaré growth, if we take b sufficiently large

log \frac{(ω̂_{t} + \sqrt{−1}∂ ¯∂ψ_{c})^{n}}{Ω_{ν}} − ψ_{c} ≥ 0 on K × [0, ∞)

holds. It is easy to see that we can take b independent of c and t. Then we have

\frac{∂(v−ψ_{c})}{∂t} ≥ log \frac{(ω̂_{t}+\sqrt{−1}∂ ¯∂v)^{n}}{(ω̂_{t}+\sqrt{−1}∂ ¯∂ψ_{c})^{n}} − (v − ψ_{c}) on K × [0, T)
v − ψ_{c} = 0  on ∂K × [0, T)
v − ψ_{c} = −ψ_{c}  on K × {0}

Since

log \frac{(ω̂_{t} + \sqrt{−1}∂ ¯∂v)^{n}}{(ω̂_{t} + \sqrt{−1}∂ ¯∂ψ_{c})^{n}} = ∫_{0}^{1} Δ̇_{a}(v − ψ_{c})da,

where Δ̇_{a} is the Laplacian with respect to the Kähler form

ω̂_{t} + \sqrt{−1}∂ ¯∂\{(1 − a)ϕ_{c} + av\}

, by the maximum principle, we obtain

v ≥ ψ_{c} on K × [0, T).

On the other hand, trivially

Δ̂_{t}v ≥ −n on K × [0, T)

holds, where Δ̂_{t} is the Laplacian with respect to the Kähler form ω̂_{t}. Let h be the C^{∞}-function on K̄ × [0, T) such that

\{ \begin{matrix}Δ̂_{t}h & = & −n & on K × [0, T) \\ h & = & 0 & on ∂K × [0, T). \end{matrix}

Then by the maximum principle, we have

v ≤ h on K × [0, T).

Hence we have

ψ_{c} ≤ v ≤ h on K × [0, T). (14)

Now to fix C^{k}-norms on X_{ν} − D, we shall construct a complete Kähler-Einstein form on X_{ν} − D.

We quote the following theorem.

Theorem 11. ( [12]) Let M̄ be a nonsingular projective manifold and let B be an effective divisor with only simple normal crossings. If KM̄ + B is ample, then there exists a unique (up to constant multiple) complete Kähler-Einstein form on M = M̄ − B with negative Ricci curvaure.

By the construction of D, D is a divisor with simple normal crossings and K_{X_{ν}} +D is ample. Hence by Theorem 11, there exists a complete Kähler-Einstein form ω_{D} on X_{ν} − D such that

ω_{D} = −Ric_{ω_{D}}.

Then we have

∥ dv ∥≤ max{∥ dh ∥, ∥ dψ_{c} ∥} on ∂K × [0, T),

where ∥ ∥ is the pointwise norm with respect to ω_{D}. To make this estimate independent of K, we need to use special properties of ω_{D}.

Definition 6. Let V be an open set in C^{n}. A holomorphic map from V into a complex manifold M of dimension n is called a quasi-coordinate map iff it is of maximal rank everywhere on V. (V; Euclidean coordinate of C^{n}) is called a local quasi-coordinate of M.

Lemma 23. (cf. [12, p.405, Lemma 2 and pp. 406-409] ) There exists a family of local quasi-coordinates V = {(V; v^{1}, …, v^{n})} of X_{ν} −D with the following properties.

(1) X_{ν} − D is covered by the images of (V; v^{1}, …, v^{n})’s.

(2) The completment of some open neighbourhood of D is covered by a finite number of (V, v^{1}, …, v^{n})’s which are local coordinate in the usual sense.

(3) Each V, as an open subset of the complex Euclidean space C^{n}, contains a ball of radius 1/2.

(4) There exists positive constants c_{D} and A_{k}(k = 0, 1, 2, …) independent of V ’s such that at each (V, v^{1}, … v^{n}), the inequalities:

\frac{1}{c_{D}} (δ_{ij}) < (g_{ij̄}^{D}) < c_{D}(δ_{ij}),

| (∂^{|p|+|q|}/∂v^{p}∂v̄^{q})g_{ij̄}^{D} |< A_{|p|+|q|}, for any multiindices p and q

hold, where g_{ij̄}^{D} denote the components of ω_{D} with respect to v^{i}’s.

Definition 7. (M̄, B) be a pair of smooth projective variety of dimension n and a divisor with simple normal crossings on it. A complete Kähler metric ω_{M} on M = M̄ − B is said to have bounded Poincaré growth on (M, B) if for any polydisk Δ^{n} = {(z_{1}, …, z_{n}) ∈ C^{n} || z_{i} |< 1(1 ≤ i ≤ n)} in M̄ such that

Δ^{n} ∩ B = {(z_{1}, …, z_{n}) ∈ Δ^{n} | z_{1} ⋯ z_{k} = 0}(k ≤ n),

ω_{M} | Δ^{n} is quasi-isometric to

ω_{P} = ∑_{i=1}^{k} \frac{\sqrt{−1}dz_{i} ∧ dz̄_{i}}{| z_{i} |^{2} (log | z_{i} |)^{2}} + ∑_{i=k+1}^{n} \sqrt{−1}dz_{i} ∧ dz̄_{i}

on every compact subset of Δ^{n} and every covariant derivative of ω_{M} | Δ^{n} is bounded on every compact subset of Δ^{n}.

Then by the construction of ω_{D}, we have:

Lemma 24. ( [12, pp.400-409]) ω_{D} has bounded Poincaré growth. on (X_{ν}, B).

Remember the definition of φ in Lemma 19. Then the following lemma is trivial.

