A geometric approach to problems in birational geometry
See allHide authors and affiliations

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved September 15, 2008 (received for review July 18, 2008)
Abstract
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.
In this article, we initiate a program to study problems in birational geometry. This approach will be more geometric than other more algebraic approaches. Most of the arguments can, however, be phrased in a purely algebraic way. It is quite likely that some of them can be applied to deal with the geometry over different ground fields.
1. Introduction
Given a projective variety M, we shall study the geometric information provided by the pluricanonical space H^{0}(M,mK_{M}). Note that the minimal model program, as led by Mori, Kawamata, Kollár, and others, has achieved great success. Although the earlier workers had solved the problem for threefolds completely, the spectacular finite generation question was recently solved by several people, using different approaches: the analytic approach of Siu (1) and the algebraic approach of Birkar, Cascini, Hacon, and McKernan (2). In our approach, instead of using the full canonical ring, we shall focus our study on the pluricanonical space for a fixed m.
Ideally, we would like to determine the birational type of our algebraic variety based on the information on this space only. Any birational transformations of algebraic manifolds will induce a linear map between the corresponding pluricanonical spaces (for each fixed m). The plurigenera are, of course, invariant under the birational transformations. But more importantly, there are other finer invariants that are preserved by these transformations. The most important ones are the natural normlike functions (called norms in this introduction) induced by integrating over M the mth root of the product of a m−pluricanonical form and its conjugate. The norm defines an interesting geometry that was not explored extensively before.
In our work, we shall initiate a program to study this geometry. The first major questions we address are the following:

Torellitype theorem. Given 2 algebraic varieties M and M′, suppose there is a linear map that defines an isometry (with respect to the norm mentioned above) between the 2 normed vector spaces H^{0}(M,mK_{M}) and H^{0}(M′,mK_{M′}). We claim that with a few exceptional cases of M and M′, the linear isometry is induced by a birational map between M and M′. This can be considered as a Torellitype theorem in birational geometry.
We call this kind of theorem a Torellitype theorem because the classical Torelli theorem says that the periods of integrals determine an algebraic curve. This remarkable theorem was generalized to higherdimensional algebraic varieties. The most notable one was the work of PiatetskyShapiro and Shafarevich (3) for algebraic K3 surfaces, which was generalized to Kähler K3s by Burns and Rapoport (4), where they proved the injectivity of period maps. The surjectivity of period maps for K3 surfaces was done using Ricci flat metrics by Siu (5) and Todorov (6) following the work of Kulikov (7) and of Perrson and Pinkham (8). This phenomenon of subjectivity is known to be rather generic, and in many cases the period map can be proved to have degree one for hypersurfaces [see, e.g., Donagi (9)].

Existence. Characterize geometrically and algebraically those normed vector spaces that can be realized as the pluricanonical spaces of some algebraic varieties as above. Hopefully, there may be some effective way to construct the birational models of these varieties.

Computation. In the case of the classical Torelli theorems, the periods can be effectively computed by methods dating back to Picard, Leray, Dwork, and others. We hope to calculate these normed spaces effectively too. Some differential geometric methods will be brought in.

Relations with questions of GIT and other invariants. Making use of the pluricanonical series, we are able to form new invariant (pseudo)metrics on the algebraic manifolds. There should be some relationship between these metrics and other wellknown canonical metrics such as Kähler–Einstein metrics. We hope to build a link between our approach with other metrical approaches to algebraic geometry.
In this article, we shall prove that when mK_{M} has no base point and defines a birational map, the normed space is indeed powerful enough to determine the birational type of the algebraic varieties. We can achieve this when m is large enough (depending on the dimension of M only). Indeed, we prove a Torelli theorem that is described in 1 under rather general assumptions. We should say that in case the manifold is 1dimensional and m = 2, the problem was treated by Royden in his study of the biholomorphic transformations of Teichmüller space (10). We think that it is possible to generalize Royden's work to higherdimensional manifolds.
