THE GEOMETRIC MINIMAL MODELS OF ANALYTIC SPACES

SHIHOKO ISHII AND PIERRE MILMAN

Abstract. This paper shows that an analytic space X has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to X factors. This is called the geometric minimal model of X and characterized by the contraction property of rational curves. Some other properties such as functoriality, direct product property, quotient property of the geometric minimal model are also studied here. The relation of the geometric minimal model with Mori’s minimal model is shown.

1. Introduction

It is well known that a two dimensional singularity (X, x) has a good model called the minimal resolution f : Y → X, that is :

(1) all resolutions of (X, x) factor through f and

(2) Y itself is non-singular.

For the higher dimensional case, such a resolution no longer exists. Then, seeking a good model of a higher dimensional singularity in the Minimal Model Conjecture, people modify the conditions (1) and (2) into a condition (3) which is equivalent to (1) and (2) in the twodimensional case.

(3) Y has ℚ-factorial terminal singularities and K_{Y} is f-nef.

As is known, such a model exists for a three-dimensional singularity ([11]), but the existence for the higher dimensional case is still conjecture.

On the other hand, the minimal resolution f : Y → X of a twodimensional singularity is also characterized as follows:

(1) all resolutions of (X, x) factor through f and

(2′) f : Y → X is maximal among those satisfying (1).

This paper shows that such a model also uniquely exists for the higher dimensional case.

We also prove that the following model f : Y → X uniquely exists for any dimension:

(1′) all proper surjective morphisms X̃ → X from non-singular X̃ factor through f and (2′′) f : Y → X is maximal among those satisfying (1′). This model is called the geometric minimal model of X and is proved

to coincide with the model satisfying (1) and (2′) if dim X ≠ 2.

The geometric minimal model is characterized by the contraction of rational curves: let X̃ → X be a proper surjective morphism from non-singular X̃, then every rational curve on X̃ which is mapped to a point on X is mapped to a point on the geometric minimal model Y; on the contrary, the maximal model Y → X with the following property is the geometric minimal: there is a proper bimeromorphic morphism Z → Y such that every rational curve on Z which is mapped to a point on X is mapped to a point on Y. By this, we can see that the geometric minimal model of a toric variety is the identity and the geometric minimal model of a cone over an Abelian variety is the corresponding line bundle over the Abelian variety.

In §2, we prove the existence of a maximal model through which all resolutions factor. The proof is done under a more general setting, and the existence of the geometric minimal model is also proved.

In §3, we prove the characterization of the geometric minimal model by the contraction of rational curves. The coincidence of the geometric minimal model and the model with (1) and (2′) for dim ≠ 2 is also proved here.

In §4, we prove miscellaneous properties of the geometric minimal model: the functoriality, the liftability of group actions, the quotient property, the direct product property, and the relation with Mori’s minimal model for 3-dimensional case.

Throughout this paper, an analytic space X is an analytic ℂ-space in Hironaka’s sense [9], i.e.,

(i) for every point x of X, there exists an open neighbourhood such that the restriction of X onto it is ℂ-isomorphic to a local analytic ℂ-space,

(ii) the underlying topological space of X is countable at infinity, i.e., X is a union of countably many compact subsets, and

(iii) X is a Hausdorff space.

Analytic spaces are always assumed to be irreducible and reduced except for Lemma 2.4.

Professor Hajime Tsuji pointed out that the technique used to prove Theorem 3.4 in the preliminary version is the same as that used in [8], thus the proof can be shortened by quoting [8], for which the authors are grateful. The authors also would like to thank Professors Kenji Matsuki, Hironobu Maeda, Noboru Nakayama and David Mathieu for providing useful information.

2. The greatest common factor of a category of proper surjective morphisms

Definition 2.1. For an analytic space X, denote by A(X) the category whose objects are proper surjective morphisms X′ → X from analytic spaces X′ to X and whose morphisms are morphisms over X.

In this category, we introduce an order between the objects according to the existence of morphisms: i.e., (X′ → X) ≥ (X′′ → X), if there is a morphism X′ → X′′ over X. In this case we say that (X′ → X) is greater than (X′′ → X).

Definition 2.2. For a subcategory F of A(X), an object f : Y → X of A(X) is called a common factor of F, if every object of F is greater than f.

A maximal object among the common factors of F is called a greatest common factor of F. We sometimes say that Y is a common factor or a greatest common factor instead that Y → X is.

For an object X̃ → X and a common factor Y → X of F, there is a morphism X̃ → Y over X by definition. We call this morphism a factorizing morphism. If Y → X is bimeromorphic, then a factorizing morphism is unique.

