j. differential geometry 61 (2002) 147-171

D-EQUIVALENCE AND K-EQUIVALENCE

YUJIRO KAWAMATA

Abstract

Let X and Y be smooth projective varieties over ℂ. They are called Dequivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.

1. Introduction

Let X be a smooth projective variety. We denote by D(X) = D^{b}(Coh(X)) the derived category of bounded complexes of coherent sheaves on X (in §6, we shall consider a generalization where X has singularities). It is known that D(X) has a structure of a triangulated category.

Definition 1.1. Let X and Y be smooth projective varieties. They are called D-equivalent if their derived categories D(X) and D(Y) of bounded complexes of coherent sheaves are equivalent as triangulated categories, i.e., there exists an equivalence of categories Φ : D(X) → D(Y) which commutes with the translations and sends any distinguished triangle to a distinguished triangle. They are called K-equivalent if they are birationally equivalent and if there exists a smooth projective variety Z with birational morphisms f : Z → X and g : Z → Y such that the pull-backs of the canonical divisors are linearly equivalent: f^{∗}K_{X} ∼ g^{∗}K_{Y}.

Received 07/03/2002.

We shall consider the following conjecture which predicts that the D and K-equivalences coincide for birationally equivalent varieties.

Conjecture 1.2. Let X and Y be birationally equivalent smooth projective varieties. Then the following are equivalent.

(1) There exists an equivalence of triangulated categories D(X) ≅ D(Y).

(2) There exists a smooth projective variety Z and birational morphisms f : Z → X and g : Z → Y such that f^{∗}K_{X} ∼ g^{∗}K_{Y}.

The category of coherent sheaves Coh(X) reflects the biregular geometry of X, but we expect that the derived category D(X) captures more essential properties such as its birational geometry.

A derived category is a purely algebraic object. But one can sometimes recover the geometry from it:

Theorem 1.3 ([2]). Let X be a smooth projective variety. Assume that K_{X} or −K_{X} is ample.

(1) Let Y be another smooth projective variety. Assume that there exists an equivalence of categories Φ : D(X) → D(Y) which commutes with the translations. Then there is an isomorphism ϕ : X → Y.

(2) The group of isomorphism classes of exact autoequivalences of D(X) is isomorphic to the semi-direct product of Aut(X) and Pic(X) ⊕ ℤ.

We shall prove a generalization of Bondal-Orlov’s theorem in this paper:

Theorem 1.4 (= Theorem 2.3). Let X and Y be smooth projective varieties. Assume that the bounded derived categories of coherent sheaves on them are equivalent as triangulated categories: D(X) ≅ D(Y). Then the following hold:

(0) dim X = dim Y. Let n be the common dimension.

(1) If K_{X} (resp. −K_{X}) is nef, then K_{Y} (resp. −K_{Y}) is also nef, and an equality on the numerical Kodaira dimension ν(X) = ν(K_{Y}) (resp. ν(X, −K_{X}) = ν(Y, −K_{Y})) holds.

D-equivalence and K-equivalence 149

(2) If κ(X) = n, i.e., X is of general type, or if κ(X, −K_{X}) = n, then X and Y are birationally equivalent. Moreover, there exist birational morphisms f : Z → X and g : Z → Y from a smooth projective variety Z such that f^{∗}K_{X} ∼ g^{∗}K_{Y}.

We also consider the following conjecture:

Conjecture 1.5. For a given smooth projective variety X, there exist only finitely many smooth projective varieties Y up to isomorphisms such that D(Y) is equivalent to D(X) as a triangulated category.

We shall give an affirmative answer for surfaces in §3 by extending a result of Bridgeland and Maciocia [7]:

Theorem 1.6 (= Theorems 3.1 and 3.2). Let X be a smooth projective surface. Then there exist at most finitely many smooth projective surfaces Y up to isomorphism such that the derived categories D(X) and D(Y) are equivalent as triangulated categories. Moreover, if X contains a (−1)-curve but is not isomorphic to a relatively minimal elliptic rational surface, then any such Y is isomorphic to X.

The above conjecture can be regarded as a generalization of the conjecture which predicts that there exist only finitely many minimal models up to isomorphisms in a fixed birational equivalence class ([11]). Note that we do not assume the minimality of X in Conjecture 1.5.

We consider the reverse direction from K-equivalence to D-equi-valence in the latter half of the paper. We collect some facts from minimal model theory in §4, and we calculate some examples in arbitrary dimension in §5. In the case of dimension 3, we have a complete answer even for the case of singular varieties:

Theorem 1.7 (= Theorems 4.6 and 6.5). Let X and Y be normal projective varieties of dimension 3 having only ℚ-factorial terminal singularities, and let X and Y be their canonical covering stacks. Assume that X and Y are K-equivalent. Then the bounded derived categories of coherent orbifold sheaves D(X) and D(Y) are equivalent as triangulated categories.

Acknowledgement. The author would like to thank Tom Bridgeland, Jiun-Cheng Chen, Akira Ishii, Keiji Oguiso, Burt Totaro and Jan Wierzba for useful discussions or comments and the anonymous referee for suggestions.

2. From D-equivalence to K-equivalence

We need the concept of Fourier-Mukai transformation:

Definition 2.1. Let X and Y be smooth projective varieties, and let p_{1} : X × Y → X and p_{2} : X × Y → Y be projections. For an object e ∈ D(X × Y), we define an integral functor Φ_{X→Y}^{e} : D(X) → D(Y) by

Φ_{X→Y}^{e} (a) = p_{2∗}(p_{1}^{∗}(a) ⊗ e)

for a ∈ D(X), where p_{1}^{∗} and ⊗ are the right derived functors and p_{2∗} is the left derived functor. An integral functor is said to be a Fourier-Mukai transformation if it is an equivalence.

The following theorem by Orlov is fundamental for the proof of Theorem 2.3.

Theorem 2.2 ([19]). Let Φ : D(X) → D(Y) be a functor of bounded derived categories of coherent sheaves which commutes with the translations and sends any distinguished triangle to a distinguished triangle. Assume that Φ is fully faithful and has a right adjoint. Then there exists an object e ∈ D(X × Y) such that Φ is isomorphic to the integral functor Φ_{X→Y}^{e}. Moreover, e is uniquely determined up to isomorphism.

The following theorem guarantees that the D-equivalence implies the K-equivalence at least for general type varieties.

Theorem 2.3. Let X and Y be smooth projective varieties. Assume that the bounded derived categories of coherent sheaves on them are equivalent as triangulated categories: D(X) ≅ D(Y). Then the following hold:

(0) dim X = dim Y. Let n be the common dimension.

(1) If K_{X} (resp. −K_{X}) is nef, then K_{Y} (resp. −K_{Y}) is also nef, and an equality on the numerical Kodaira dimension ν(X) = ν(Y) (resp. ν(X, −K_{X}) = ν(Y, −K_{Y})) holds.

(2) If κ(X) = n, i.e., X is of general type, or if κ(X, −K_{X}) = n, then X and Y are birationally equivalent. Moreover, there exist birational morphisms f : Z → X and g : Z → Y from a smooth projective variety Z such that f^{∗}K_{X} ∼ g^{∗}K_{Y}.