We shall study 2 by using a more differential geometric approach. We here outline what kind of metric we can obtain. At every point η_{0} ∈ H^{0}(M,mK_{M}) and η_{1}, η_{2} ∈ H^{0}(M,mK_{M}), viewed as 2 tangent vectors at η_{0} in H^{0}(M,mK_{M}), we define a hermitian metric
where
2. Pseudonorms on H^{0}(M,mK_{M}) and their asymptotic properties
2.1. The Pseudonorm 〈〈 〉〉_{m}.
Let M be a complex manifold of dimension n. To every η ∈ H^{0}(M,mK_{M}) we can associate a real nonnegative continuous (n,n)form on M, denoted as 〈η〉_{m}, as follows:
Let u = {(U, (w_{U}^{j} = u_{U}^{j} + iv_{U}^{j})_{j = 1}^{n})} be an open cover of M of coordinate charts. If η_{U} = η_{U}(dw_{U}^{1} ∧ … ∧ dw_{U}^{n})^{⊗m} with η_{U} ∈ 𝒪_{M}(U), we can define on U a real nonnegative continuous (n,n) form and can verify that {〈η_{U}〉_{m}}_{U∈u} does give a globally defined form, denoted as 〈η〉_{m}. It is routine to see that this definition does not depend on the choice of u.
If M is compact, we define and will abbreviate it as 〈〈η〉〉 if m is clear in the context. Therefore, for a compact complex manifold M we have defined a function and will call it the pseudonorm associated to mK_{M}.
From the fact that a + b^{α} ≤ a^{α} + b^{α} for any 0 < α < 1 and a, b ∈ C we can verify the triangle inequality 〈〈η_{1} + η_{2}〉〉 ≤ 〈〈η_{1}〉〉 + 〈〈η_{2}〉〉 for any η_{1},η_{2} ∈ H^{0}(M,mK_{M}). From the definition 〈〈η〉〉 = 0 if and only if η = 0 ∈ H^{0}(M,mK_{M}). However,
We define a metric space structure on H^{0}(M,mK_{M}) using 〈〈 〉〉 by H^{0}(M,mK_{M}) so metrized will be denoted as (H^{0}(M,mK_{M}), 〈〈 〉〉).
If φ : M′ → M is a birational map, then the induced isomorphism φ : (H^{0}(M,mK_{M}),〈〈 〉〉) → (H^{0}(M′,mK_{M}′),〈〈 〉〉) is an isometry.
2.2. A Local Asymptotic Expansion.
We will state the main local asymptotic result (C.Y.C., unpublished work) and then deduce from it the global one in the next section, namely the asymptotic property of 〈〈 〉〉.
We first settle the notation as follows: Assume that l(A,B) = l_{1} = … = l_{μ(A,B)} < l_{μ(A,B) + 1} ≤ … ≤ l_{n}. Notice that l(A,B) and μ(A,B) only depend on the multiindices A and B. If A and B are clear in our arguments we will denote l(A,B) and μ(A,B) by l and μ, respectively.
We abbreviate (x_{1},y_{1}, … , x_{n},y_{n}), (z_{1}, … , z_{n}), z_{1}^{a1} … z_{n}^{an}, z_{1}^{b1} … z_{n}^{bn}, and dx_{1}dy_{1} … dx_{n}dy_{n} as (X,Y), Z, Z^{A}, Z^{B}, and dX dY, respectively. Let
Theorem 2.2.1. where c(A,B,ϕ) is a real number depending on ϕ. In the last case we have c(A,B,ϕ) ≥ 0, and
Remark 2.2.2. More specific information when
In 2.4, we will apply this result to obtain the main global result. Let η_{0},η ∈ H^{0}(M,mK_{M}). We hope to describe the asymptotic behavior of 〈〈η_{0} + t_{η}〉〉 as t → 0.
2.3. The Characteristic Index and Indicatrix.
Before getting into the deduction of the global asymptotic expansion, we introduce several quantities measuring how singular a divisor is at a point in the ambient space.
Let M be a smooth variety, D a nonzero effective divisor on M. (In 2.4 D will be chosen to be {η_{0} = 0} for the η_{0} ∈ H^{0}(M,mK_{M}) we consider.) We first choose a log resolution π : M̃ → M for the pair (M,D) and write where E runs over all irreducible subvarieties of M̃ of codimension 1.
Definition 2.3.1. (1) For every x ∈ M the log canonical threshold of D at x, denoted as lct(D,x), is given by We also have the global log canonical threshold of D, It is clear that lct(D) = min_{x∈M} lct(D,x).
(2) For every x ∈ M the log canonical multiplicity of D at x, denoted as μ(D,x), is given by
(3) The characteristic index of D at x is the pair (lct(D,x), μ(D,x)). Consider the following total order: The global characteristic index of D, denoted as (lct(D),μ(D)), is given by
(4) We define the characteristic indicatrix of (M,D), denoted as C(D), to be the set of points achieving global characteristic index, i.e. Notice that in general C(D) is different from the minimal log canonical centers of D.