Theorem 2.3. Assume that in a subcategory F of A(X) there is an object X′ → X which is bimeromorphic.

Then, for every common factor of F there is a greatest common factor which is greater. Moreover, a greatest common factor is unique.

If for every object X′ → X of F, X′ is normal, then the greatest common factor of F is normal.

To prove the theorem, we need the following lemma and definition.

Lemma 2.4. Let ⋯ E_{i+1} ⟶^{φ_{i+1}} E_{i} ⟶^{φ_{i}} E_{i−1} ⋯ → E_{0} be a sequence

of proper surjective morphisms of reduced (not necessarily irreducible) analytic spaces. For each i ≥ 0, let ψ_{i} : E → E_{i} be a proper surjective morphism of reduced analytic spaces. Assume the commutativity ψ_{i} = φ_{i+1} ∘ ψ_{i+1} for every i ≥ 0.

Then, for every compact subset K ⊂ E_{0}, there exists a number r ≥ 0 such that φ_{i} restricted onto the inverse image of K is of finite fibers for all i ≥ r.

Proof. Since we restrict onto a compact subset, we may assume that the number of irreducible components of E is finite. Then, we may assume that the number of irreducible components of E_{i}’s is constant, say s, because they are bounded by that of E. Let E_{i}^{j} (j = 1, .., s) be the irreducible components of E_{i} and φ_{i+1}(E_{i+1}^{j}) = E_{i}^{j}. Note that for every j there is an irreducible component E^{j} of E which is mapped onto E_{i}^{j}. Then, for each j, we may assume that dim E_{i}^{j}’s are constant for all i, because they are bounded by dim E. Now let us prove the lemma by induction on n = dim E.

Assume n = 1. Then dim E_{i}^{j} = 0 for all i or dim E_{i}^{j} = 1 for all i. Both cases are trivial.

Assume n ≥ 2 and the lemma holds true for the dimension less than n. Since it is sufficient to prove the finiteness of φ_{i} on each irreducible component E_{i}^{j}’, we may assume that E_{i}’s and E are all irreducible. Let dim E_{i} = k and Φ_{i} = φ_{1} ⋯ φ_{i−1} ∘ φ_{i}. Define D_{i} ⊂ E_{i} as {x ∈ E_{i} | dim_{x} Φ_{i}^{−1} ∘ Φ_{i}(x) > 0}, then it is a proper closed subset of E_{i}. Next define D ⊂ E as {y ∈ E | dim_{y} ψ_{0}^{−1} ∘ ψ_{0}(y) > n − k}, then it is also a proper closed subset in E and satisfies D_{i} ⊂ ψ_{i}(D). This inclusion is proved as follows:

For any point x ∈ D_{i}, take an irreducible component D_{i0} of Φ_{i}^{−1} ∘ Φ_{i}(x) containing x, then dim D_{i0} > 0. Let D′ ⊂ E be an irreducible closed subset dominating D_{i0} and y ∈ D′ a point corresponding to x. Then, according to a general theory, dim D′ ≥ dim D_{i0} + n − k. By D′ ⊂ ψ_{0}^{−1} ∘ψ_{0}(y), we obtain that dim_{y} ψ_{0}^{−1} ∘ψ_{0}(y) > n−k, which yields x ∈ ψ_{i}(D).

Now we obtain a sequence ⋯ ψ_{i}(D) → ψ_{i−1}(D) → ⋯ → ψ_{0}(D) of proper surjective morphisms of reduced schemes which are dominated by D of dimension < n. By induction hypothesis, ψ_{i}(D) → ψ_{i−1}(D) is of finite fibers for i >> 0. On the other hand φ_{i} : E_{i} → E_{i−1} is of finite fibers outside of D_{i}, therefore φ_{i} is of finite fibers on the whole of E_{i} for i >> 0.

Definition 2.5. Let K be a compact subset of X. We say that a common factor Y → X is maximal on K, if for every common factor Z → X greater than Y we have Z|_{K} = Y |_{K}, where |_{K} means the inverse image of K.