Proof. By Theorem 2.2, there exists an object e ∈ D(X × Y) such that Φ_{X→Y}^{e} : D(X) → D(Y) is an equivalence. Let

e^{∨} = RHom_{O_{X×Y}} (e, O_{X×Y}),

the the derived dual object. By the Grothendieck duality, the right and left adjoint functors of Φ = Φ_{X→Y}^{e} are given by Φ_{Y →X}^{e^{∨}⊗p_{1}^{∗}ω_{X}[dim X]} and Φ_{Y →X}^{e^{∨}⊗p_{2}^{∗}ω_{Y} [dim Y]}.

Since Φ is an equivalence, the right and left adjoint functors of Φ = Φ_{X→Y}^{e} are isomorphic. By Theorem 2.2 again, we have an isomorphism of objects

e^{∨} ⊗ p_{1}^{∗}ω_{X}[dim X] ≅ e^{∨} ⊗ p_{2}^{∗}ω_{Y} [dim Y].

It follows immediately that dim X = dim Y.

Let H^{i}(e^{∨}) be the cohomology sheaves, Γ the union of the supports of the H^{i}(e^{∨}) for all i, Γ = ⋃_{j} Z_{j} the decomposition to irreducible components, and let ν_{j} : Z̃_{j} → Z_{j} be the normalizations. We take a Z_{j} and assume that it is an irreducible component of the support of H^{i}(e^{∨}). By taking the determinant of both sides of the isomorphism

ν_{j}^{∗}(H^{i}(e^{∨}) ⊗ p_{1}^{∗}ω_{X}) ≅ ν_{j}^{∗}(H^{i}(e^{∨}) ⊗ p_{2}^{∗}ω_{Y})

we obtain

ν_{j}^{∗}p_{1}^{∗}ω_{X}^{⊗m_{j}} ≅ ν_{j}^{∗}p_{2}^{∗}ω_{Y}^{⊗m_{j}}

where m_{j} is the rank of ν_{j}^{∗}H^{i}(e^{∨}).

(1) Since Φ_{X→Y}^{e} is an equivalence, the projections p_{1}|_{Γ} : Γ → X and p_{2}|_{Γ} : Γ → Y are surjective. Let Z_{1} be an irreducible component of Γ which dominates Y. If K_{X} is nef, then m_{1}ν_{1}^{∗}p_{1}^{∗}K_{X} ∼ m_{1}ν_{1}^{∗}p_{2}^{∗}K_{Y} is also nef, hence so is K_{Y}. We have also ν(X) ≥ ν(Z̃_{1}, ν_{1}^{∗}p_{2}^{∗}K_{Y}) = ν(Y), thus ν(X) = ν(Y). The case where −K_{X} is nef is proved similarly.

(2) If κ(X) = n, then there exist an ample ℚ-divisor A and an effective ℚ-divisor B on X such that K_{X} ∼_{ℚ} A+B by Kodaira’s lemma. Let Z_{1} be an irreducible component of Γ which dominates X. Then the projection p_{2}|_{Z_{1}} : Z_{1} → Y is quasi-finite on Z_{1}\p_{1}^{−1}(Supp(B)). Indeed, if there exists a curve C which is contained in Z_{1}∩p_{2}^{−1}(y) for a point y ∈ Y but not entirely in p_{1}^{−1}(Supp(B)), then we have (p_{2}^{∗}K_{Y} · C) = 0 while (p_{1}^{∗}K_{X} · C) ≥ (p_{1}^{∗}A · C) > 0, a contradiction. Since dim X = dim Y = n, it follows that dim Z_{1} = n and Z_{1} also dominates Y.

We claim that the set Γ ∩ p_{1}^{−1}(x) consisits of 1 point for a general point x ∈ X. Indeed, the previous argument showed already that Γ ∩ p_{1}^{−1}(x) is a finite set. If it is not connected, then the natural map Hom_{D(X)}(O_{x}, O_{x}) → Hom_{D(Y)}(Φ(O_{x}), Φ(O_{x})) is not surjective, a contradiction. Therefore, Z_{1} is a graph of a birational map. If we take Z to be any resolution of Z_{1}, then the conclusion holds.

The case where κ(X, −K_{X}) = n is proved similarly. q.e.d.

Remark 2.4.

(0) The differential geometric picture of the above proof is that the kernel object e of the Fourier-Mukai transformation cannot spread itself if the Ricci curvature is non-vanishing.

(1) In the case where K_{X} or −K_{X} is ample, we can also reprove Theorem 1.3 (2) by a similar argument as above.

Indeed, if we take B = 0, then Z_{1} becomes a graph of an isomorphism, say h. Now e can be considered as a complex of sheaves on X so that we have Φ(O_{x}) ≅ h(e ⊗_{O_{X}} O_{x}) for any x ∈ X, where the tensor product is taken in D(X). Since

Hom_{D(X)}^{p}(Φ(O_{x}), Φ(O_{x})) = 0

for any p < 0, it follows that there exists an integer i_{0} such that e[i_{0}] is a sheaf. Since

Hom_{D(X)}(Φ(O_{x}), Φ(O_{x})) = ℂ

e[i_{0}] is invertible.

We note that we did not assume in Theorem 1.3 that Φ sends any distinguished triangle to a distinguished triangle.

(2) We can extend Theorem 1.3 (2) to the case where X admits quotient singularities if K_{X} generates the local class group at any point as in [12]. Namely, let X be the smooth stack which lies naturally above X and let D(X) = D^{b}(Coh(X)) be the derived category of bounded complexes of coherent sheaves on X (see §6). Then Auteq(D(X)) is isomorphic to the semi-direct product of Aut(X) and Pic(X) ⊕ ℤ. The proof is the same as in [2].

On the other hand, if K_{X} does not generate the local class group, then the group of autoequivalences is much larger. For example, if Y is a smooth projective minimal surface of general type and X is its canonical model, then D(X) is equivalent to D(Y). If C is an exceptional curve of the resolution Y → X, then O_{C}(−1) is a 2-spherical object in D(Y) and generates an autoequivalence of infinite order ([21], see also §4).

(3) If ν(X) = ν(Y) = 0 in Theorem 2.3 (1), then K_{X} ∼ 0 if and only if K_{Y} ∼ 0 because Φ commutes with the Serre functors. More generally, it is known that the orders of the canonical divisors coincide ([7] Lemma 2.1).

3. Fourier-Mukai partners of surfaces

We have a complete picture of D and K-equivalences for surfaces. We start with the case of minimal surfaces:

Theorem 3.1 ([7]). Let X be a smooth projective surface. Assume that there is no (−1)-curve on X. Then there exist at most finitely many smooth projective surfaces Y such that the derived categories D(X) and D(Y) are equivalent as triangulated categories.

We note that there are Fourier-Mukai partners which are not birationally equivalent in the case of abelian or K3 or elliptic surfaces ([16], [17], [19], [20], [7], [10]). It is rather surprising that the existence of a (−1)-curve reduces the symmetry drastically:

Theorem 3.2. Let X be a smooth projective surface. Assume that there exists a (−1)-curve on X. Then there exist at most finitely many smooth projective surfaces Y such that the derived categories D(X) and D(Y) are equivalent as triangulated categories. Moreover, if X is not isomorphic to a relatively minimal elliptic rational surface, then any such Y is isomorphic to X.