The total order defined here is adapted to the comparison of the asymptotic order of functions of the form
The log canonical thresholds is well defined, namely it is independent of the choice of log resolutions. In fact and the multiplier ideal sheaves J(M,cD) do not depend on the log resolution we choose.
This also gives the basic inequality (by taking a blowup Bl_{x}(M) → M followed by a log resolution).
Remark 2.3.1. In the rest of the article we do not need μ(D,x) and C(D) to be independent of the choice of log resolution. For each divisor D we can simply choose a fixed resolution to define μ(D,x) and C(D). However, they can indeed be defined in terms of some resolution free ideal sheaves, hence are both independent of the choice of resolution (C.Y.C., unpublished work). Instead of giving a formal proof of this independence here, we would like to point out that one can see this, at least analytically, by using Theorem 2.2.1 for the case χ = ϕ ≡ 1 and that pulling back a differential form by an analytic modification does not change its integral.
Remark 2.3.2. The characteristic index is a finer measurement of singularity than lct is. For example, lct alone can not tell between a reduced nonsmooth s.n.c. divisor and a smooth divisor. Higher characteristic indices correspond to worse singularities. In this sense, the characteristic indicatrix C(D) is the set of points at which the pair (M, D) is the most singular.
2.4. The Asymptotic Property of 〈〈 〉〉_{m}.
In this subsection we assume M to be compact. We first give the local setting. As in 2.1, let u = {(U, (w_{U}^{j})_{j = 1}^{n})} be a finite open cover of coordinate charts on M. We choose a log resolution π : M̃ → M for (M,D_{η0} = {_{η0} = 0}) and a finite refinement v = {(V, Z_{V} = X_{V} + iY_{V})} of π^{−1}u = {π^{−1}U} formed by charts in M̃, where Z_{V} and (X_{V},Y_{V}) abbreviate (z_{V}^{1}, … , z_{V}^{n}) and (x_{V}^{1},y_{V}^{1}…,x_{V}^{n},y_{V}^{n}) respectively. Let τ : v → u be such that π(V) ⊂ τ(V). Finally, we choose a partition of unity {χ_{V}(X_{V},Y_{V})} subordinate to v. v and {χ_{V}} can be so chosen that
(i) the image of Z_{V} : V → C^{n} is Δ_{0} = {(z_{1}, …, z_{n}) ∈ C^{n} z_{j} < 1 for all j};
(ii) if U = τ (V), then
for some nonvanishing j_{V} ∈ 𝒪 (
(iii) following the notation in (ii), we have
and
where ϕ_{V} and c_{V} ∈ 𝒪 (
(iv) for each V we have l_{V}^{1} = … = l_{V}^{μV} < l_{V}^{μV+1} ≤ … ≤ l_{V}^{n}, where l_{V}^{j} =
(v) χ_{V} (0,0) ≠ 0 for every V.
Remark 2.4.1. In (ii) and (iii) and in the following proof of Theorem 2.4.2 we will have to consider two different kinds of pullbacks via π : M̃ → M of elements in H^{0}(M, K_{M}), and it is important not to mix them up. The first one is π* : H^{0}(M, mK_{M}) → H^{0}(M̃,mK^{M̃}) which acts on K_{M} as the usual pullback of differential forms via the map π. The second one is π** : H^{0}(M,mK_{M}) → H^{0}(M̃,π*(mK_{M})), the usual pullback map from the sections of a vector bundle to those of its pullback bundle via a map.
In terms of the v and χ_{V} chosen above we can write Our main asymptotic result for 〈〈 〉〉_{m} is the following
Theorem 2.4.2. Given η_{0}, η ∈ H^{0}(M,mK_{M}), let (l,μ) = (lct(D_{η0}),μ(D_{η0})) and C(D_{η0}) be defined as in 2.3. We have where c(η_{0}, η) is a real number depending on η_{0} and η. In the last case we have c(η_{0}, η) ≥ 0, and
Proof: Following the notation at the beginning of 2.2, for each V ∈ v we obtain correspondingly a pair (l_{V},μ_{V}). It is clear that (l,μ) = sup_{V} (l_{V},μ_{V}) according to the total order we introduced in 2.3(3). For each V, applying Theorem 2.2.1 to the case
For the statement about c(η_{0}, η), notice that, by (2.1), only those V with (l_{V},μ_{V}) = (l,μ) will contribute to c(η_{0}, η). More precisely, By Theorem 2.2.1 we know that c(η_{0},η) ≥ 0 and We know that ι_{μ}(M̃_{D,μ}) (see (2.2)) is defined by z_{V}^{1} = … = z_{V}^{μ} = 0 in every such V. Regarding the conditions (ii) and (iii) above satisfied by the v we choose, the last statement is equivalent to saying that p**η vanishes on ι_{μ}(M̃_{D,μ}). This is the same as saying that η vanishes on πι_{μ}(M̃_{D,μ}) = C(D_{η0}).