Lemma 25. φ^{−1} ∥ dφ ∥ is uniformly bounded on X_{ν} − D.

We note that v satisfies the following differential inequality.

Δ_{D}v ≥ −tr_{ω_{D}} ω̂_{t} on K × [0, T),

where Δ_{D} is the Laplacian with respect to ω_{D}. Let h_{D} be the solution of the Dirichlet problem

\{ \begin{matrix}Δ_{D}h_{D} & = −tr_{ω_{D}} ω̂_{t} & on K × [0, T) \\ h_{D} & = 0 & on ∂K × [0, T) \end{matrix}

Then by the maximum principle, we have

v ≤ h_{D}

holds on K × [0, T). Hence

ψ_{c} ≤ v ≤ h_{D}

holds on K × [0, T) by (12). Hence by the maximum principle, we have

∥ dv ∥≤ max{∥ dψ_{c} ∥, ∥ dh_{D} ∥}on ∂K × [0, T).

By the standard boundary estimate for the second order linear elliptic equations (cf. [6]), we see that h_{D} is smooth on K̄. By using the standard elliptic estimate and Lemma 23, it is easy to obtain an estimate for ∥ dh_{D} ∥ on K. But in this case, since ω_{φ} = \sqrt{−1}∂ ¯∂φ is a complete Kähler form of Poincaré growth, we can find a negative constant b′ independent of c and t such that

b′Δ_{D}(φ − c) ≤ −tr_{ω_{D}} ω̂_{t}

holds. Then by the maximum principle, we see that

h_{D} ≤ b′(φ − c) on K

holds.

Then since b and b′ are independent of c and t, by Lemma 25, we have:

Lemma 26. There exists a positive constant C′_{1} independent of c ≥ c_{0} such that

∥ dv ∥≤ C′_{1}c on ∂K × [0, T)

where ∥ ∥ is the norm with respect to the Kähler form ω_{D}.

Remark 6. As you have seen above, in the proof of Lemma 26, the use of h_{D} is not unnecessary. We can use a b′ψ_{c} instead of h_{D} from the first. The reason why we have used h_{D} here is that the method can be applicable more general situations.

6.7. C^{1}-estimate on K. Let π_{α} : X_{ν} ⟶ P^{n} be the generic projection constructed in 6.3. And let

Z = Re(∑_{i} β_{i} \frac{∂}{∂(z_{i}/z_{0})}),

where (β_{1}, …, β_{n}) ∈ C^{n} − {O}. Then

θ = π_{α}^{∗}(Z)

is a holomorphic differential operator on X_{ν} − D which is meromorphic on X_{ν}. By operating θ to (11), we have

\frac{∂(θv)}{∂t} = Δ̃(θv) − θv + θ log \frac{ω̂^{n}}{Ω_{ν}} on K × [0, T)
θv = θv  on ∂K × [0, T)
θv = 0  on ∂K × {0},

where Δ̃ is the Laplacian with repect to

ω = ω̂ + \sqrt{−1}∂ ¯∂v.

Then by the maximam principle and Lemma 25, we get

∥ θv ∥≤ C_{1}(θ, K),

where

C_{1}(θ, K) = max\{ sup_{∂K×[0,T)} ∥ θv ∥, sup_{K×[0,T)} ∥ θ log \frac{ω̂^{n}}{Ω_{ν}} ∥\}.

Then since ∥ d log(ω̂^{n}/Ω_{ν}) ∥ is bounded on X_{ν} − D, if we take m sufficiently large and π_{α}(1 ≤ α ≤ m) properly, we get :

Lemma 27. There exists a positive constant C_{1}(K) which depends on c ≥ c_{0} such that

∥ dv ∥≤ C_{1}(K) on K × [0, T)

holds.

The estimate is getting worse if the point goes far from the boundary because θ has a pole along D.

We set

K_{c}(ε) = K_{c} − K_{c−ε}.

Then by the above argument and the construction of D in 6.3, we obtain the following estimate.

Lemma 28. There exists positive constants C_{1} and A_{1} independent of c ≥ c_{0} such that

∥ dv ∥≤ C_{1}c on K(e^{−A_{1}c}) × [0, T).

Remark 7. This idea is inspired by the idea in [4].

6.8. C^{2}-estimate on ∂K. In this subsection, we follow the argument in [2, pp. 218-223] and prove:

Lemma 29. There exists a positive constant C′_{2} independent of c ≥ c_{0} such that

∥ \sqrt{−1}∂ ¯∂v ∥≤ exp(C′_{2}c) on ∂K × [0, T)

Let P be a point on ∂K. Choose a coordinates z_{1}, …, z_{n} with origin at P such that

(1) dg_{ij̄}^{D}(P) = 0 and g_{ij̄}^{D}(P) = δ_{ij}.

(2) There exists a positive number b̂ such that

r = b̂(φ − c)

satisfies r_{z_{α}}(0) = 0 for α < n, r_{y_{n}}(0) = 0, r_{x_{n}} = −1, where

z_{α} = x_{α} + \sqrt{−1}y_{α}

and

r_{z_{α}} = \frac{∂r}{∂z_{α}}

and so on. We set s_{1} = x_{1}, s_{2} = y_{1}, …, s_{2n−3} = x_{n−1}, s_{2n−2} = y_{n−1}, s_{2n−1} = y_{n} = s, s′ = (s_{1}, …, s_{2n−1}). By ∂ ¯∂-Poincaré lemma, we choose a smooth function ϕ defined on a open neighbourhood U of P such that

ω̂_{t} = \sqrt{−1}∂ ¯∂ϕ

holds on U. Let g be a function defined by

g = ϕ + v.