Proof. We use the notation of the proof of Theorem 2.3. Let C be a (−1)-curve and Γ_{C} = p_{1}^{−1}(C)∩Γ. Since −K_{X}|_{C} is ample, the projection p_{2}|_{Γ_{C}} : Γ_{C} → Y is a finite morphism. We have two possibilities that dim Γ_{C} = 1 or 2.

Assume first that dim Γ_{C} = 1. We take an irreducible component Z_{1} of Γ which dominates X, and let Z_{1,C} = p_{1}^{−1}(C)∩Z_{1} and C′ = p_{2}(Z_{1,C}). We know that dim Z_{1,C} = dim C′ = 1. It follows that dim Z_{1} = 2 and the projection p_{1}|_{Z_{1}} : Z_{1} → X is generically finite, hence a birational morphism as in the proof of Theorem 2.3.

If Z_{1} dominates Y, then the other projection p_{2}|_{Z_{1}} : Z_{1} → Y is also birational, and X and Y are K-equivalent through Z_{1}. Hence X and Y are isomorphic (cf. Lemma 4.2).

Otherwise, we have p_{2}(Z_{1}) = C′. There exists an open dense subset U ⊂ X such that p_{1} induces an isomorphism p_{1}^{−1}(U)∩Γ = p_{1}^{−1}(U)∩Z_{1} → U. Take two distinct points x_{1}, x_{2} ∈ U which correspond to the same point y ∈ C′, i.e., y = p_{2}(p_{1}^{−1}(x_{1}) ∩ Γ) = p_{2}(p_{1}^{−1}(x_{2}) ∩ Γ). Then both Φ(O_{x_{1}}) and Φ(O_{x_{2}}) are supported at y, hence Hom_{D(Y)}^{p}(Φ(O_{x_{1}}),Φ(O_{x_{2}})) ≠ 0 for some p, a contradiction.

Assume next that dim Γ_{C} = 2. Then p_{2}|_{Γ_{C}} : Γ_{C} → Y is dominant. Since (K_{X} · C) < 0, we deduce that −K_{Y} is nef and ν(Y, −K_{Y}) = 1. Hence −K_{X} is also nef and ν(X, −K_{X}) = 1 by Theorem 2.3. By the classification of surfaces, such a surface is isomorphic to either a minimal elliptic ruled surface or a rational surface with Euler number 12. Since X has a (−1)-curve, X is a rational surface. By [7] Proposition 2.3, Y is also a rational surface.

We have the possibilities that dim Γ = 2 or 3. If dim Γ = 2, then we obtain our result as before. If dim Γ = 3, then X and Y are dominated by families of curves whose intersection numbers with the canonical divisors vanish. Thus X and Y are relatively minimal rational elliptic surfaces. By [7] Proposition 4.4, we obtain our result. Here we note that the proof there works also for relatively minimal elliptic surfaces of negative Kodaira dimension. q.e.d.

We can extend some of the above argument to higher dimensional case:

Proposition 3.3. Let X and Y be smooth projective varieties. Assume that κ(X) ≥ 0 but K_{X} is not nef, and that there is an extremal contraction morphism ϕ : X → W which contracts a prime divisor D to a point. Assume that the derived categories D(X) and D(Y) are equivalent as triangulated categories. Then X and Y are birational and K-equivalent.

Proof. We use the notation of the proof of Theorem 2.3. The proof is similar to that of Theorem 3.2.

Let Γ_{D} = p_{1}^{−1}(D) ∩ Γ. If dim Γ_{D} = n − 1 for n = dim X, then there exists an irreducible component Z_{1} of Γ of dimension n which dominates X. Then it follows that X and Y are birational and K-equivalent as in the proof of Theorem 3.2.

Assume that dim Γ_{D} ≥ n. Since −K_{X}|_{D} is ample, the projection p_{2}|_{Γ_{D}} : Γ_{D} → Y is a finite morphism. Hence dim Γ_{D} = n, and −K_{Y} is nef with ν(Y, −K_{Y}) = n − 1, a contradiction to κ(X). q.e.d.

Remark 3.4. We cannot expect similar statements for other types of contractions. For example, let A be an abelian surface, Â its dual, and S a smooth projective surface which contains a (−1)-curve. Let X = A × S and Y = Â × S. Then X has a divisorial contraction, D(X) ≅ D(Y), but X and Y are not birational in general.

4. Flops and minimal models

We consider normal varieties which are not necessarily smooth in this section.

Definition 4.1. Let X and Y be normal quasiprojective varieties whose canonical divisors are ℚ-Cartier divisors. A birational map α : X −−→ Y is said to be crepant if there exists a smooth quasiprojective variety Z with birational projective morphisms f : Z → X and g : Z → Y such that α ∘ f = g and f^{∗}K_{X} ∼_{ℚ} g^{∗}K_{Y}.

Lemma 4.2. Let α : X −−→ Y be a crepant birational map between quasiprojective varieties with only terminal singularities. Then α is an isomorphism in codimension 1; i.e., there exist closed subvarieties E ⊂ X and F ⊂ Y of codimension at least 2 such that α induces an isomorphism X \ E ≅ Y \ F.

Proof. Since X has only terminal singularities, a prime divisor D on Z is mapped by f to a subvariety of codimension at least 2 on X if and only if it appears in the relative canonical divisor K_{Z/X} = K_{Z} − f^{∗}K_{X} as an irreducible component. Since a similar statement holds for g, our assertion follows from the equality K_{Z/X} = K_{Z/Y}. q.e.d.

Definition 4.3. A projective variety X with only canonical singularities is called minimal if K_{X} is nef.

The minimality of a variety is characterized by the minimality of its canonical divisor:

Lemma 4.4. Let X and Y be normal projective varieties whose canonical divisors are ℚ-Cartier divisors. Assume that X and Y are birationally equivalent, X has only canonical singularities and that K_{X} is nef. Then the inequality K_{X} ≤ K_{Y} holds in the following sense: Let Z any smooth projective variety with projective birational morphisms f : Z → X and g : Z → Y. Then there exists a positive integer m such that m(g^{∗}K_{Y} − f^{∗}K_{X}) is linearly equivalent to an effective divisor. In particular, any birational map between minimal varieties is crepant.

Proof. We write f^{∗}K_{X} +A = g^{∗}K_{Y} +B, where A and B are effective divisors without common irreducible components. Since X has only canonical singularities, we may assume that codim g(Supp(B)) ≥ 2.

Assuming that B ≠ 0, we shall derive a contradiction. Let H and M be very ample divisors on Y and Z, respectively, and let n = dim Y and d = dim g(Supp(B)). We consider a generic surface section

S = g^{∗}H_{1} ∩ ⋯ ∩ g^{∗}H_{d} ∩ M_{1} ∩ ⋯ ∩ M_{n−d−2}

for H_{i} ∈ |H| and M_{j} ∈ |M|. By the Hodge index theorem, we have (g^{∗}H^{d}·M^{n−d−2}·B^{2}) < 0, while (g^{∗}H^{d}·M^{n−d−2}·B·(f^{∗}K_{X}+A−g^{∗}K_{Y})) ≥ 0 because K_{X} is nef and (g^{∗}H^{d}·g^{∗}K_{Y} ·M^{n−d−2}·B) = 0, a contradiction.

q.e.d.