Proof of Theorem 2.3. First note that all common factors of F are bimeromorphic to X by the assumption of the theorem. Let Y → X be a common factor of F. We will prove that for a compact subset K, there exists a common factor Y′ → X greater than Y and maximal on K. If Y is not maximal on K, there is a common factor Y_{1} → X greater than Y = Y_{0} and not equal to Y_{0} → X on K. Next, if Y_{1} is not maximal on K, there is Y_{2} → X as above. Successively we obtain a sequence:

⋯ → Y_{i} ⟶^{φ_{i}} Y_{i−1} → ⋯ ⟶^{φ_{1}} Y_{0}

of common factors of F. Let X′ → X be an object of F and ψ_{i} : X′ → Y_{i} the factorizing morphism for every i. Since Y_{i}’s are bimeromorphic to X, the commutativity φ_{i} ∘ ψ_{i} = ψ_{i−1} follows for every i. Take the normalizations X′^{∗} and Y_{i}^{∗} of X′ and Y_{i}, respectively. Then by the lemma, there is an integer r = r(K) such that Y_{i+1}^{∗} → Y_{i}^{∗} ( every i > r) is a proper bimeromorphic morphism between normal spaces, and its restriction onto the inverse images of K is of finite fibers, which is therefore an isomorphism by the Zariski Main Theorem (for analytic version, see [6, 4.9]). Hence, on K, every common factor in

⋯ → Y_{i} ⟶^{φ_{i}} Y_{i−1} → ⋯ ⟶^{φ_{r+1}} Y_{r}

is dominated by the common normalization π_{i} : Y^{∗} → Y_{i}. Therefore, the coherent O_{Y_{r}}-module π_{r}_{∗}O_{Y^{∗}} has an ascending chain of submodules:

O_{Y_{r}} ⊂ ρ_{r+1}_{∗}O_{Y_{r+1}} ⊂ ... ⊂ ρ_{i}_{∗}O_{Y_{i}} ⊂ ...,

where ρ_{i} : Y_{i} → Y_{r} is the composite φ_{r} ∘ ⋯ ∘ φ_{i}. By the coherency of π_{r}_{∗}O_{Y^{∗}} this chain should terminate at a finite stage. Thus, we obtain a common factor Y′ → X greater than Y and maximal on K.

Since X is countable at infinity, X = ⋃_{i=1}^{∞} K_{i} with K_{i} compact and K_{i} ⊂ K_{i+1}. Take a sequence

⋯ → Y′_{i} ⟶^{φ′_{i}} Y′_{i−1} → ⋯ ⟶^{φ′_{1}} Y_{0} = Y

of common factors such that Y′_{i} → X is maximal on K_{i}. Then,

⋯ → Y′_{i} ⟶^{φ′_{i}} Y′_{i−1} → ⋯ ⟶^{φ′_{n+1}} Y′_{n}

are all isomorphic on K_{n}.

Now, as in [10, 4.1.7], we can define an analytic space Y_{∞} = lim_{⟵}_{r}Y′_{r},

which is again a common factor of F and greater than all Y′_{i}’s. This Y_{∞} is a greatest common factor of F. Indeed, if Z → X is a common factor greater than Y_{∞}, then for every K_{i}, Z|_{K_{i}} = Y′_{i}|_{K_{i}} = Y_{∞}|_{K_{i}} by the maximality of Y′_{i} on K_{i}. Hence Z = Y_{∞}.

Let Y → X and Y′ → X be greatest common factors. Then the unique component Z of Y ×_{X} Y′ dominating Y and Y′ is again a common factor of F and greater than Y and Y′. By the maximality of Y and Y′, Z coincides with Y and Y′, which yields the uniqueness of the greatest common factor.

If every object of F is normal, the normalization of a common factor is also a common factor. So the greatest common factor is normal. □

Remark 2.6. Instead of Lemma 2.4, we can prove the theorem differently: Consider a sequence of common factors as in the proof of Theorem 2.3. Each Y_{i} is dominated by a common object X̃ ⟶^{f_{i}} Y_{i}. Let J_{i,a} be the defining ideal of f_{i}^{−1}f_{i}(a) in X̃. On a compact subset K on Y_{0}, consider the Hilbert-Samuel function H_{i,a} of O_{X̃,a}/J_{i,a} for a ∈ f_{0}^{−1}(K). One can easily prove that there exists n(K) such that for any n ≥ n(K) and a ∈ f_{0}^{−1}(K), the equality H_{n,a} = H_{n(K),a} holds by means of Zariski upper-semicontinuity ([2, lemmas 7.1, 7.2], [5, Lemma 3.10 and Definition 3.11]) and the stabilization property of Hilbert-Samuel function ([3, Theorem 5.2.1]). This equality yields the equality of the ideals J_{n,a} = J_{n(K),a}, and therefore Y_{n} → Y_{n(K)} is finite on K. The rest of the proof follows in the same way as above.

Remark 2.7. Consider the case that X is an algebraic variety. Theorem 2.3 holds also for the category Alg(X) whose objects are proper surjective morphisms of algebraic varieties to X. The proof is easier. In Lemma 2.4, there is a number r so that φ_{i} (i ≥ r) is of finite fibers for the global E_{i}’s. Therefore one can prove Theorem 2.3 immediately by showing the ascending chain condition, without using Y_{∞}.