It is clear that to estimate \sqrt{−1}∂ ¯∂v(P) is equivalent to \sqrt{−1}∂ ¯∂g(P) because ω̂_{t} is uniformly bounded with respect to ω_{D} on X_{ν} − D by a constant independent of t. . Moreover by Lemma 24, the convariant derivatives of ω̂_{t} of any order with respect to ω_{D} is uniformly bounded with respect to the norm defined by ω_{D} on X_{ν} − D by a constant independent of t. Then by Lemma 23 , we may assume that U contains a ball of radius 1/2c_{D} with center P and any derivatives of ϕ of a fixed order with respect to (z_{1}, …, z_{n}) is bounded by a constant independent of c ≥ c_{0}, if we allow (U, z_{1}, …, z_{n}) to be a quasi-coordinate. Since the estimate is completely local, this does not cause any trouble in our estimate in this subsection. Hence the C^{2}-estimate of v on ∂K is reduced completely to the C^{2}-estimate of g on ∂K.

Sublemma 1. There exists a positive constant Ċ_{2} independent of c ≥ c_{0} such that

| g_{s_{i}s_{j}}(0) |≤ Ċ_{2} i, j ≤ 2n − 1.

For r near 0 we may represent g as

g = ϕ + σr.

Then

g_{x_{n}}(0) = ϕ_{x_{n}} − σ(0),

so that by Lemma 26, σ(0) ≤ C′_{1}c. Hence

g_{s_{i}s_{j}}(0) = ϕ_{s_{i}s_{j}} + σ(0)r_{s_{i}s_{j}}

holds. We note that r_{s_{i}s_{j}} = O(1/c) because of the normalization. Hence we get the sublemma.

Sublemma 2. There exists a positive constant C̈_{2}independent of c ≥ c_{0} such that

| g_{s_{i}x_{n}}(0) |≤ exp(C̈_{2}c)

The proof of Sublemma 2 is a little bit technical.

Writing the Taylor expansion of r up to second order we obtain:

r = Re(−z_{n} + ∑ a_{ij}z_{i}z_{j}) + ∑ b_{ij̄}z_{i}z̄_{j} + O(| z |^{3}).

Introducing new coordinates of the form

\begin{aligned}z′_{n} & = z_{n} − ∑ a_{ij}z_{i}z_{j}, \\ z′_{k} & = z_{k} for k ≤ n − 1, \end{aligned}

we can write

r = −Re(z′_{n}) + ∑ c_{ij̄}z′_{i}z̄′_{j} + O(| z |^{3}). (15)

It is clear that (c_{ij̄}) is positive definite.

We define T_{i} in a neighbourhood of 0 by

T_{i} = \frac{∂}{∂s_{i}} − \frac{r_{s_{i}}}{r_{x_{n}}} \frac{∂}{∂x_{n}} , for i = 1, …, 2n − 1;

then T_{i}r = 0 nad we have T_{i}(g − ϕ) = 0 on r = 0. We show that for suitable ε > 0, in the region

S_{ε} = {x ∈ U | r(x) ≤ 0, x_{n} ≤ ε},

where U is a neighbourhood of the origin, we set

w = ±T_{i}(g − ϕ) + (g_{s} − ϕ_{s})^{2} − Ax_{n} + B | z |^{2}

We claim :

(a) For B sufficiently large, L̃w ≥ 0;

(b)On ∂S_{ε}, if A is sufficiently large, w ≤ 0 holds.

To prove (a) set

a = −r_{s_{i}}/r_{x_{n}}

and consider (we use summation convention)

L̃(T_{i}g) = T_{i} log Ψ(z, g(z)) + g^{pq̄}a_{p}g_{x_{n}q̄} + g^{pq̄}aq̄g_{x_{n},p} + g^{pq̄}a_{pq̄}g_{x_{n}}, (16)

where

L̃ = Δ̃ − \frac{∂}{∂t},

and

Ψ(z, g(z)) = \frac{Ω_{ν}}{(\sqrt{−1})^{n}dz_{1} ∧ dz̄_{1} ∧ … ∧ dz_{n} ∧ dz̄_{n}} exp(−gϕ).

Obeserve that g^{pq̄}g_{nq̄} = δ_{n}^{p} and that

\frac{∂}{∂x_{n}} = 2\frac{∂}{∂z_{n}} + \sqrt{−1}\frac{∂}{∂s}

so that

g_{x_{n}q̄} = 2g_{nq̄} + \sqrt{−1}g_{sq̄}.

Thus the second term on the right-hand side of (16) is of the form

a_{n} + g^{pq̄}a_{p}g_{tq̄} = O(1 + (∑ g^{iī})^{1/2}(g^{pq̄}g_{ps}g_{q̄s})^{1/2}).

A similar estimate holds for the third term on the right of (16) while the forth term is O(∑ g^{iī}). Thus by Lemma 19 and the arithmetic-geometric mean inequality, we have

±L̃T_{i}g ≤ −CΨ^{−1/n} − g^{pq̄}g_{ps}g_{q̄s}.

Further

\begin{aligned}L̃(g_{s} − ϕ_{s})^{2} & = 2g^{pq̄}g_{ps}g_{q̄s} + 2(g_{s} − ϕ_{s})(∂_{s} log Ψ − L̃ϕ_{s}) \\ & ≥ 2g^{pq̄}g_{ps}g_{q̄s} − CΨ^{−1/n} \end{aligned}

holds on S_{ε} by the C^{1}-estimate (Lemma 28) and the arithmetic-geometric mean inequality (if we take ε sufficiently small). Hence we find

\begin{aligned}L̃w & ≥ B ∑ g^{iī} − CΨ^{−1/n} \\ & ≥ 0 on S_{ε} × [0, T), \end{aligned}

if B is sufficiently large. B depends on the C^{1}-estimate of v on S_{ε} × [0, T) and which is uniform with the weight φ by Lemma 27. This completes the proof of (a).