3. Identifying the Images of Rational Maps φ_{mKM}
We still assume M to be compact. In this section we are going to use Theorem 2.4.2 to study the image of the rational map φ = φ_{mKM} associated to the linear system mK_{M}.
Let B = BsmK_{M}. First we recall the definition of φ. It is given by Notice that φ is defined only for x ∈ M  B. (Otherwise {η η(x) = 0} = H^{0}(M,mK_{M}) is not a hyperplane.) In general, for any hyperplane H in H^{0}(M,mK_{M}) we have and BsH  B = φ^{−1}(H). Therefore
Question: Given H in the image of φ, can we characterize H by a subset of the hyperplane in H^{0}(M,mK_{M}) it represents and metrical properties of 〈〈 〉〉?
Idea: If we can find η_{0} ∈ H^{0}(M,mK_{M}) such that 2lct(D_{η0}) +
Definition 3.1. We say that property (CS) (standing for “concentrating singularities”) holds for mK_{M} if for a generic H in the image of φ there exists η_{0} ∈ H^{0}(M,mK_{M}) such that 2lct(D_{η0}) +
The following is the main ingredient in using metrical properties of pseudonorms to identify images of rational maps of the form we consider above.
Lemma 3.1. Let M, M′ be compact complex manifolds. If (CS) holds for both mK_{M} and mK_{M}′ and is a linear isometry, then the isomorphism induced by ι, maps the closure of the image of φ_{mKM} isomorphically onto that of φ_{mKM′}.
Proof: By symmetry, it suffices to prove that I maps a generic point in the image of φ_{mKM} into that of φ_{mKM′}.
By (CS), for a generic H in the image of φ we select a section η_{0} ∈ H^{0}(M,mK_{M}) such that 2lct(D_{η0}) +
By the definition of I and the fact that ι is a linear isometry, By the first ⇔ in (3.1), showing that I(H) is in the image of φ′ = φ_{mKM′} is equivalent to showing that BsI(H)  B′ ≠ ϕ, where B′ = BsmK_{M′}.
By Theorem 2.4.2, C(D_{ιη0}) ⊆ BsI(H), hence it suffices to prove C(D_{ιη0}) ⊈ B′. Assume this to be false, i.e. C(D_{ιη0}) ⊆ B′. Because ι is an isometry, ιη_{0} has the same asymptotic behavior as that of η_{0}, hence 2lct(D_{ιη0}) +
Remark 3.1. The more detailed asymptotic expansions which are mentioned in Remark 2.2.2 actually allow us to remove the condition 2lct(D_{η0}) +
4. Birational Equivalence between Smooth Varieties of General Type
In this section M will be a smooth compact complex manifold such that the rational map φ_{mKM} maps M to its image birationally for sufficiently large m. We want to know for which r ∈ N (CS) holds for rK_{M}.
In case rK_{M} maps M birationally to its image, the condition (CS) admits an equivalent statement in terms of points in M instead of those in the image. It is clear that in this case (CS) can be restated in the following way:
(CS) For a generic point x in M there exists η_{0} ∈ H^{0}(M, rK_{M}) such that
for any y ≠ x and (2lct(D_{η0}) +
Definition 4.1. (i) For any x ∈ M we define (ii)
Remark 4.1. (ii) in particular implies that φ = φ_{rKM} maps M birationally to its image. Indeed for a generic x ∈ M
Lemma 4.1. If r ∈ S_{M} then (CS) holds for rK_{M}.
Proof: (ii) and Bertini's theorem imply that for a generic x ∈ M there is η_{0} ∈ H^{0}(M,rK_{M}) such that mult_{x}η_{0} ≥
Lemma 4.2. S_{M} is a semigroup, i.e. (r_{1} + r_{2}) ∈ S_{M} if r_{1}, r_{2} ∈ S_{M}.