Example 2.8. There are many subcategories of A(X) which satisfy the condition of the theorem.

Res(X) : the category of resolutions of the singularities of X.

Sm(X) : the category of proper surjective morphisms X′ → X, where X′ is non-singular. Here, we note that the dimension of X′ may be bigger than the dimension of X.

C(X) (resp. Lt(X), T (X)) : the category of proper bimeromorphic morphisms X′ → X with canonical singularities (resp. log-terminal singularities, terminal singularities) on X′.

These are of course nonempty categories due to the results of [9], [1], [4], [5] and [13].

We can also think of the following categories for a toric variety X (cf., 2.7):

Res(X)_{tor} : the category of toric resolutions of the singularities of X.

Sm(X)_{tor} : the category of toric proper surjective morphisms X′ → X, where X′ is non-singular.

We may expect that the greatest common factors of Sm(X) and of Sm(X)_{tor} are different even for the same X.

Example 2.9. For a two-dimenstional analytic space X, the greatest common factor of Res(X) is the minimal resolution.

Definition 2.10. The greatest common factor of Sm(X) is called the geometric minimal model of X.

Definition 2.11. For an algebraic variety X, the greatest common factor of Sm(X) in the category Alg(X) is called the algebraic geometric minimal model which is not studied in the following sections (cf. 2.7).

3. The geometric minimal model and rational curves

In this section we study some properties of the geometric minimal model of an analytic space X.

First, we state a basic lemma, which is well known and used often in this paper.

Lemma 3.1. Let f : Z → X be a proper surjective morpphism and g : Z → Y a morphism. Assume X is normal and g is constant on f^{−1}(x) for every x ∈ X.

Then, there is a morphism φ : X → Y such that g = φ ∘ f.

Proof. Let Γ ⊂ X × Y be the reduced image of the proper morphism (f, g) : Z → X × Y and Φ : Γ → X the restriction of the projection X×Y → X onto Γ. Then, Φ^{−1}(x) consists of one point. Now we obtain that Φ : Γ → X is a bijective proper bimeromorphic morphism. Since X is normal, Φ is an isomorphism by Zariski’s Main Theorem.

The following is a basic property of the geometric minimal model.

Theorem 3.2. Let X be an analytic space of dimension ≥ 1, φ : Y → X a common factor of Sm(X) and f : X′ → X an arbitrary object of Sm(X).

Then every rational curve l ⊂ X′ which is mapped to a point on X is mapped to a point on Y by the factorizing morphism X′ → Y.

Proof. It is sufficient to construct a commutative diagram with f, γ and f ∘ α = γ ∘ β being in Sm(X)

X̃ ⟶^{β} X′′
α ↓  γ ↓
X′ ⟶^{f} X

such that there is a curve l̃ on X̃ with α(l̃) = l and β(l̃) = p a point. This is because γ and f factor through φ : Y → X, and the image of l on Y coincides with the image of l̃ and therefore with the image of p.

Now we construct X̃ and l̃. First we may assume that dim X′ = n ≥ 3. In fact, if dim X′ = 1, then there is no such rational curve. If dim X′ = 2, take the product X′×ℙ^{1}. Then, the composite X′×ℙ^{1} ⟶^{p_{1}} X′ → X is also an object of Sm(X), where p_{1} is the projection to the first factor. Take a section l′ of p_{1} over l and replace X′, l with X′ ×ℙ^{1}, l′. Now dim X′ ≥ 3; then there is a proper bimeromorphic morphism α_{1} : X_{1} → X′ with a non-singular rational curve l_{1} ⊂ X_{1} corresponding onto l such that the normal sheaf N_{l_{1}/X_{1}} = O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−1) ⊕ ⋯ ⊕ O_{ℙ^{1}}(−1) by [8]. The following construction is also found in [8]: take the blow-up α_{2} : X̃ → X_{1} with the center l_{1} and define α := α_{1} ∘ α_{2}.

Then the exceptional divisor E of α_{2} is ℙ^{1}×ℙ^{n−2} and its normal sheaf is p_{1}^{∗}O_{ℙ^{1}}(−1) ⊗ p_{2}^{∗}O_{ℙ^{n−2}}(−1), where p_{i} is the projection to the i-th factor and n is the dimension of X′. Then, there is a proper bimeromorphic morphism β : X̃ → X′′ whose restriction onto E is the projection E ⟶^{p_{2}} ℙ^{n−2}. By the normal bundle of E, we can see that X′′ is nonsingular. Since f ∘ α is constant on β^{−1}(x) for every x ∈ X′′, we obtain a morphism γ : X′′ → X with γ ∘ β = f ∘ α by Lemma 3.1. Now, take a section l̃= ℙ^{1} × {P } ⊂ E over l_{1}, where P is a point of ℙ^{n−2}. Then β(l̃) is a point, which shows that l̃ is a curve we wanted.