To prove (b), consider first ∂S_{ε} ∩ ∂K. Here we write x_{n} = ρ(s_{1}, …, s_{2n−1}) and from (13), we deduce that

ρ(t) = ∑_{ij<2n} b_{ij̄}s_{i}s_{j} + O(| s′ |^{3}) (17)

with (b_{ij̄}) positive definite. Thus on ∂K near O we have

x_{n} ≥ a | z |^{2},

where a is uniformly bounded from below by a positive constant times 1/c where by the construction of K. Also,

g(s, ρ(s)) = ϕ(s, ρ(s)),

so that

| g_{s} − ϕ_{s} |^{2}≤ C | s |^{2}≤ Cρ.

Taking A large we obtain (b). By Lemma 28. if we take a positive constant C̈_{2} sufficiently large, we may assume that A is bounded from above by a constant times exp(C̈_{2}c) for some positive constant C̈_{2} independent of c ≥ c_{0}. By the maximum principle and (a),(b),

w ≤ 0 on S_{ε}

holds.

In view of the maximum principle, we have

| (T_{i}g)_{x_{n}}(0) |≤ A.

This completes the proof of Sublemma 2.

Using, still the special coordinate above we see that to finish our proof of Lemma 29 , we have only to establish the estimate

| g_{x_{n}x_{n}}(O) |≤ exp(Ć_{2}c) (18)

for some constant Ć_{2} independent of c ≥ c_{0}. By the previous estimates:

| g_{s_{i}s_{j}}(O) |≤ Ċ_{2} (1 ≤ i, j ≤ 2n − 1),
| g_{s_{i}x_{n}} |≤ exp(C̈_{2}c) (1 ≤ i ≤ 2n − 1),

it suffices to prove

| g_{nn̄}(O) |≤ exp(C′_{2}c)

for some Ĉ_{2} independent of c ≥ c_{0}. We may solve the equation

det(g_{ij̄})(O) = \frac{Ω_{ν}}{ω_{D}^{n}} (O)

for g_{nn̄}(O). Then since there exists a positive constant C^{♭} such that

exp(−C^{♭}c) ≤ \frac{ω_{D}^{n}}{Ω_{ν}} ≤ exp(C^{♭}c)

holds on X_{ν} − D, we see that (16) follows from (17) provided we know the following sublemma.

Sublemma 3. ( [2, pp. 221-223]) The (n − 1) by (n − 1) matrix

(g_{z_{α}z̄_{β}}(O))_{α,β<n} ≥ C_{3}(\frac{Ω_{ν}}{ω_{D}^{n}})^{\frac{1}{n}} \frac{1}{c}I

for some C_{3}; here I is the (n − 1) by (n − 1) identity matrix. C_{3} is independent of c ≥ c_{0} The proof of this sublemma is very technical. After subtraction of a linear function we may assume that ϕ_{s_{j}}(O) = 0, j ≤ 2n − 1.

To prove Sublemma 3, it suffices to prove

∑_{α,β<n} γ_{α}γ̄_{β}g_{z_{α}z̄_{β}}(O) ≥ C_{3} | γ |^{2}

which we shall do for γ = (1, 0, …, 0). We shall show that

g_{1¯1}(O) ≥ C_{4}, (19)

where C_{4} is a positive constant. Let g̃ = g − λx_{n} with λ so chosen that

(\frac{∂^{2}}{∂s_{1}^{2}} + \frac{∂^{2}}{∂s_{2}^{2}} )g̃(s_{1}, …, s_{2n−1}, ρ(s_{1}, …, s_{2n−1})) = 0 at O,

i.e.

0 = g_{1¯1}(O) + g̃_{x_{n}}(O)ρ_{1¯1}(O) = g_{1¯1}(O) + (g_{x_{n}}(O) − λ)ρ_{1¯1}(O). (20)

Using the fact that any real homogeneous cubic polynomial in (s_{1}, s_{2}) admits the unique decomposition

Re(α(s_{1} + \sqrt{−1}s_{2})^{3} + β(s_{1} + \sqrt{−1}s_{2})(s_{1} + \sqrt{−1}s_{2})^{2}),

we find on expanding g̃ |_{∂K∩U} in a Taylor series, in s_{1}, …, s_{2n−1},

\begin{aligned}g̃ |_{∂K∩U}= Re ∑_{2}^{n−1} a_{j}z_{1}z̄_{j} + Re(az_{1}s) & +Re(p(z_{1}, …, z_{n−1}) + βz_{1} | z_{1} |^{2}) \\ & +O(s_{3}^{2} + … + s_{2n−1}^{2}), \end{aligned}

where p is a holomorphic cubic polynomial.

With the aid of (13), we may replace the term βz_{1} | z_{1} |^{2} to (ρ_{z_{1}z̄_{1}}(O))^{−1}βz_{1}x_{n}, if we change the a_{j}, a and p. Thus by changing the a_{j}, a and p appropriately we may obtain the inequality:

g̃ |_{∂K∩U}≤ Re p(z) + Re ∑_{2}^{n} a_{j}z_{1}z̄_{j} + C ∑_{j=2}^{n} | z_{j} |^{2} .

Let ǧ = g̃ − Re p(z) and observe that ǧ satisfies

det(ǧ_{jk̄}) = det(g̃_{jk̄}) = Ψ(z, g(z)).