Proof: Condition (i) obviously holds for (r_{1} + r_{2}) if it does for r_{1} and r_{2}. As for condition (ii), for x in some Zariski open subset U ⊆ M we have BsV(r_{1},x) = BsV(r_{2},x) = {x} since r_{1} and r_{2} ∈ S_{M}. We want to show that for x ∈ U, y ∉ BsV(r_{1} + r_{2},x) if y ≠ x. By Bertini's theorem we can find η_{j} ∈ V(r_{j},x) such that η_{j}(y) ≠ 0 for j = 1,2. Let η = η_{1} ⊗ η_{2} ∈ H^{0}(M, (r_{1} + r_{2})K_{M}). We have
by the fact that
Lemma 4.3. Suppose BsmK_{M} = ϕ and φ_{mKM} maps M onto its image in ℙH^{0}(M,mK_{M})* birationally. Then νm ∈ S_{M} for any integer ν ≥ 2n + 1.
Proof: Condition (i) in the definition of S_{M} obviously holds. Only (ii) needs to be verified.
Since φ_{mKM} maps M to its image birationally, we can find Zariski open subsets U_{0} and U of M and the image of φ_{mKM} respectively such that φ_{mKM} : U_{0} = φ_{mKM}^{1}(U)
We want to show that y ∉ BsV(νm,x) if y ≠ x (i.e. (ii)) for x ∈ U_{0}. By the choice of U_{0} it is clear that Bs{η ∈ H^{0}(M,mK_{M})η(x) = 0} = {x}. Therefore, for any y ≠ x there exists η ∈ H^{0}(M,mK_{M}) such that η(x) = 0 and η(y) ≠ 0. Taking η_{0} = η^{⊗ν} ∈ H^{0}(M,νmK_{M}), we have η ∈ V(νm,x) since when ν > 2n + 1. So y ∉ BsV(νm,x).
Lemma 4.4. Let M be a nonsingular complex projective variety of general type and of dimension n. Let d ∈ N be such that BsmdK_{M} = ϕ for m ≥ m_{0}. Then there exists r_{0} ∈ N depending only on m_{0} and n such that rd ∈ S_{M} if r ≥ r_{0}.
Proof: It is proved in refs. 11 and 12 that for each n ∈ N there exists m_{n} ∈ N such that if M is a smooth projective variety of general type and of dimension n then the rational map φ_{mKM} maps M to its image birationally for any m ≥ m_{n}.
Choose distinct prime numbers m, m′, ν and ν′ such that m, m′ ≥ max {
Our main theorem is the following
Theorem 4.1. Let M and M′ be smooth complex projective varieties of general type and of dimension n and d ∈ N such that BsmdK_{M} = BsmdK_{M′} = ϕ for m ≥ m_{0}. Let r_{0} ∈ N as given by Lemma 4.4.
If for some r ≥ r_{0} we have a linear isometry then there exists a unique birational map ψ : M → M′ and c ∈ C with c = 1 such that cι = ψ*, the isomorphism induced by ψ.
Proof: Lemma 4.1 and Lemma 4.4 together imply the (CS) holds for ρdK_{M} and ρdK_{M′} if ρ ≥ r_{0}. By Lemma 4.4 and Remark 4.2 φ_{rdKM} and φ_{rdKM′} map M and M′ birationally to their images respectively. Denote the isomorphism induced by ι as The assumption and Lemma 3.1 implies that I identifies the images of φ_{rdKM} and φ_{rdKM′}. Therefore we obtain a unique birational map ψ making the following diagram of rational maps commutative:
Let ψ*: H^{0}(M, rdK_{M}) → H^{0}(M′, rdK_{M′}) be the isomorphism induced by ψ. It is an isometry with respect to 〈〈 〉〉_{rd}. Since ψ* and ι both induce I : ℙH^{0} (M,rdK_{M})* → ℙH^{0} (M′,rdK_{M′})*, there is c ∈ C such that cι = ψ*. Both ι and ψ* are isometries with respect to those 〈〈 〉〉s, hence c = 1.
Using this theorem we can obtain several uniform results. For example, in the case n = 2, we can even have r_{0} depending only on n. The reason is that it is enough to prove the theorem for M and M′ both minimal models. By the classical results due to Bombieri and Kodaira BsmK_{M} = BsmK_{M′} = ϕ if m ≥ 5. The proof of Lemma 4.4 shows that S, the additive semigroup of N generated by {ab  a,b ∈ N, a ≥ 5, b ≥ 6}, is contained in S_{M}. It is not hard to see that m ∈ S for any m ≥ 75, and hence r_{0} can be chosen to be 75. Then we can take d = 1 and m_{0} = 5 in Theorem 4.1 and get the following
Theorem 4.2. Given a linear isometry for some m ≥ 75, there exists a unique pair of a birational map ψ : M′ → M and a complex number c of unit length such that ψ*, the isomorphism induced by ψ, is equal to cι.
For higher dimensions, in the same spirit we obtain the following
Theorem 4.3. There exists r_{0} ∈ N which depends on n, such that for any two smooth complex projective varieties M and M′ of general type and of dimension n which both admit smooth minimal models, if for some r ≥ r_{0} we have a linear isometry then there exists a unique birational map ψ : M → M′ and a unique complex number c of unit length such that ψ*, the isomorphism induced by ψ, is equal to cι.
Proof: As remarked in the paragraph before Theorem 4.2 we may assume that M and M′ are both minimal models, i.e. K_{M} and K_{M′} are both nef.
Kollár's effective base freeness theorem ([8], 1.1 Theorem) says that if a log pair (X,Δ) is proper and klt of dimension n, L a nef Cartier divisor on X, and a ∈ N such that aL  (K_{X} + Δ) is nef and big, then 2(n + 2)!(a + n)L is base point free. Applying this to the case X = M (resp. M′), Δ = 0, L = K_{M} (resp. K_{M′}) and a ≥ 2 we have that Bs2m(n + 2)!K_{M} = Bs2m(n + 2)!K_{M′} = ϕ if m ≥ n + 2.
Therefore we may take d = 2(n + 2)! and m_{0} = n + 2 in Lemma 4.4 and Theorem 4.1, and then the theorem follows.
Remark 4.2. It is shown in ref. 2 that every variety of general type admits a minimal model. However in the proof above the smoothness of the minimal models are required.
Here, in order to illustrate how the main idea goes, we only deal with the case when suitable base point free conditions hold. The presence of base loci is another technical issue. By a careful analysis and modification of the results in 2, a suitable use of the effective base point freeness, and the existence of minimal models for varieties of general type, we are still able to say something for the general case. The following theorems 4.4 and 4.5 are the precise results whose proofs can be found in C.Y.C. and S.T.Y., unpublished work.
We first recall some facts about the minimal models. It is known that every projective manifold X of general type admits a minimal model Y with K_{Y} QCartier (2). The index of Y is defined as j_{Y} = min{jjK_{Y} is Cartier}. It is also known that any two birational minimal models have the same index. Hence we can define the index of a projective manifold to be that of any of its minimal models. We have the following
Theorem 4.4 (C.Y.C. and S.T.Y., unpublished work) For every natural number j there exists r_{n,j} which depends only on n and j such that given any two ndimensional projective manifolds M and M′ of general type with indices j, and a linear isometry for some r ≥ r_{n,j}, there exists a unique birational map ψ : M′ → M and a unique complex number c of unit length such that the induced map ψ* is equal to cι.
The number r_{n,j} in this theorem depends not only on the dimension n but also on the index of minimal models. To get a uniform result in higher dimensional cases, we need to introduce some objects here. Let for any m,r ∈ N, where the map is the canonical one. V(M,m,r) inherits from (H^{0}(M,rmK_{M}), 〈〈 〉〉_{rm}) a pseudonorm, still denoted as 〈〈 〉〉_{rm}. It is clear that (V(M,m,r), 〈〈 〉〉_{rm}) is a birational invariant.
Recall also the definition of m_{n} in the proof of Lemma 4.4, which is a number such that ϕ_{mKM} maps M birationally to its image for every m ≥ m_{n}. With these notions, we can also prove the following result :
Theorem 4.5. Given a linear isometry for some r ≥ 2n + 1 and m ≥ m_{n}, there exists a unique birational map ψ : M′ → M and a unique complex number c of unit length such that the induced map ψ* is equal to cι.
Remark 4.3. Many of the results in this article have a more general version for L + mK_{M} where L is a hermitian line bundle (C.Y.C., unpublished work).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: cychi{at}math.harvard.edu

Author contributions: S.T.Y. and C.Y.C. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
 © 2008 by The National Academy of Sciences of the USA
References
 ↵
 Siu YT
 ↵
 Birkar C,
 Cascini P,
 Hacon C,
 McKernan J
 ↵
 PiatetskiiShapiro I,
 Shafarevic I
 ↵
 Burns D,
 Rapoport M
 ↵
 Siu YT
 ↵
 Todorov S
 ↵
 Kulikov VS
 ↵
 Persson U,
 Pinkham H
 ↵
 Donagi R
 ↵
 Royden HL
 ↵
 Hacon C,
 McKernan J
 ↵
 Takayama S

 Kollár J