Theorem 3.3. Let X be an analytic space of dimension ≥ 3, φ′ : Y′ → X a common factor of Res(X) and f : X′ → X an arbitrary object of Res(X).

Then every rational curve l ⊂ X′ which is mapped to a point on X is mapped to a point on Y′ by the factorizing morphism X′ → Y′.

Proof. Since all objects in Res(X) are of dimension ≥ 3, we can construct a commutative diagram in Res(X) in the same way as in the proof of the previous theorem.

Theorem 3.4. Let Y → X be a bimeromorphic object of A(X) satisfying the following condition (R):

(R) There exists a proper bimeromorphic morphism Ỹ → Y such that every rational curve l ⊂ Ỹ which is mapped to a point in X is mapped to a point in Y.

Then Y → X is a common factor of Sm(X).

Proof. Let X′ → X be an arbitrary object of Sm(X). Then, X′ ×_{X} Ỹ → X′ is bimeromorphic on a component Z of X′ ×_{X} Ỹ. Then, by [9, Main Theorem II’] and [10, Corollary 1], for every point x ∈ X′ there are a relatively compact neighbourhood U_{r}, (r ≥ 1) of x, a nonsingular analytic space U_{1}, bimeromorphic morphisms f : U_{1} → U_{r} and U_{1} → Z|_{U_{r}} such that f is the composite f_{r−1} ∘f_{r−2} ∘⋯∘f_{1} of blow-ups f_{i} : U_{i} → U_{i+1} with non-singular centers and the following diagram is commutative:

U_{1} ⟶^{f_{1}} U_{2} → ... ⟶^{f_{r−1}} U_{r} ⊂ X′
↓
Z  ↓
↓
Ỹ → Y  →  X.

Therefore, f_{1}^{−1}(y) = ℙ^{s} (s ≥ 0) for every y ∈ U_{2}.

For a rational curve l ⊂ f_{1}^{−1}(y) ⊂ U_{1}, if the image of l on Ỹ is not a point then it is a rational curve. So l is mapped to a point in Y by the condition (R), which yields the existence of a morphism U_{2} → Y by Lemma 3.1. In the same way we induce the morphisms U_{i} → Y for i = 2, .., r. Now we complete the proof of the statement that X′ → X factors through Y → X.

Corollary 3.5. Let X be an analytic space of dimension ≠ 2.

Then, the greatest common factor of Res(X) coincides with the geormetric minimal model.

Proof. Since Res(X) ⊂ Sm(X), it follows that the greatest common factor Y′ → X of Res(X) is greater than the geometric minimal model Y → X. For dim X = 1, Y and Y′ are normal and bimeromorphic to X, therefore they coincide.

For dim X ≥ 3, it is sufficient to prove that Y′ → X is a common factor of Sm(X). This follows from Theorem 3.3 and Theorem 3.4.

Theorem 3.6. The geometric minimal model of an analytic space X is the maximal bimeromorphic object among the objects satisfying (R) in A(X).

Proof. This is clear by Theorem 3.4, because the geometric minimal model also satisfies (R) by Theorem 3.2.

Corollary 3.7. If Y → X is the geometric minimal model, then for every non-trivial bimeromorphic morphism Z → Y, there is a rational curve on Z which is mapped to a point on Y.

Proof. If Y → X is the geometric minimal model, then Z is not a common factor of Sm(X) by the maximality of Y. Therefore, by Theorem 3.4, for a resolution Z̃ → Z of singularities of Z there is a rational curve l ⊂ Z̃ which is mapped to a point on X and to a curve l′ on Z. This curve l′ is a required rational curve on Z.

Now, by the above theorem, the following is obvious.

Corollary 3.8. If X is an analytic space of dimension two, then the geometric minimal model of X is a surface obtained by contracting all rational exceptional curves of the minimal resolution.

Example 3.9. Let X be an affine cone over a non-singular variety A, which means that there is a line bundle Y over A with the zero section of negative normal bundle and the zero section is contracted to a point of X in Y. We will denote the zero section of Y also by A.

(1) If A is an Abelian variety or the direct product of non-singular curves of positive genera. Then Y is a resolution of the singularities on X and also the geometric minimal model of X, since A has no rational curve (see Theorem 3.4).