Recall that Ψ(z, g(z)) ≥ δ > 0 on a neighbourhood of O, where δ depends on Ω_{ν}/ω_{D}^{n}(O) and the C^{1}-estimate of v on the neighbourhood. With ε small we see that in the set S_{ε}, we have Ψ(z, g(z)) ≥ δ. Let

h = −δ_{0}x_{n} + δ_{1} | z |^{2} +\frac{1}{B} ∑_{2}^{n} | a_{j}z_{1} + Bz_{j} |^{2} .

We wish to show that with the suitable choice of δ_{0}, δ_{1}, B > 0 we have h ≥ ǧ on ∂S_{ε}. First observe that if B is sufficiently large and δ_{0} so small that −δ_{0}x_{n} + δ_{1} | z |^{2}≥ 0 on ∂S_{ε} ∩ ∂K (the dependence of δ_{0} and δ_{1} is controlled by the Levi form of ∂K). By Lemma 28, if we take B sufficiently large, we have

ǧ ≤ h on ∂S_{ε}.

The function h is plurisubharmonic and the lowest eigenvalues of the complex Hessian (h_{ij̄}) are bounded independently by δ_{1} while the other eigenvalues are bounded independently of δ_{1}.

Hence choosing δ_{1} equal to small const. times δ^{1/n}

det(h_{ij̄}) ≤ δ in S_{ε}

holds. By the mximum principle

ǧ ≤ h on S_{ε}

holds. Hence by the maximum principle

ǧ_{x_{n}}(O) ≤ h_{x_{n}}(O) = −δ_{0}.

The desired inequality follows from (19). This completes the proof of Lemma 29.

6.9. C^{2}-estimate on K. Using the C^{2}-estimate on ∂K, we shall obtain a C^{2}- estimate inside K. The method here is the same as in [26].

Let H be a smooth function on X_{ν} − D defined by

H = exp(δ_{ν})(∏_{k} ∥ λ_{k} ∥ ∏_{k}(log \frac{1}{∥ λ_{k} ∥})^{−1})^{ε}

where δ_{ν} is the one in 4.2 and ∥ λ_{k} ∥’s are the ones in 4.6 and ε is a sufficiently small positive number such that

ω_{H} = ω̂_{t} + \sqrt{−1}∂ ¯∂log H

is a complete Kähler form on (X_{ν} − D) × [0, ∞] which is quasi-isometric to ω_{D} on X_{ν} − D, i.e., there exists a positive constant C(D, H) > 1 such that

\frac{1}{C(D, H)}ω_{H} ≤ ω_{D} ≤ C(D, H)ω_{H}

holds on X_{ν} − D. We note that ω_{H} have bounded Poincaré growth so that the bisectional curvature of ω_{H} is bounded between two constants uniformly on (X_{ν} − D) × [0, ∞].

We set

v_{H} = v − log H = u − (1 − e^{−t^{4}})ξ_{ν} − log H,
Ω_{H} = H · Ω_{ν}.

48 HAJIME TSUJI

Then v_{H} satisfies the equation

\frac{∂v_{H}}{∂t} = log \frac{(ω_{H} + \sqrt{−1}∂ ¯∂v_{H})^{n}}{Ω_{H}} − v_{H} on K × [0, T).

By Lemma 20 and Lemma 22 v_{H} satisfies the C^{0}-estimate: Lemma 30. For every sufficiently small positive number ε

\begin{matrix}v_{H} ≥ & C_{0}^{−}(ε) − log H + (1 − e^{−t^{4}})εδ_{ν} \\ v_{H} ≤ & C_{0}^{+}(1 − e^{−t}) − (1 − e^{−t^{4}})ξ_{ν} − log H \end{matrix}

holds on K ×[0, T), where C_{0}^{+}, C_{0}^{−}(ε) are constants in Lemma 20 and 22 respectively.

We have the following lemma. Lemma 31. ( [26, Lemma 3.2])

H^{−C}e^{Cv}(Δ̃ − \frac{∂}{∂t})(e^{−Cv}H^{C}tr_{ω_{H}}ω) ≥

(−Δ_{H} log \frac{ω_{H}^{n}}{Ω_{H}} − n^{2} inf_{i≠j} R_{iījj̄}^{H} − n) + C(n − \frac{1}{C} − \frac{∂v_{H}}{∂t})tr_{ω_{H}}ω −

e^{−t}tr_{ω_{H}}((ω_{0} − ω_{∞}) · ω) + (C + inf_{i≠j} R_{iījj̄}^{H}) exp(\frac{1}{n − 1}(−\frac{∂v_{H}}{∂t} − v_{H} + log \frac{ω_{H}^{n}}{Ω} ))(tr_{ω_{H}}ω)^{\frac{n}{n−1}} ,

holds on K ×[0, T), where tr_{ω_{H}}((ω_{0} −ω_{∞})·ω) is defined as in Lemma 3.4, inf_{i≠j} R_{iījj̄}^{H} denotes the infimum of the bisectional curvarue of ω_{H} on (X_{ν} − D) × [0, ∞] and C is a positive constant such that

C + inf_{i≠j} R_{iījj̄}^{H} > 1

holds.

The proof of this lemma is the same as one of Lemma 3.2 in [26]. Hence we omit it. Lemma 32. If we take C sufficiently large, then there exists a positive constant C_{2} independent of c ≥ c_{0} such that

H^{C}tr_{ω_{D}}ω ≤ C_{2} on K × [0, T)

holds.

Proof. By the definition H has zero of order at least r_{j}^{ν}/2 along R_{j}^{ν} (cf. 6.2). Then by Lemma 29, if we take C sufficiently large, there exists a constant C̃_{2} independent of c ≥ c_{0} such that

H^{C}tr_{ω_{D}}ω ≤ C̃_{2} on ∂K × [0, T).

holds. Suppose e^{−Cv}H^{C}tr_{ω_{H}}ω takes its maximum at P_{0} ∈ K × {t_{0}}(t_{0} ∈ [0, T)) then by Lemma 31, we have

(tr_{ω_{H}}ω)(P_{0}) ≤ Ĉ_{2}

for a positive constant Ĉ_{2} independent of c ≥ c_{0} and C(if it is sufficiently large). Hence in this case we have

(H^{C}tr_{ω_{H}}ω)(P) ≤ H^{C}(P_{0}) exp(−C(v(P_{0}) − v(P)))Ĉ_{2} on K × [0, T)

holds. By Lemma 30 (since in Lemma 30, we can take ε arbitrarily small), H exp(−v) = exp(−v_{H}) is uniformly bounded from above on X_{ν} − D. Hence if we change Ĉ_{2}, if necessary, by the maximum principle for parabolic equations, we may assume that

(H^{C}tr_{ω_{H}}ω)(P) ≤ exp(Cv(P))Ĉ_{2}

holds on K × [0, T). We note that ω_{D} and ω_{H} are quasi-isometric on X_{ν} − D. Hence by Lemma 22, if we take C sufficiently large, there exists a positive constant C_{2} independent of c ≥ c_{0} such that

H^{C}tr_{ω_{D}}ω ≤ C_{2} on K × [0, T)

holds. Q.E.D.

By [23], the higher order interior estimate on K×[0, T) follows. As for the boundary estimate of u, we just need to follow the argument in [2]. This completes the proof of Theorem 10.

6.10. Costruction of ω_{E}. Let u_{∞} be as in Theorem 10. Then by the construction

ω_{K} = ω_{∞} + \sqrt{−1}∂ ¯∂u_{∞}

is a Kähler-Einstein form on K̄ = K̄_{c}. We may assume that c > 1. Let us take the exahustion {K_{lc}}_{l=1}^{∞} of X_{ν} − D and let

ω_{l}^{ν} = ω_{K_{lc}}.

Then by Lemma 20 and Lemma 32 and the regularity theorem in [23], we have the following lemma.

Lemma 33. There exists a subsequence of {ω_{l}^{ν}}_{l=1}^{∞} which converges uniformly on every compact subset of X_{ν} − D in C^{∞}-topology to a Kähler-Einstein form ω^{ν} on X_{ν} − D.

Although ω^{ν} is a Kähler-Einstein form, it is not enough good for our purpose. Let us consider the linear system | m!νL | and construct ξ_{m!ν} as before. We denote

ξ_{m!ν} by ξ^{(m)} for simplicity. Since X_{m!ν} − F^{m!ν} are all biholomorphic to X − SBs(L), we may consider {ξ^{(m)}} as a family of functions on X_{ν} − D. Let us denote X_{m!ν} by X^{(m)} for simplicity and define X_{(1)} = X.

Lemma 34. We can construct {ξ^{(m)}} so that

ξ^{(1)} ≤ ξ^{(2)} ≤ … ≤ ξ^{(m)} ≤ ξ^{(m+1)} ≤ …

holds on X_{ν} − D.

Proof. Let I_{μ} denote the ideal sheaf of the base scheme Bs | μL |. Then clearly

I_{μ_{1}+μ_{2}} ↪ I_{μ_{1}} ⊗_{O_{X}} I_{μ_{2}}

holds. Hence inductively we can construct a morphism

μ_{m} : X_{(m)} ⟶ X_{(m−1)} (m ≥ 2)

such that

(1) f_{(m)} := μ_{m} ∘ ⋯ ∘ μ_{1} : X_{(m)} ⟶ X_{(0)}(= X) is a resolution of Bs | m!νL |.

(2) The fixed part of | f_{(m)}^{∗}(m!L) | is a divisor with normal crossings on X_{(m)}.

An explicit construction of {ξ^{(m)}}_{m=1}^{∞} is as follows. Let V^{(1)} = {η_{i}^{(1)}}_{i=0}^{N(1)} be a basis of H^{0}(X, O_{X}(νL)). By induction for each m ≥ 1, we can construct a finite subset

V^{(m)} = {η_{1}^{(m)}, …, η_{N(m)}^{(m)}}

in H^{0}(X, O_{X}(m!νL) with the following properties.

(1) V^{(m)} spans H^{0}(X, O_{X}(m!νL)).

(2) V^{(m)} contains all the elements of the form:

(η_{i_{1}}^{(m−1)})^{⊗a_{1}} ⊗ ⋯ ⊗ (η_{i_{m}}^{(m−1)})^{⊗a_{m}},

∑_{i=1}^{m} a_{i} = m, a_{i} ≥ 0, 0 ≤ i_{1} < … < i_{m} ≤ N(m − 1).

Now we set

ξ^{(m)} = \frac{1}{m!ν} log( ∑_{i=0}^{N(m)} \frac{(\sqrt{−1})^{m!νn^{2}}η_{i}^{(m)} ∧ η̄_{i}^{(m)}}{Ω^{m!ν}}).

We may consider ξ^{(m)} as a function on X_{ν} − D. Then by the construction

ξ^{(1)} ≤ ξ^{(2)} ≤ … ≤ ξ^{(m)} ≤ …

holds and

ω_{∞} + \sqrt{−1}∂ ¯∂ξ^{(m)}

is a smooth semipositive form on X_{(m)} and positive on X_{ν} − D. This completes the proof of the lemma. Q.E.D.

Now we consider the following Dirichlet problem.

\frac{∂u^{(m)}}{∂t} = log \frac{(ω_{t}+\sqrt{−1}∂ ¯∂u^{(m)})^{n}}{Ω} − u^{(m)} on K × [0, T_{m})
u^{(m)} = (1 − e^{−t^{4}})ξ^{(m)}  on ∂K × [0, T_{m})
u^{(m)} = 0  on K × {0},

where T_{m} is the maximal existence time for smooth solution on K̄.

Lemma 35. The followings are true.

(1) T_{m} is infinite and u_{∞}^{(m)} = lim_{t→∞} u^{(m)} exists in C^{∞}-topology on K̄.

(2) ω_{1}^{(m)} := ω_{∞} + \sqrt{−1}∂ ¯∂u_{∞}^{(m)} is a Kähler form on K.

(3) If we define a sequence of Kähler forms {ω_{l}^{(m)}}_{l=1}^{∞} in the same manner as the definition of {ω_{l}^{ν}}_{l=1}^{∞} above, then there exists a subsequence of {ω_{l}^{(m)}} which converges uniformly on every compact subset of X_{ν} − D uniformly in C^{∞}- topology.

Proof. The only difference between the above equation and the equation (10) is that ∇^{k}ξ^{(m)}(k ≥ 1) is bounded with respect to ω_{D} with weights different from before. It is easy to find such weights. In fact, there exists a positive constant C(m, k) depending only on m and k such that H^{C(m,k)} ∥ ∇^{k}ξ^{(m)} ∥ is bounded by a positive constant on X_{ν} − D. Hence the previous argument is valid with some weight with respect to H. Hence the first assertion is trivial.

Then by replacing K to K_{lc}, we get a sequence of Kähler-Einstein form {ω_{l}^{(m)}}_{l=1}^{∞} which are defined on K_{lc} respectively (ω_{l}^{(1)} = ω_{l}^{ν}).

We would like to find a subsequence of {ω_{l}^{(m)}}_{l=1}^{∞} which converges in C^{∞}-topology on every compact subset of X_{ν} − D.

For the first, replacing ξ_{ν} by ξ^{(m)}, completely analogous estimate as Lemma 22 holds for u^{(m)} with the perturbation δ^{(m)} completely analogous to δ_{ν}. As for the C^{2}-estimate, by the proof of Lemma 35, Lemma 32 also holds for ω_{l}^{m} if we replace C and C_{2} to appropriate constants independent of l. The rest of the proof is the same as in the proof of Lemma 33. Q.E.D.

Taking subsequence, if necessary, we obtain ω^{(m)} as before, where ω^{(0)} = ω^{ν}.

52 HAJIME TSUJI

We shall consider ω^{(m)} as a d-closed positive (1, 1)-current on X by

ω^{(m)} := ω_{∞} + \sqrt{−1}∂ ¯∂log \frac{(ω^{(m)})^{n}}{Ω} ,

where ∂ ¯∂ is taken in the sense of a current. This definition is well defined by the C^{0}-estimate (Lemma 22) and clearly

[ω^{(m)}] = 2πc_{1}(L)

holds. Now we want to show that Proposition 2.

ω_{E} = lim_{m→∞} ω^{(m)}

exists in the sense of a d-closed positive (1, 1)-current on X_{ν}. Lemma 36. There exists a positive constant C_{3} independent of m such that

(ω^{(m)})^{n} ≤ C_{3}ω_{D}^{n} on X_{ν} − D

holds.

Proof. Since ω_{D} is a complete Kähler-Einstein form on X_{ν} − D, by applying the maximum principle to the function

log \frac{(ω^{(m)})^{n}}{ω_{D}^{n}} ,

we obtain the lemma. Q.E.D.

Hence {(ω^{(m)})^{n}} is uniformly bounded from above. For the next we shall show: Lemma 37. For every m ≥ 1, we have that

(ω^{(m)})^{n} ≤ (ω^{(m+1)})^{n} on X_{ν} − D

holds.

Proof. u^{(m+1)} − u^{(m)} satisfies the following equations:

\frac{∂(u^{(m+1)}−u^{(m)})}{∂t} = log \frac{(ω_{t}+\sqrt{−1}∂ ¯∂u^{(m+1)})^{n}}{(ω_{t}+\sqrt{−1}∂ ¯∂u^{(m)})^{n}} − (u^{(m+1)} − u^{(m)}) on K × [0, ∞)
u^{(m+1)} − u^{(m)} = (1 − e^{−t^{4}})(ξ^{(m+1)} − ξ^{(m)})  on ∂K × [0, ∞)
u^{(m+1)} − u^{(m)} = 0  on K × {0}.

Since

log \frac{(ω_{t} + \sqrt{−1}∂ ¯∂u^{(m+1)})^{n}}{(ω_{t} + \sqrt{−1}∂ ¯∂u^{(m)})^{n}} = ∫_{0}^{1} Δ_{a}^{(m,m+1)}(u^{(m+1)} − u^{(m)})da,

where Δ_{a}^{(m,m+1)}(a ∈ [0, 1]) is the Laplacian with respect to the Kähler form

ω_{t} + \sqrt{−1}∂ ¯∂\{(1 − a)u^{(m)} + au^{(m+1)}\},

we see that this equation is of parabolic type. We note that ξ^{(m+1)} ≥ ξ^{(m)} on X_{ν} − D by the construction. Then by the maximum principle we obtain

u^{(m)} ≤ u^{(m+1)}on K × [0, ∞).

We set

u_{∞}^{(m)} = lim_{t→∞} u^{(m)}.

Then we have

ω_{1}^{(m)} = ω_{∞} + \sqrt{−1}∂ ¯∂u_{∞}^{(m)}

and

(ω_{1}^{(m)})^{n} = exp(u_{∞}^{(m)})Ω

on K. Hence we see that

(ω_{1}^{(m)})^{n} ≤ (ω_{1}^{(m+1)})^{n}

holds on K. By replacing K to K_{lc} and repeating the same argument, we see that

(ω_{l}^{(m)})^{n} ≤ (ω_{l}^{(m+1)})^{n}

holds on K_{lc}. By letting l tend to infinifty, we completes the proof of the lemma. Q.E.D.

Hence {(ω^{(m)})^{n}}_{m=1}^{∞} is monotone increasing and bounded from above uniformly on every compact subset of X_{ν} − D

ω_{E}^{n} := lim_{m→∞}(ω^{(m)})^{n}

exists.

We shall define a d-closed positive (1, 1) current ω_{E} on X by

ω_{E} = \sqrt{−1}∂ ¯∂log ω_{E}^{n}.

ω_{E} is well defined by the C^{0}-estimates in the last subsection. Then by the construction it is clear that [ω_{E}] = 2πc_{1}(K_{X}) and (K_{X}, (ω_{E}^{n})^{−1}) is an AZD. This completes the proof of Theorem 7.

References

1. T. Aubin, Équation du type Monge-Ampere sur les variété Kähleériennes compactes, C.R.Acad. Sci. paris Sér. A-B 283(1976), 11 9-121.

2. Caffarelli, L.-Kohn J.J.-Nirenberg L.-Spruck J., The Dirichlet problem for nonlinear secondorder elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic equations, Comm. Pure and Appl. Math. 38, 209-252 (1985).

3. Carlson, J.-Griffith, P.H., A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. 95 (1972), 557-584.

4. T. Fujita, Approximating Zariski decomposition of big line bundles ,Kodai Math. J. 17 (1994), 1-3.

5. T. Fujita, On Kähler fibre spaces over curves, J. Math. Soc. of Japan 30 (1978), 779-794.

6. Gilberg -Trudinger, N.S., Elliptic partial differential equations of 2nd order, 2nd edition, Springer-Verlag.

7. M. Gromov, Kähler hyperbolicity and L^{2}-Hodge theory. Jour. Diff. Geom. 33 (1991), 263-292.

8. Hamilton, R.S., Three manifolds with positive Ricci curvature, J. Diff. Geom. 17, 255-306 (1982).

9. Hörmander, L., An Introduction to Complex Analysis in Several Variables 3-rd ed., North-Holland (1990).

10. S. Iitaka, Deformation of compact complex surfaces I,II and III, in Global Analysis, papers in honor of K. Kodaira, Princeton Univ. Press (1969), 267-272, J. Math. Soc. Japan 22 (1970),247-261, ibid. 23 (1971), 692-705.

11. Y. Kawamata, Kodaira dimension of algebraice fibre spaces over curves, Invent. Math. 66 (1982), 57-71.

12. S. Kobayashi, Hyperbolic manifolds and meromorphic mappings, Dekker (1970).

13. J. Kollar, S. Mori, Classification of three dimensional flips, J. of A.M.S. 5 (1992), 533-702.

14. S. Kobayashi, T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), 7-16.

15. S. Kobayashi, T. Ochiai, Mappings into compact complex manifolds with negative first Chern class, Jour. Math. Soc. of Japan 23 (1971), 137-148.

16. K. Kodaira, Complex manifolds and deformation of complex structures, Springer(1986).

17. K. Kodaira - D.C. Spencer, A theorem of completeness fpr complex analytic fibre spaces, Acta Math. 100 (1958),281-294.

18. M. Levine, Pluricanonical divisors on Kähler manifolds, Inv. Math. 74 (1983), 293-303.

19. A. M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990), 549-596.

20. N. Nakayama, Invariance of plurigenera of algebraic varieties under minimal model conjecture, Topology 25(!986), 237-251.

21. T, Ohsawa- K. Takegoshi, L^{2}-extension of holomorphic functions, Math. Z. 195 (1987), 197-204.

22. G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, Jour. Diff. Geometry 32 (1990), 99-130.

23. Trudinger, N.S.: Fully nonlinear elliptic equation under natural structure conditions, Trans. Am. Math. Soc. 272, 751-769 (1983).

24. Tsuji. H.: Existence and Degeneration of Kähler-Einstein Metrics on Minimal Algebraic Varieties of General Type, Math. Ann. 281, 123-133 (1988).

25. H. Tsuji, Analytic Zariski decomposition, Part 1, Existence, preprint (1993).

26. H. Tsuji, On the universal coverings of projective manifolds of general type, to appear in Kodai Math. J. (1996).

27. M. Wright, The Kobayashi pseudometric on algebraic manifolds of general type and in defor-

mation of complex manifolds, Trans. A.M.S. 232 (1977), 357-370.

28. S.-T. Yau, Intrinsic measures of compact complex manifolds, Math. Ann. 212 (1975), 317-329.

29. S.-T. Yau, On the Ricci curvature of compact Kähler manifold and the complex Monge-Ampére equation I, Comm. Pure Appl. Math. 31 (1978), 339-411.

30. S.-T. Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100(1978),197-203.

Hajime Tsuji, Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ohokayama Megro 152, Tokyo, Japan

Current address: 1000 Centennial Drive, Berkeley, CA 94720, U.S.A.