We consider a special kind of crepant birational maps called flops:

Definition 4.5. Let X and Y be quasiprojective varieties with only canonical singularities, and D a ℚ-Cartier divisor on X. A birational map α : X −−→ Y is said to be a D-flop, or simply a flop, if there exist a normal quasiprojective variety W and crepant birational projective morphisms ϕ : X → W and ψ : Y → W which satisfy the following conditions:

(1) ϕ = ψ ∘ α.

(2) ϕ and ψ are isomorphisms in codimension 1.

(3) D is ϕ-ample, and for any ℚ-Cartier divisor A on X, there exist a ℚ-Cartier divisor A_{0} on W and a rational number r such that A ∼_{ℚ} ϕ^{∗}A_{0} + rD.

(4) Let D′ be the strict transform of D on Y. Then −D′ is ψ-ample, and for any ℚ-Cartier divisor B on Y, there exist a ℚ-Cartier divisor B_{0} on W and a rational number r′ such that B ∼_{ℚ} ψ^{∗}B_{0} + r′D′.

We can define flops of complex analytic spaces instead of quasiprojective varieties in a similar way. In this case, X and Y are complex analytic spaces which are relatively projective over a complex analytic space W.

Any crepant birational map between projective varieties with only ℚ-factorial terminal singularities is expected to be decomposed into a sequence of flops:

Theorem 4.6. Let α : X −−→ Y be a crepant birational map between projective varieties of dimension 3 with only ℚ-factorial terminal singularities. Then α is decomposed into a sequence of flops.

Proof. We may assume that the subvariety E of Lemma 4.2 is purely 1-dimensional. We may also assume that any irreducible component of E is the image of a curve on Z which is mapped to a point on Y. Since α is crepant, we have K_{X}|_{E} ∼_{ℚ} 0. Let H be an ample Cartier divisor on Y such that H − K_{Y} is still ample, and let H′ be its strict transform on X. By construction, any curve C such that (H′ · C) ≤ 0 is contained in E. We run the minimal model program with respect to K_{X} +ϵH′, where ϵ is a small positive number, for only those extremal rays on which H′ is nonpositive. Then the associated extremal curves are contained in E, so we obtain an H′-flop. We denote the result after the flop again by the same letters such as X, E and H′. After a finite flops, we have no more extremal rays on which H′ is nonpositive. Then H′ becomes nef and big. Since H′ is ample outside E, H′ − K_{X} is also nef and big, By the base point free theorem, we obtain a birational morphism X → Y, which should be an isomorphism. q.e.d.

5. From K-equivalence to D-equivalence

The following is a special case of the implication from (2) to (1) in Conjecture 1.2:

Conjecture 5.1. Let X and Y be smooth projective varieties and α : X → W ← Y a flop. Then there exists an equivalence of triangulated categories Φ : D(X) → D(Y).

The examples in this section suggest that the integral functor Φ_{X→Y}^{e} for the structure sheaf e = O_{X×_{W} Y} of the subscheme X ×_{W} Y ⊂ X × Y might work.

We consider the following two examples of flops in this section:

Example 5.2.

(1) A standard flop. Let X be a smooth projective variety of dimension 2m + 1 for some positive integer m, and E a subvariety of X. Assume that E ≅ ℙ^{m}, and N_{E/X} ≅ O_{ℙ^{m}}(−1)^{m+1}. Let f : Z → X be the blowing-up with center E. Then the exceptional divisor G is isomorphic to ℙ^{m} × ℙ^{m} and can be blown-down to another direction, so that we obtain a birational morphism g : Z → Y and a subvariety F = g(G) ≅ ℙ^{m}. There is a projective variety W with contraction morphisms ϕ : X → W and ψ : Y → W whose exceptional loci are E and F, respectively, and such that w_{0} = ϕ(E) = ψ(F) is the only singular point of W. Then α = g ∘ f^{−1} = ψ^{−1} ∘ ϕ is a flop.

(2) Mukai’s flop. Let W_{0} be a generic hypersurface section of W in (1) through the singular point w_{0}. Let X_{0} = ϕ^{−1}(W_{0}), Y_{0} = ψ^{−1}(W_{0}), ϕ_{0} = ϕ|_{X_{0}}, and ψ_{0} = ψ|_{Y_{0}}. Then X_{0} and Y_{0} are smooth, and α_{0} = ψ_{0}^{−1} ∘ ϕ_{0} is a flop. The inverse image Z̃_{0} = f^{−1}(X_{0}) = g^{−1}(Y_{0}) is reducible with 2 irreducible components G and Z_{0}, where Z_{0} is smooth. The restrictions f_{0} = f|_{Z_{0}} and g_{0} = g|_{Z_{0}} are again birational morphisms, and α_{0} = g_{0} ∘ f_{0}^{−1}. We set G_{0} = G ∩ Z_{0}. Then f_{0}(G_{0}) = E and g_{0}(G_{0}) = F.

We need the following concepts:

Definition 5.3. A set Ω of objects of D(X) is said to a spanning class if the following hold for any a ∈ D(X).

(1) Hom^{p}(a, ω) = 0 for all p ∈ ℤ and all ω ∈ Ω implies that a ≅ 0

(2) Hom^{p}(ω, a) = 0 for all p ∈ ℤ and all ω ∈ Ω implies that a ≅ 0.

For example, the set of point sheaves {O_{P} } for a smooth projective variety is a spanning class ([3] Example 2.2).

Definition 5.4. A Serre functor S_{X} : D(X) → D(X) is an autoequivalence of triangulated categories which induces bifunctorial isomorphisms

Hom_{D(X)}(a, b) → Hom_{D(X)}(b, S_{X}(a))^{∗}

for a, b ∈ D(X).

If a Serre functor exists, then it is unique up to isomorphisms. If X is smooth and projective, then S_{X}(a) = a ⊗ ω_{X}[dim X] is a Serre functor.

In order to prove that a functor Φ : D(X) → D(Y) to be fully faithful, it is sufficient to check it for the spanning class ([3] Theorem 2.3):

Φ : Hom^{p}(ω_{1}, ω_{2}) ≅ Hom^{p}(Φ(ω_{1}), Φ(ω_{2}))

for all p ∈ ℤ and all ω_{1}, ω_{2} ∈ Ω. Moreover, by [5] Theorem 2.3, provided that Φ = Φ_{X→Y}^{e} is fully faithful, it is an equivalence if and only if it commutes with the Serre functor. Theorefore, in order to prove our conjecture, we may consider locally over an analytic neighborhood of a point of W and replace the given flop by any other flop which is analytically isomorphic to the original one. If Φ is proved to be fully faithful, then it is automatically an equivalence in our case.