(2) If A is a K3-surface, then there is a rational curve with the positive self-intersection number. Then, the whole of A should be contracted to a point in the geometric minimal model; therefore the geometric minimal model of X is the identity.

(3) Let A be a ruled surface with q ≥ 1 and p : A → C the ruling, then there is a contraction Y → Y′ of the zero section A to a curve C in Y by [7, Theorem 1]. Then Y′ is the geometric minimal model of X.

Example 3.10. Let X ⊂ ℂ^{4} be a hypersurface defined by a homogeneous polynomial f such that the hyperplane in ℙ^{3} defined by f is non-singular and hyperbolic. Then, the blow-up φ : Y → X of X at the origin is the geometric minimal model of X, because there is no rational curve in a fiber of φ and Y is non-singular.

Example 3.11. Let X be a toric variety, then the geometric minimal model is the identity. Indeed, for every toric resolution f : X′ → X, each fiber f^{−1}(x) of f is a connected union of rational varieties, so it should be mapped to a point on the geometric minimal model.

4. Miscellaneous properties on the geometric minimal models

Proposition 4.1. (Functoriality) Let X and X′ be analytic spaces and φ : Y_{X} → X and φ′ : Y_{X′} → X′ the geometric minimal models of X and X′, respectively.

Then a proper surjective morphism f : X → X′ is uniquely lifted to f̃: Y_{X} → Y_{X′} such that φ′ ∘ f̃ = f ∘ φ.

Proof. The composition with f gives an injection Sm(X) → Sm(X′). Therefore the greatest common factor of the image of Sm(X) is greater than that of Sm(X′).

Example 4.2. Let π : X → S be a proper surjective morphism to a surface S. Let S′ be a surface obtained by contracting all rational exceptional curves of the minimal resolution of S. If the geometric minimal model of X is the identity (for example, X is toric), then π factors through S′.

Corollary 4.3. Let X be an analytic space and φ : Y → X the geometric minimal model of X.

Then every automorphism σ : X → X is lifted to σ′ : Y → Y such that φ ∘ σ′ = σ ∘ φ.

Proof. By Proposition 4.1, σ is lifted to σ′ : Y → Y and σ^{−1} is lifted to σ′′ : Y → Y. By the uniqueness of the lifting, σ′ and σ′′ are the inverse morphism of each other.

By this corollary, we obtain that every action of a finite group on X is lifted to the geometric minimal model. What follows is a discussion of the quotient by a finite group.

Proposition 4.4. Let X be an analytic space and G a finite group acting on X. Let φ : Y_{X} → X and φ′ : Y_{X/G} → X/G be the geometric minimal models of X and X/G, respectively.

Then there is a morphism Y_{X}/G → Y_{X/G} over X/G.

Proof. By the functoriality of the gemometric minimal models (cf. Proposition 4.1), there is a morphism Y_{X} → Y_{X/G}. By the uniqueness of this morphism, it is a G-equivariant morphism with the trivial action on Y_{X/G}. Hence the morphism factors through Y_{X}/G.

The following is obvious from the above proposition.

Corollary 4.5. Let X be an analytic space and G a finite group acting on X. If the geometric minimal model of X is the identity, then the geometric minimal model of X/G is also the identity.

Example 4.6. Let X = ℂ^{n}/G by a finite group G, then the geometric minimal model of X is the identity.

Example 4.7. The equality Y_{X}/G ≃ Y_{X/G} does not hold in general. Indeed, take a line bundle Y′ = V(O_{ℙ^{1}}(m)) over ℙ^{1}, where m is a positive integer. Let π : C → ℙ^{1} be a double cover from an elliptic curve C and G = ℤ/(2) the Galois group. Then G acts on Y = V(π^{∗}O_{ℙ^{1}}(m)) and also on X, which is a variety obtained by contracting the zero section of Y. The quotient X/G has a resolution Y′, therefore it is the quotient of ℂ^{2} by a finite group of order m. Hence the geometric minimal model of X/G is the identity, but the geometric minimal model of X is Y, and Y/G ≃ Y′.

In order to prove the direct product property of the geometric minimal model, we state the following lemma.

Lemma 4.8. Let X and X′ be analytic spaces, F a subcategory of A(X) satisfying the condition of Theorem 2.3. Let Y_{X} be the greatest common factor of F and f : Z → X′ an object of A(X′). Let F × Z be the category of the morphisms X̃ × Z → X × X′ with X̃ ∈ F.

Then Y_{X} × Z is the greatest common factor of F × Z.

