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In this section, we will consider complexes of sheaves of abelian groups and the bounded derived categories Db(X, Z) thereof.

Let X and Xbe smooth varieties of dimension n, Y a varieties, and f∶ X → Y and f∶ X→ Y be proper birational morphisms. Suppose that X and Xare K-equivalent through f′−1○f. That is to say, given any common resolution g∶ Z → X and g∶ Z → X, the relative canonical divisors Kgand Kg are equal. We suppose further that f and fare semismall in the sense that dim X×Y X= n and dim X×Y X= n. In this case f′−1○ f is refered to as a semismall K-equivalence.

One important example we shall bear in mind is that when f∶ X → Y is a semismall crepant resolution and f∶ X→ Y is a flop, X and Xare K-equivalent through f′−1○f.

In this case f′−1○ f is refered to as a semismall flop.

A stratum T ∈ T is called f-relevant if dim f−1(T ) ×Tf−1(T ) = n.

The goal of this section is to prove the following 2.1.

Proposition 2.1. — For a sufficiently fine stratification T of Y , on each rele-vant stratum T ∈ T of dimension d, there is an isomorphism Rn−dfZf−1(T )Rn−dfZf′−1(T ).

Taking the dual local system,

Corollary 2.2. — Under the assumptions of 2.1, let i∶ T → Y be the embedding.

There is an isomorphism Rn−di!fZX≅ Rn−di!fZX

Here is a plan of the proof.

Given an initial stratificationT , we shall recursively study the isomorphism and at the same time refine the stratification and the common resolution Z if necessary.

To be more precise, assume the comparison is done on a union of strata, which is Zariski open in Y . We shall pick a stratum dense, say T , in the complement, blow up Z accordingly, and establish the comparison on a Zariski open subset of T in case T is relevant, by means of arc spaces. Then we shall refine the stratification by

splitting up T into the Zariski open subset and some other smooth Zariski locally closed subvarieties. By noetherian induction, this will prove 2.1.

Let T ⊆ Y be a Zariski locally closed smooth conncected subvariety, of dimension d.

Suppose that Rn−dfZ and Rn−dfZ are (classically) locally constant constructible, and that f and f are flat over T .

We begin with some observations on the irreducible components. Consider the projection

πm,f−1(T )∶ Lm(X)∣

f−1(T )→ f−1(T )

from the space of mth truncated arcs with origin in f−1(T ), to f−1(T ).

Lemma 2.3. — Under the above assumptions, the irreducible components ofLm(X)∣

f−1(T )

correspond to those of f−1(T ) under πm,f−1(T ), and that m,f−1(T ),!∶ Z[2mn] ≅ Z.

Proof. — Let E⊂ f−1(T ) be an irreducible component. Then Lm(X)∣

E

is a Zariski locally trivial Amn-fibration over E, and hence is irreducible. Consequently, the irreducible components of Lm(X)∣

f−1(T ) correspond to those of f−1(T ) under the projection πf−1(T ),m∶ Lm(X)∣

f−1(T ) → f−1(T ). Since it is a Amn-fibration, by the base change property for exceptional push-forwards, the projection πm,f−1(T ) induces

m,f−1(T ),!∶ Z[2mn] ≅ Z.

In regards to 2.3, we have

m,f−1(T ),!∶ Z[2mn] ≅ Z on f−1(T ) ⊆ X as well as

m,f′−1(T ),!∶ Z[2mn] ≅ Z on f′−1(T ) ⊆ X and

m,h−1(T ),!∶ Z[2mn] ≅ Z on h−1(T ) ⊆ Z.

The situation is indicated below

Lm(Z)∣

h−1(T )

Lm(X)∣

f−1(T ) h−1(T ) Lm(X)∣

f′−1(T )

f−1(T ) f′−1(T )

Lm(g) πm,h−1(T ) Lm(g)

πm,f −1(T ) g

g

πm,f ′ −1(T )

The next step is to establish relations between the irreducible components of Lm(Z)∣

h−1(T ) and those of Lm(X)∣

f−1(T ). We shall see that each component of Lm(X)∣

f−1(T )is dominated by one certain irreducible component ofLm(Z)∣

h−1(T ). For each irreducible component F ⊆ Lm(Z)∣

h−1(T ), let δ(F ) = min {k ∶ Lm(Z)≤k∩ F ≠ ∅} .

Lemma 2.4. — Under the above assumptions, assume in addition that m≥ 2δ(F )+1 for all irreducible components F ⊆ Lm(Z)∣

h−1(T ). Then δ(F ) = dim F −dim Lm(g)(F ) ≥ dim F− (n + d)/2 + mn, and that the equality holds exactly when F dominates an ir-reducible component of maximal dimension (n + d)/2 + mn.

Proof. — By 1.1, the additional assumption m≥ 2δ(F ) + 1 make arcs L (Z)≤δ(F ) of contact order≤ δ(F ) all stabilised(i.e. there is a piecewise affine fibration structure), in particular, Lm(g) is a Zariski locally trivial Aδ(F )-fibration when restricted to Lm(Z)≤δ(F )with image Lm(X)≤δ(F ). Therefore

dim(Lm(Z)≤δ(F )∩ F ) − dim (Lm(X)≤δ(F )∩ Lm(g)(F )) = δ(F ).

On the other hand, Lm(Z)≤δ(F )∩ F is Zariski open and dense in F , and that Lm(X)≤δ(F )∩ Lm(g)(F ) is Zariski open and dense in Lm(g)(F ), so

δ(F ) = dim (Lm(Z)≤δ(F )∩ F ) − dim (Lm(X)≤δ(F )∩ Lm(g)(F ))

= dim F − dim Lm(g)(F ).

Recall that the assumption on semismallness forces that dim f−1(T ) ≤ (n + d)/2,

Corollary 2.5. — Under the assumptions of 2.4, Lm(g) induces a bijection {irreducible components E ⊆ Lm(X)∣

f−1(T )∶ dim E =n+ d 2 + mn}

←→ {irreducible components F ⊆ Lm(Z)∣

h−1(T )∶ δ(F ) = dim F −n+ d 2 − mn}

Proof. — In regards to 2.4, we are left to prove that each irreducible component ELm(X)∣

f−1(T ) is dominated by exactly one irreducible component ofLm(Z)∣

h−1(T ). This is clear by the surjectivity of Lm(g) and the fact that Lm(g) is an Aδ(F ) -fibration over an Zariski open set of E.

For convenience, we may and we shall refine the resolution as Z→ Z, which is an isomorphism outside the closure h−1(T ), such that h−1(T ) is a sum of divisors, and that dim F= (n−1)+mn for all F . The condition then becomes δ(F ) ≥ (n−d)/2−1 =∶ δ.

We rename Z∶= Z.

2.5 describes explicitly the relation between components of importance, the diffi-culty to the comparison being thatLm(g) is only an Aδ-fibration on an Zariski open subset. This difficulty is evitable by restricting everything to an Zariski open subset of T .

Lemma 2.6. — Under the assumptions of 2.4, U is Zariski open and dense in T .

Proof. — Put

so that U = V ∩ V. Then it suffices to verify that V and V are Zariski open and dense in T . We shall do that for V only.

We may assume that T is irreducible.

Firstly, V is nonempty. If V = ∅, then by the piecewise fibration structure 1.1 dim(Lm(Z)

By 2.3, the former sheave is

R(n−d)+2mn(f ○ πm,f−1(T ))!Z

Lm(X)

f −1(T )

≅ Rn−dfZf−1(T ),

which is locally constant constructible by assumption. According to the semiconti-nuity of quotients of locally constant sheaves, the set of t∈ T at which the stalk is zero

By restricting to Lm(Z)≤δtrivial Aδ-fibrations. Therefore there are Zariski open dense V ⊆ Lm(X)≤δ

f−1(U)and V⊆ Lm(X)≤δ

f′−1(U)such thatLm(g)≤δand Lm(g)≤δ are trivial Aδ-fibrations.

Set ̃W = Lm(g)−1(V )∩Lm(g)−1(V) ⊆ Lm(Z)≤δ

h−1(U)which is also Zariski open and dense, with complement ̃Wc⊆ Lm(Z)≤δ

f′−1(U)be the corresponding complements.

Finally, pick W ⊆ U such that the fibre of f ○ πm,f−1(U)∶ Wc → U and of fπm,f′−1(U)∶ W′c→ U over each point of W is of dimension < (n − d)/2.

Lemma 2.7. — Under the assumptions of 2.4, in the above diagramme, there is an isomorphism on W , Rn−dfZf−1(W )≅ Rn−dfZf′−1(W ).

Proof. — In this context, Leray spectral sequence for Rπm,f−1(U),!○ RLm(g)≤δ! then gives

R2mn+2δm,f−1(U)○ Lm(g)≤δ)!Z̃W ≅ R2mnπm,f−1(U),!(RLm(g)≤δ! Z̃W) (1)

Applying RLm(g)≤δ! to the relative sequence

and the latter is isomorphic to

Rn−df

Combining equations 1, 2, 3, 4 and 5, we get

⎛⎜

Therefore

Rn−dfZf−1(W )≅ (Rn−dfZf−1(U)) ∣

W ≅ (Rn−dfZf′−1(U)) ∣

W ≅ Rn−dfZf′−1(W ). We may redefine T as W and refine the stratificationT accordingly, so that

Rn−dfZ≅ Rn−dfZ.

From the arguments above, one sees that

Proposition 2.8. — Under the assumptions of 2.4, the semismallness of f implies that of f, and that dim X×Y X= n. Moreover, if a stratum T is f-relevant, then it is also f-relevant.

Finally, we can prove 2.1.

Proof of 2.1. — As is sketched right after the statement of 2.1, the proof proceeds by recurrence on strata.

We assign a partial order on the set of strata. There is a relation between two strata T ≤ T if T ⊆ T.

Suppose that T ∈ T is a stratum, and that on every T∈ T with T ≤ Tbut T ≠ T, the comparison Rn−dfZT ≅ Rn−dfZT has been established. Apply 2.7 to T , so that there is a resolution Z1 → Z and an Zariski open W ⊆ T with Rn−dfZWRn−dfZW. Refine the stratification on T so that W is one of the strata. This finishes the recursive step.

Clearly by the noetherian assumption on Y , the recurrence eventually stops. That finishes the proof.

Proposition 2.9. — In coefficient Q, we have RfQ≅ RfQ

Consequently, if X and X are projective, then the Chow motives in Q-coefficients are isomorphic

M(X)Q≅ M(X)Q

Proof. — Tensoring with Q the isomorphisms provided by 2.1, we have Rn−dfQ≅ Rn−df(Z ⊗LfQ) ≅ Rn−dfZLQ

≅ Rn−dfZLQ≅ Rn−df(Z ⊗Lf′∗Q) ≅ Rn−dfQ.

Then

IC(T, Rn−dfQ) ≅ IC (T, Rn−dfQ) .

By the explicit decomposition theorem of [BM], RfQ[n] ≅ ⊕

T∈T

IC(T, Rn−dfQ) ≅ ⊕

T∈T

IC(T, Rn−dfQ) ≅ RfQ[n].

An argument in [dCM2], which we will reproduce in the next section, shows that M(X)Q≅ M(X)Q.

As the decomposition theorem is available only with coefficients in fields of character-istic 0, in order to extend the result to more general coefficients, we shall study the extensions of perverse sheaves across strata in the next section and forth.

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