Proposition 5.5 ([1]). In Example 5.2 (1), Z is isomorphic to the fiber product X ×_{W} Y which is a closed subscheme of X × Y, and the functor

g_{∗}f^{∗} = Φ_{X→Y}^{O_{Z}} : D(X) → D(Y)

is an equivalence of triangulated categories.

Proof. We may replace X, Y and Z by the total space of the vector bundles N_{E/X}, N_{F/Y} and N_{G/Z}, respectively. We denote by O_{X}(k), O_{Y} (l) and O_{Z}(k, l) the pull-backs of O_{E}(k), O_{F} (l) and O_{G}(k, l), respectively. The set of objects

{O_{X}(−k) ∈ D(X)|k = 0, 1, …, m}

spans D(X). Since K_{Z/X} ∼ mG, we have

O_{X}(−k) ⟶^{f^{∗}} O_{Z}(−k, 0) ≅ O_{Z}(0, k)(kG) ⟶^{g_{∗}} O_{Y} (k).

We have

Hom^{p}(O_{X}(−k_{1}), O_{X}(−k_{1})) ≅ Hom^{p}(O_{Y} (k_{1}), O_{Y} (k_{1})) ≅ 0

for p ≠ 0 and k_{1}, k_{2} = 0, 1, …, m by the vanishing theorem, and

Φ_{X→Y}^{O_{Z}} : Hom(O_{X}(−k_{1}), O_{X}(−k_{1})) ≅ Hom(O_{Y} (k_{1}), O_{Y} (k_{1}))

because X and Y are isomorphic in codimension 1. Therefore, Φ_{X→Y}^{O_{Z}} is an equivalence by the remarks preceding to the proposition. q.e.d.

Lemma 5.6 ([8]). Let π_{X} : X → S and π_{Y} : Y → S be smooth projective morphisms from smooth quasiprojective varieties to a smooth quasiprojective curve. Let s_{0} ∈ S be a point, and let X_{0} = π_{X}^{−1}(s_{0}) and Y_{0} = π_{X}^{−1}(s_{0}) be fibers. Let i_{X_{0}} : X_{0} → X, i_{Y_{0}} : Y_{0} → Y and i_{X×_{S}Y} : X ×_{S} Y → X × Y be the embeddings. Let e ∈ D(X ×_{S} Y) be an object, and let e_{0} = e ⊗ O_{X_{0}×Y_{0}} and e′ = i_{X×_{S}Y ∗}(e). Then there is an isomorphism of functors from D(X_{0}) to D(Y):

i_{Y_{0}∗} ∘ Φ_{X_{0}→Y_{0}}^{e_{0}} ≅ Φ_{X→Y}^{e′} ∘ i_{X_{0}∗}.

Proof. Let i_{X_{0}×Y_{0}} : X_{0} × Y_{0} → X ×_{S} Y be the embedding. Let p_{1} : X ×_{S} Y → X, p_{2} : X ×_{S} Y → Y, p_{1,0} : X_{0} × Y_{0} → X_{0} and p_{2,0} : X_{0} × Y_{0} → Y_{0} be projections. For a ∈ D(X_{0}), we have

i_{Y_{0}∗} ∘ Φ_{X_{0}→Y_{0}}^{e_{0}}(a) ≅ i_{Y_{0}∗}p_{2,0∗}(p_{1,0}^{∗}(a) ⊗ e_{0}) ≅ p_{2∗}i_{X_{0}×Y_{0}∗}(p_{1,0}^{∗}(a) ⊗ e_{0})
≅ p_{2∗}(i_{X_{0}×Y_{0}∗}p_{1,0}^{∗}(a) ⊗ e) ≅ p_{2∗}(p_{1}^{∗}i_{X_{0}∗}(a) ⊗ e)
≅ Φ_{X→Y}^{e′} ∘ i_{X_{0}∗}(a).

q.e.d.

Corollary 5.7. In Example 5.2 (2), the functor

Φ_{X_{0}→Y_{0}}^{OZ̃_{0}} : D(X_{0}) → D(Y_{0})

is an equivalence of triangulated categories.

Proof. Since Z = X ×_{W} Y is a subscheme of X ×_{S} Y, we have the following isomorphisms:

i_{X_{0}∗}Φ_{Y_{0}→X_{0}}^{OZ̃_{0}(mG)}Φ_{X_{0}→Y_{0}}^{OZ̃_{0}} ≅ Φ_{Y →X}^{O_{Z}(mG)}Φ_{X→Y}^{O_{Z}} i_{X_{0}∗} ≅ i_{X_{0}∗}.

For any a ∈ D(X_{0}), let b ∈ D(X_{0}) be the cone of the natural morphism

a → Φ_{Y_{0}→X_{0}}^{OZ̃_{0}(mG)}Φ_{X_{0}→Y_{0}}^{OZ̃_{0}}(a).

Then i_{X_{0}∗}(b) ≅ 0, hence b ≅ 0. q.e.d.

The following concept is useful for constructing autoequivalences of derived categories.

Definition 5.8 ([21]). An object s ∈ D(X) is called n-spherical if

Hom_{D(X)}^{p}(s, s) ≅ \begin{cases}ℂ & if p = 0, n \\ 0 & otherwise. \end{cases}

The twisting functors T_{s}, T′_{s} : D(X) → D(X) are defined such that the following triangles are distinguished:

RHom_{X}(s, a) ⊗ s → a → T_{s}(a) → RHom_{X}(s, a) ⊗ s[1]
T′_{s}(a) → a → RHom_{ℂ}(RHom_{X}(a, s), s) → T′_{s}(a)[1]

where RHom_{X} denotes the derived global Hom. If s is n-spherical for n = dim X, then T_{s} and T′_{s} are equivalences and T_{s} ∘ T′_{s} ≅ Id_{D(X)}.

Example 5.9.

(1) O_{E} in Example 5.2 (1) is a (2m + 1)-sherical object. Indeed, since N_{E/X} ≅ O_{E}(−1)^{m+1}, we have

Ext_{O_{X_{0}}}^{p} (O_{E}, O_{E}) ≅ ⋀^{p}(O_{E}(−1)^{m+1}).

Hence

Hom_{D(X)}^{p}(O_{E}, O_{E}) ≅ \begin{cases}ℂ & if p = 0, 2m + 1 \\ 0 & otherwise. \end{cases}

(2) O_{E} in Example 5.2 (2) is not a 2m-sherical object. Indeed, since N_{E/X_{0}} ≅ Ω_{E}^{1}, we have

Ext_{O_{X_{0}}}^{p} (O_{E}, O_{E}) ≅ Ω_{E}^{p}.

Hence

Hom_{D(X_{0})}^{p}(O_{E}, O_{E}) ≅ \begin{cases}ℂ & if p = 0, 2, …, 2m \\ 0 & otherwise. \end{cases}

There is some relationship between the flops and the twistings.

Example 5.10. If m = 1 in Example 5.2 (1), then there are isomorphisms

Φ_{Y →X}^{O_{Z}} ∘ Φ_{X→Y}^{O_{Z}} (O_{X}(−k)) ≅ T′_{O_{E}(−1)}(O_{X}(−k))

for k = 0, 1.

Indeed, we have

\begin{matrix}O_{X} & ⟶^{Φ_{X→Y}^{O_{Z}}} & O_{Y} & ⟶^{Φ_{Y →X}^{O_{Z}}} & O_{X} \\ O_{X}(−1) & ⟶^{Φ_{X→Y}^{O_{Z}}} & O_{Y} (1) & ⟶^{Φ_{Y →X}^{O_{Z}}} & I_{E}(−1) \end{matrix}

where I_{E} is the ideal sheaf of E in X. On the other hand,

RHom_{X}(O_{X}, O_{E}(−1)) = 0, RHom_{X}(O_{X}(−1), O_{E}(−1)) = ℂ

hence

T′_{O_{E}(−1)}(O_{X}) = O_{X}, T′_{O_{E}(−1)}(O_{X}(−1)) = I_{E}(−1).

Example 5.11. In Example 5.2 (2), there are isomorphisms

Φ_{X_{0}→Y_{0}}^{O_{Z_{0}}} ∘ (Φ_{X_{0}→Y_{0}}^{OZ̃_{0}})^{−1}(O_{Y_{0}}(k)) ≅ T_{O_{F} (−1)}(O_{Y_{0}}(k))

for k = 0, 1, …, m.

Indeed, since

Hom_{D(Y_{0})}^{p}(O_{F} (−1), O_{Y_{0}}(k))
≅ Hom_{D(Y_{0})}(O_{Y_{0}}(k), O_{F} (−1)[2m − p])^{∗}

≅ \begin{cases}0 & if k = 0, …, m − 1 \\ 0 & if k = m and p ≠ m \\ ℂ & if k = m and p = m. \end{cases}

we have

T_{O_{F} (−1)}(O_{Y_{0}}(k)) ≅ O_{Y_{0}}(k)

for k = 0, 1, …, m − 1, and

O_{F} (−1)[−m] → O_{Y_{0}}(m) → T_{O_{F} (−1)}(O_{Y_{0}}(m)) → O_{F} (−1)[−m + 1]

is a distinguished triangle, where the first arrow is nontrivial. On the other hand, we have an exact sequence

0 → O_{X} → O_{X} → O_{X_{0}} → 0

where the first arrow is the multiplication by an equation of W_{0} ⊂ W. Hence

Φ_{X_{0}→Y_{0}}^{OZ̃_{0}}(O_{X_{0}}(−k)) ≅ O_{Y_{0}}(k)

for k = 0, 1, …, m. If k = 0, 1, …, m − 1, then we also have

O_{X_{0}}(−k) ⟶^{f_{0}^{∗}} O_{Z_{0}}(−k, 0) ≅ O_{Z_{0}}(0, k)(kG_{0}) ⟶^{g_{0∗}} O_{Y_{0}}(k)

because K_{Z_{0}/Y_{0}} = (m − 1)G_{0}.

For k = m, we have an exact sequence

0 → O_{Z_{0}}(0, m)((m − 1)G_{0}) → O_{Z_{0}}(0, m)(mG_{0}) → ω_{G_{0}}(0, m) → 0.

Since g_{0∗}(ω_{G_{0}}) ≅ ω_{F} [−m + 1] ≅ O_{F} (−m − 1)[−m + 1], we obtain a distinguished triangle

O_{F} (−1)[−m] → O_{Y_{0}}(m) → g_{0∗}f_{0}^{∗}(O_{X_{0}}(−m)) → O_{F} (−1)[−m + 1].

D-equivalence and K-equivalence 163

We claim that the first arrow is nontrivial as an element of

Hom_{D(Y_{0})}(O_{F} (−1)[−m], O_{Y_{0}}(m)) ≅ ℂ.

Indeed, if not, then we would have Hom_{D(Y_{0})}(O_{F} (−1)[−m], O_{Y_{0}}(m))

≅ Hom_{D(Y_{0})}^{1}(g_{0∗}f_{0}^{∗}(O_{X_{0}}(−m)), O_{Y_{0}}(m))

but

Hom_{D(Y_{0})}^{1}(g_{0∗}f_{0}^{∗}(O_{X_{0}}(−m)), O_{Y_{0}}(m))
≅ Hom_{D(Z_{0})}^{1}(f_{0}^{∗}(O_{X_{0}}(−m)), g_{0}^{!}O_{Y_{0}}(m))
≅ Hom_{D(Z_{0})}^{1}(O_{Z_{0}}(0, m)(mG_{0}), O_{Z_{0}}(0, m)((m − 1)G_{0}))
≅ H^{1}(Z_{0}, O_{Z_{0}}(−G_{0})) ≅ 0

a contradiction. Therefore, we have the desired isomorphisms.

Proposition 5.12. In Example 5.2 (2), if m ≥ 2, then the functor

g_{0∗}f_{0}^{∗} = Φ_{X_{0}→Y_{0}}^{O_{Z_{0}}} : D(X_{0}) → D(Y_{0})

is not an equivalence.

Proof. Let us write Φ = Φ_{X_{0}→Y_{0}}^{O_{Z_{0}}} and a = O_{X_{0}}(−m). We consider a spectral sequence

E_{2}^{p,q} = ⨁_{i∈ℤ}Ext^{p}(H^{i}(Φ(a)), H^{q+i}(Φ(a))) ⇒ Hom_{D(Y_{0})}^{p+q}(Φ(a), Φ(a))

given by the last line of [24] 4.6.10. We have

H^{q}(Φ(a)) ≅ \begin{cases}O_{Y_{0}}(m) & if q = 0 \\ O_{F} (−1) & if q = m − 1 \\ 0 & otherwise \end{cases}

and

Ext^{p}(O_{Y_{0}}(m), O_{Y_{0}}(m)) ≅ Ext^{p}(O_{X_{0}}(−m), O_{X_{0}}(−m)) = 0 for p ≠ 0

Ext^{p}(O_{Y_{0}}(m), O_{F} (−1)) ≅ \begin{cases}ℂ & if p = m \\ 0 & otherwise \end{cases}

Ext^{p}(O_{F} (−1), O_{Y_{0}}(m)) ≅ \begin{cases}ℂ & if p = m \\ 0 & otherwise \end{cases}

Ext^{p}(O_{F} (−1), O_{F} (−1)) ≅ \begin{cases}ℂ & if p = 0, 2, …, 2m \\ 0 & otherwise. \end{cases}

Then the terms E_{2}^{p,0} for p = 2, …, 2m−2 survive, hence Hom_{D(X_{0})}^{p}(a, a) and Hom_{D(Y_{0})}^{p}(Φ(a), Φ(a)) are not isomorphic for these p. q.e.d.

Remark 5.13. After this paper was written, Jan Wierzba informed us that Corollary 5.7 and Proposition 5.12 were already proved by Namikawa [18], though the proofs are different. Combining with a result in [9] or [26] (see also [14]), we obtain the implication from (2) to (1) in Conjecture 1.2 in the case of symplectic projective manifolds of dimension 4.

6. Flops of terminal 3-folds

We shall deal with singular verieties in this section.

The smoothness of the given varieties is an important assumption for the study of derived categories. For example, any coherent sheaf on a smooth projective variety has a finite locally free resolution, hence the Serre functor exists.

We can compare our situation with the deformation theory of maps from curves to varieties. The latter is not applicable to singular varieties because the smoothness assumption is essential for a good obstruction theory. However it provides deep results such as the theory of rationally connected varieties.

We can still deal with singular varieties as if they are smooth in some cases:

(1) If X is a variety with only quotient singularities, then we consider a smooth stack X above X as a natural substitute (cf. [12]).

(2) If X has only hypersurface singularities, then we embed X into a smooth variety by deformations (cf. [8]).

(3) If X is a normal crossing variety, then we replace X by its smooth hypercovering (cf. [15]).

We consider a mixture of (1) and (2) in this section.

Definition 6.1. Let X be a normal quasiprojective variety such that the canonical divisor K_{X} is a ℚ-Cartier divisor. Each point x ∈ X has an open neighborhood U_{x} such that m_{x}K_{X} is a principal Cartier divisor on U_{x} for a minimum positive integer m_{x}. The canonical covering π_{x} : Ũ_{x} → U_{x} is a finite morphism of degree m_{x} from a normal variety which is etale in codimension 1 and such that KŨ_{x} is a Cartier divisor. The canonical coverings are etale locally uniquely determined, thus we can define the canonical covering stack X as the stack above X given by the collection of canonical coverings π_{x} : Ũ_{x} → U_{x}.

We denote by D(X) = D^{b}(Coh(X)) the derived category of bounded complexes of coherent orbifold sheaves on X (cf. [12]).

The following was suggested by Burt Totaro.

Proposition 6.2. Let X be a normal projective variety such that the canonical divisor K_{X} is a ℚ-Cartier divisor. Then there exists an embedding

ϕ : X → ℙ(a_{1}, …, a_{N})

to a weighted projective space such that the stack structure on X induced from the natural smooth stack structure of ℙ(a_{1}, …, a_{N}) coincides with the one defined by the canonical coverings.

Proof. Let H be an ample Cartier divisor such that K_{X} + H is still ample as a ℚ-Cartier divisor. The ring R = ⨁_{m=0}^{∞} H^{0}(X, m(K_{X} + H)) is a finitely generated algebra over ℂ. Let x_{1}, …, x_{N} be a set of homogeneous generators of R of degree a_{1}, …, a_{N}. Then we obtain an embedding of X to a weighted projective space

ϕ : X → ℙ(a_{1}, …, a_{N}).

Since K_{X} + H is ample, g.c.d.(a_{1}, …, a_{N}) = 1.

We claim that

g.c.d.(a_{1}, …, ǎ_{i}, …, a_{N}) = 1

for any i = 1, …, N, i.e., the sequence of integers (a_{1}, …, a_{N}) is wellformed. Indeed, suppose that (a_{2}, …, a_{N}) = c ≠ 1. Let m be a sufficiently large integer which is not divisible by c, and consider an exact sequence

0 → O_{X}((m − a_{1})(K_{X} + H)) → O_{X}(m(K_{X} + H)) → F_{m} → 0

given by the multiplication by x_{1}, where F_{m} is a sheaf on X_{1} = div(x_{1}). By assumption, we have H^{0}(X, (m − a_{1})(K_{X} + H)) ≅ H^{0}(X, m(K_{X} + H)), while H^{0}(X_{1}, F_{m}) ≠ 0 and H^{1}(X, (m−a_{1})(K_{X} +H)) = 0 for large m, a contradiction.

Let us fix a point p ∈ X. Then there exists a homogeneous coodinate, say x_{1}, such that x_{1}(p) ≠ 0. We have a commutative diagram

U ⟶^{ϕ} U_{x_{1}} ⟶^{⊂} ℙ(a_{1}, …, a_{N})
π_{U} ↑  ↑^{π_{1}}
Ũ \underrightarrow{ϕ̃} Ũ_{x_{1}}

where U is a small open neighborhood of p, U_{x_{1}} is the open subset of ℙ(a_{1}, …, a_{N}) defined by x_{1} ≠ 0, π_{U} : Ũ → U is a canonical covering, and π_{1} : Ũ_{x_{1}} → U_{x_{1}} is the natural covering from an affine space with coordinates

x_{2}x_{1}^{−a_{2}/a_{1}}, …, x_{N}x_{1}^{−a_{N} /a_{1}}.

Note that both π_{U} and π_{1} are etale in codimension 1.

Since x_{1}(p) ≠ 0, we may choose a branch of x_{1}^{1/a_{1}} on sufficiently small U. Then ϕ can be lifted to a morphism ϕ̃: Ũ → Ũ_{x_{1}} which we can check to be etale. Therefore, the two stack structures coincide. q.e.d.

Remark 6.3.

(1) By the proposition, any coherent orbifold sheaf on the canonical covering stack X has a surjection from a locally free orbifold sheaf on X. But the Serre functor for the category D(X) does not exist in general.

(2) Totaro ([22]) proved the following resolution theorem: on a smooth orbifold whose coarse moduli space is a separated scheme, any coherent orbifold sheaf has a finite resolution by locally free orbifold sheaves.

We still have a good spanning class for terminal 3-folds:

Lemma 6.4. Let X be a normal projective variety of dimension 3 with only terminal singularities, m_{x} the index of K_{X} at x ∈ X, and X the canonical covering stack of X. Then the set {O_{x}(iK_{X})|x ∈ X, 0 ≤ i < m_{x}} is a spanning class of D(X).

Proof (cf. [3] Example 2.2 and [8] Lemma 3.4). Let a be a nonzero object of D(X). Take a point x_{0} in the support of a, and let q_{0} be the maximal value of q such that H^{q}(a)_{x_{0}} ≠ 0. Then there exists an integer i_{0} such that Hom(H^{q_{0}}(a), O_{x_{0}}(i_{0}K_{X})) ≠ 0. Then

Hom_{D(X)}^{−q_{0}}(a, O_{x_{0}}(i_{0}K_{X})) ≠ 0.

If the support of a is not contained in the singular locus of X, then we take the above point x_{0} from the smooth locus of X. By the Serre duality, we have Hom^{n+q_{0}}(O_{x_{0}}, a) ≠ 0, where n = dim X. Otherwise, let q_{1} be the minimal value of q such that H^{q}(a)_{x_{0}} ≠ 0. Since X has only isolated singularities, there exists an integer i_{1} such that Hom(O_{x_{0}}(i_{1}K_{X}), H^{q_{1}}(a)) ≠ 0. Hence Hom_{D(X)}^{q_{1}}(O_{x_{0}}(i_{1}K_{X}), a) ≠ 0.

q.e.d.

Theorem 6.5. Let X and Y be normal quasiprojective varieties of dimension 3 with only ℚ-factorial terminal singularities,

X ⟶^{ϕ} W ⟵^{ψ} Y

a flop, and X and Y the canonical covering stacks above X and Y, respectively. Then the bounded derived categories of coherent orbifold sheaves D(X) and D(Y) are equivalent as triangulated categories.

Proof. The assertion is already proved in the case where K_{X} is a Cartier divisor by Bridgeland [4] and Chen [8] (see also [23]). Indeed, it is proved that the structure sheaf O_{Z} of the fiber product Z = X ×_{W} Y is quasi-isomorphic to a finite complex of sheaves on X × Y flat over X so that the integral functor Φ_{X→Y}^{O_{Z}} : D(X) → D(Y) is defined and is an equivalence ([8] Lemma 2.1 and Proposition 4.2).

We shall give a new simpler proof, which is based on [23] §4.1, that Φ = Φ_{X→Y}^{O_{Z}} : D(X) → D(Y) is an equivalence in the case where K_{X} is a Cartier divisor. We may assume that W is a hypersurface singularity of multiplicity 2. Thus W has an involution σ such that W/⟨σ⟩ is smooth. We may take Y = X and ψ = σ ∘ ϕ.

First we prove that Φ(O_{X}) ≅ O_{Y}. Indeed, for any closed point y ∈ Y, the scheme theoretic fiber g^{−1}(y) is isomorphic to the fiber f^{−1}(ψ(y)). We have H^{0}(O_{f^{−1}(ψ(y))}) ≅ ℂ, hence the natural homomorphism O_{Y} → R^{0}g_{∗}O_{Z} is an isomorphism.

Any subscheme of Z which is mapped by g to an infinitesimal subscheme y of Y supported at y is isomorphic to a subscheme of the product C × y for a subscheme C of X which is mapped by ϕ to an infinitesimal subscheme w of W supported at ϕ(y). Since R^{1}ϕ_{∗}O_{X} = 0, it follows that R^{1}g_{∗}O_{Z} = 0.

Let C_{j} (j = 1, …, t) be the exceptional curves of ϕ, and L_{i} (i = 1, …, t) invertible sheaves on X such that (L_{i} · C_{j}) = δ_{ij}. Then L_{i} are generated by global sections for all i. We note that R^{1}ϕ_{∗}L_{i}^{∗} may not necessarily vanish.

According to [23] §4.1, we construct locally free sheaves M_{i} and N_{i} on X by the following exact sequences:

0 → O_{X}^{r_{i}} → M_{i} → L_{i} → 0
0 → N_{i} → O_{X}^{s_{i}} → L_{i} → 0

for some integers r_{i}, s_{i} such that we have the vanishing higher direct image sheaves R^{1}ϕ_{∗}M_{i}^{∗} = 0 and R^{1}ϕ_{∗}N_{i} = 0. By [23] Proposition 4.1.2, if we take r_{i} and s_{i} to be the minimal possible integers under the vanishing conditions, then we have

ϕ_{∗}N_{i} ≅ σ_{∗}ϕ_{∗}M_{i}

where we note that σ_{∗}L_{i} ≅ L_{i}^{∗}. By construction, M_{i} and N_{i}^{∗} are generated by global sections.

It follows that R^{1}g_{∗}f^{∗}N_{i} = 0 from R^{1}ϕ_{∗}N_{i} = 0 as before. We consider an exact sequence

0 → g_{∗}f^{∗}N_{i} → O_{Y}^{s_{i}} → g_{∗}f^{∗}L_{i} → 0.

Since there is a non-natural injection g_{∗}f^{∗}L_{i} → g_{∗}O_{Z}, the sheaf g_{∗}f^{∗}L_{i} is torsion free. Hence g_{∗}f^{∗}N_{i} is a reflexive sheaf. Since ψ_{∗}g_{∗}f^{∗}N_{i} ≅ ϕ_{∗}N_{i} ≅ ψ_{∗}M_{i}, we conclude that Φ(N_{i}) ≅ M_{i}.

The set of sheaves Ω = {O_{X}, N_{1}, …, N_{t}} is a spanning class of D(X). ω is locally free, ω^{∗} is generated by global sections and R^{1}ϕ_{∗}ω = 0 for any ω ∈ Ω. Hence

Hom_{D(X)}^{p}(ω_{1}, ω_{2}) = 0

for p > 0 and ω_{1}, ω_{2} ∈ Ω. Similarly we have

Hom_{D(Y)}^{p}(Φ(ω_{1}), Φ(ω_{2})) = 0.

Since X and Y are isomorphic in codimension 1, we have

Hom_{X}(ω_{1}, ω_{2}) ≅ Hom_{Y} (Φ(ω_{1}), Φ(ω_{2})).

Therefore, we have proved that Φ is an equivalence in the case where K_{X} is a Cartier divisor.

Now we consider the general case. Let W be the canonical covering stack of W. Let w ∈ W be a point, W_{w} its small neighborhood on which m_{w}K_{W} is a principal Cartier divisor, and π_{w} : W̃_{w} → W_{w} a canonical covering. Then m_{w}K_{X} and m_{w}K_{Y} are also principal Cartier divisors on X_{w} = ϕ^{−1}(W_{w}) and Y_{w} = ψ^{−1}(W_{w}), respectively, and we have corresponding canonical coverings π_{X} : X̃_{w} → X_{w} and π_{Y} : Ỹ_{w} → Y_{w}. Thus there are morphisms of stacks ϕ : X → W and ψ : Y → W. Let

Z = X ×_{W} Y

be the fiber product as a stack. Then it is a stack above Z = X ×_{W} Y where local coverings are given by

Z̃_{w} = X̃_{w} ×W̃_{w} Ỹ_{w} → Z_{w} = X_{w} ×_{W_{w}} Y_{w}.

Let 𝔣 : Z → X and 𝔤 : Z → Y be the induced morphisms.

We claim that the functor

𝔤_{∗}𝔣^{∗} : D(X) → D(Y)

is defined and is an equivalence. Indeed, over an open subset W_{w}, we know already that the integral functor

Φ_{X̃_{w}→Ỹ_{w}}^{OZ̃w} : D(X̃_{w}) → D(Ỹ_{w})

is an equivalence. Let X_{w} = X |_{X_{w}} = [X̃_{w}/G], Y_{w} = Y|_{Y_{w}} = [Ỹ_{w}/G], Z_{w} = Z|_{Z_{w}} = [Z̃_{w}/G], 𝔣_{w} = 𝔣|_{Z_{w}} and 𝔤_{w} = 𝔤|_{Z_{w}}. The Galois group G = ℤ/m_{w} acts equivariantly so that we have D(X̃_{w})^{G} ≅ D(X_{w}) and D(Ỹ_{w})^{G} ≅ D(X_{w}) (cf. [7]). Hence we have a well-defined equivalence

𝔤_{w∗}𝔣_{w}^{∗} : D(X_{w}) → D(Y_{w}).

By Lemma 6.4, we conclude the proof. q.e.d.

Remark 6.6. We note that the equivalence Φ = 𝔤_{∗}𝔣^{∗} : D(X) → D(Y) does not induce an equivalence D(X) → D(Y) of usual derived categories for singular varieties. Indeed, we can construct a similar example as in [12] Example 5.1. There is a skyscraper sheaf a ∈ D(X) supported over a non-Gorenstein singular point of X such that π_{X∗}(a) = 0 in D(X), but its image Φ(a) ∈ D(Y) has a 1-dimensional support so that π_{Y ∗}(Φ(a)) ≠ 0 in D(Y).

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