Proof. It is clear that Y_{X} ×Z is a common factor of F ×Z. By Theorem 2.3, there is the greatest common factor Y′ of F × Z. So for every object X̃ → X of F, there is a factorizing morphism φ : X̃ × Z → Y′ of X̃ × Z → X × X′. On the other hand, there is a morphism ψ : Y′ → Y_{X} × Z by the maximality of Y′. Therefore for every point z ∈ Z, X̃ × {z} → X × {f(z)} is factored as follows:

X̃ × {z} ⟶^{φ} ψ^{−1}(Y_{X} × {z}) ⟶^{ψ} Y_{X} × {z} → X × {f(z)}.

Then, by the maximality of Y_{X}, ψ^{−1}(Y_{X} × {z}) ⟶^{ψ} Y_{X} × {z} is an isomorphism. Now we obtain that ψ : Y′ → Y_{X} × Z is bijective and ψ^{−1}((y, z)) ≃ {(y, z)} for every point (y, z) ∈ Y_{X} ×Z, which yields that ψ is an isomorphism.

Proposition 4.9. (Direct product) Let X and X′ be analytic spaces, Y_{X}, Y_{X′} and Y_{X×X′} the geometric minimal models of X, X′ and X×X′, respectively.

Then Y_{X} × Y_{X′} ≃ Y_{X×X′} over X × X′.

Proof. First for any objects X̃ → X and X̃′ → X′ of Sm(X) and Sm(X′), respectively, X̃ × X̃′ → X × X′ factors through Y_{X×X′}. For every X̃′ in Sm(X′), Y_{X} ×X̃′ is the greatest common factor of Sm(X)× X̃′ by Lemma 4.8. So there is a morphism Y_{X} × X̃′ → Y_{X×X′}. Then by Lemma 4.8 again, Y_{X} × Y_{X′} is the maximal common factor of Y_{X} × Sm(X′). So there is a morphism Y_{X} × Y_{X′} → Y_{X×X′}.

In order to prove the existence of the inverse morphism Y_{X×X′} → Y_{X} × Y_{X′}, it is sufficient to prove that Y_{X} × Y_{X′} is a common factor of Sm(X ×X′). Let X̃ and X̃′ be resolutions of Y_{X} and Y_{X′}, respectively, and l a rational curve on X̃ ×X̃′ which is mapped to a point on X ×X′. By Theorem 3.4, it is sufficient to prove that l is mapped to a point on Y_{X} × Y_{X′}. If the image of l by the projection X̃ × X̃′ ⟶^{p_{1}} X̃ is not a point, then it is a rational curve, and therefore is mapped to a point on Y_{X}. In the same way, l is mapped to a point on Y_{X′} through the projection X̃ × X̃′ ⟶^{p_{2}} X̃′. Hence l is mapped to a point on Y_{X} × Y_{X′}.

Next we consider an action of a complex Lie group.

Proposition 4.10. Let G be a complex Lie group acting on an analytic space X.

Then the action is lifted to the geometric minimal model of X.

Proof. Let σ : G × X → X be the morphism defining the action, p_{1} : G × X → G the projection to the first factor and Y the geometric minimal model of X. Since the geometric minimal model of G × X is G × Y by Proposition 4.9, the automorphism (p_{1}, σ) : G × X → G × X is lifted to Φ : G×Y ˜→G×Y. The composite p_{2}∘Φ : G×Y → Y defines the lifted action of G on Y, where p_{2} : G × Y → Y is the projection to the second factor.

4.11. It is natural to ask if the geometric minimal model is a local property, i.e., if φ : Y → X is the geometric minimal model of X and U ⊂ X is open, then φ^{−1}(U) → U is the geometric minimal model of U. The following example shows that the geometric minimal model is not local.

Example 4.12. Here we construct U ⊂ X such that the geometric minimal model of X is the identity but that of U is not.

Let S be a compact rational surface with only one simple elliptic singularity p on it. Let V = S × ℙ^{1} and C ∌ p be a non-singular very ample divisor on S_{0} = S × {0}. Take the blow-up X′ → V with the center C, then the proper transform S′_{0} of S_{0} on X′ has the negative normal bundle on X′. Hence, there is a morphism f : X′ → X which contracts S′_{0} to a point Q of X. Since X′ has singularities (simple elliptic singularities)×ℙ^{1}, there is a resolution g : X̃ → X′ whose exceptional divisor E is isomorphic to (elliptic curve)×ℙ^{1}. Then, the composite f ∘g : X̃ → X factors through the geometric minimal model Y → X of X. The proper transform S′′_{0} of S′_{0} on X̃ is contracted to a point on Y, because S′′_{0} is rational. As S′′_{0} ∩ E has the trivial normal bundle in E, the other elliptic fibers are all contracted to points on Y, therefore Y must be isomorphic to X.

On the other hand, for U = X\{Q}, the restriction (f ∘g)^{−1}(U) → U is a resolution of singularities on U and has no rational curves in each fiber of f ∘ g, therefore it is the geometric minimal model.

Proposition 4.13. Let X be a three-dimensional analytic space with at worst canonical singularities.

Then the geometric minimal model of X is the identity.

Proof. By [12], there is a resolution f : X̃ → X of the singularities of X whose exceptional divisors are all ruled surfaces. Then, the geometric minimal model φ : Y → X of X is either an isomorphism or a contraction of the union C of finite curves. Assume that φ is not an isomorphism and let g : X̃ → Y be the factorizing morphism. Then, all components of C are rational, otherwise R^{1}φ_{∗}O_{Y} ≠ 0, which contradicts the edge sequence of the Leray spectral sequence:

E_{2}^{p,q} = R^{p}φ_{∗}R^{q}g_{∗}OX̃ ⟹ E^{p+q} = R^{p+q}f_{∗}OX̃.

If the curve C is not contained in the singular locus, then we can construct a resolution X̃′ → X with a rational curve l on X̃′ mapped to a point on X and mapped to a curve on Y, a contradiction. Thus, C is contained in the singular locus of Y. Note that Y is again of index one and has rational singularities. So general points of C are cDV singularities in Y. For an irreducible component C_{i} of C, take the blow-up h : Y′ → Y with the center C_{i}.

Then, in case Y has compound A_{1}, D_{n} (n ≥ 4), E_{6}, E_{7}, E_{8} singularities on general points of C_{i}, h has an irreducible exceptional divisor E and h^{−1}(p) is an irreducible rational curve for a general p ∈ C_{i}. Then, E is rational and is not contained in the singular locus of Y′. Therefore, by Theorem 3.2, E must be mapped to a point on Y, a contradiction.

In case Y has compound A_{n} (n ≥ 2) singularities at general points of C_{i}, then h^{−1}(p) is the union of two ℙ^{1} with one intersection point. This implies that an exceptional divisor E of h has a generic section on C_{i}, therefore again E is rational and is not contained in the singular locus of Y′, a contradiction in the same way as above.

Corollary 4.14. Let X be a sufficiently small neighbourhood of a logterminal singularity of dimension three.

Then the geometric minimal model of X is the identity.

Proof. We may assume that X is a cyclic quotient of an analytic space with at worst canonical singularities. Then, by Corollary 4.5 and Proposition 4.13 we obtain the statement.

Corollary 4.15. Let X be an analytic space of dimension three, Y′ → X the relative canonical model in Mori’s sense and Y → X the geometric minimal model.

Then there is a morphism φ : Y′ → Y over X. Therefore, there is a morphism Y′′ → Y over X, where Y′′ is a relative minimal model in Mori’s sense.

Proof. By 4.1, the geometric minimal model of Y′ is greater than Y over X. Here, by Proposition 4.13, the geometric minimal model of Y′ is the identity.

References

[1] J. M. Aroca, H. Hironaka & J. Vicente Desingularization Theorems, Mem. Math. del Inst. “Jorge Juan” 30, (1977) Madrid.

[2] E. Bierstone & P. Milman, Relations among analytic functions I, II, Ann, Inst. Fourier (Grenoble) 37 (1), (1987) 187-239, 37 (2), (1987) 49-77.

[3] E. Bierstone & P. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989) 801-836.

[4] E. Bierstone & P. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry, Progress in Math. 94, Birkhauser (1991) 11-30.

[5] E. Bierstone & P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128, (1997), 207-302

[6] G. Fischer, Complex Analytic Geometry, Lecture Notes in Math., 538, Springer-Verlag (1976).

[7] A. Fujiki, On the blowing down of analytic spaces, Publ. RIMS, Kyoto Univ. 10, (1975) 473-507.

[8] A. Fujiki, On the minimal models of complex manifolds, Math. Ann. 253, (1980) 111-128.

[9] H. Hironaka, Resolution of singularities of an algebraic variety over a field of chracteristic zero: I, II, Annals of Math.,79, (1964) 109-326.

[10] H. Hironaka, Flattening Theorem in complex-analytic geometry, Am. J. Math., 97, (1975) 503-547.

[11] S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1, (1988) 117-253.

[12] M. Reid, Canonical 3-folds, Proc. Algebraic Geometry Anger, Sijthoff and Nordhoff (1979) 273-310.

[13] O. Villamayor, Patching local uniformizations, Ann. Sci. l’École Norm. Sup. (4) 25, (1992) 629-677.

Shihoko Ishii: Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo, Japan

Pierre Milman: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada