In this section, we will consider complexes of sheaves of abelian groups and the bounded derived categories Db(X, Z) thereof.
Let X and X′be smooth varieties of dimension n, Y a varieties, and f∶ X → Y and f′∶ X′→ Y be proper birational morphisms. Suppose that X and X′are 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 f′are 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 X′are 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−df∗Zf−1(T ) ≅ Rn−df∗′Zf′−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!f∗ZX≅ Rn−di!f∗′ZX′
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−df∗Z and Rn−df∗′Z 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 Rπm,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
Rπm,f−1(T ),!∶ Z[2mn] ≅ Z.
In regards to 2.3, we have
Rπm,f−1(T ),!∶ Z[2mn] ≅ Z on f−1(T ) ⊆ X as well as
Rπm,f′−1(T ),!∶ Z[2mn] ≅ Z on f′−1(T ) ⊆ X′ and
Rπm,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 E⊆ Lm(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)>δ∣
By 2.3, the former sheave is
R(n−d)+2mn(f ○ πm,f−1(T ))!Z
Lm(X)∣
f −1(T )
≅ Rn−df∗Zf−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)≤δ∣ trivial 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 f′○ πm,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−df∗Zf−1(W )≅ Rn−df∗′Zf′−1(W ).
Proof. — In this context, Leray spectral sequence for Rπm,f−1(U),!○ RLm(g)≤δ! then gives
R2mn+2δ(πm,f−1(U)○ Lm(g)≤δ)!Z̃W ≅ R2mnπm,f−1(U),!(R2δLm(g)≤δ! Z̃W) (1)
Applying R2δLm(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−df∗Zf−1(W )≅ (Rn−df∗Zf−1(U)) ∣
W ≅ (Rn−df∗′Zf′−1(U)) ∣
W ≅ Rn−df∗′Zf′−1(W ). We may redefine T as W and refine the stratificationT accordingly, so that
Rn−df∗Z≅ Rn−df′∗Z.
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 ≤ T′but T ≠ T′, the comparison Rn−df∗ZT ≅ Rn−df′∗ZT has been established. Apply 2.7 to T , so that there is a resolution Z1 → Z and an Zariski open W ⊆ T with Rn−df∗ZW ≅ Rn−df′∗ZW. 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 Rf∗Q≅ Rf∗′Q
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−df∗Q≅ Rn−df∗(Z ⊗Lf∗Q) ≅ Rn−df∗Z⊗LQ
≅ Rn−df∗′Z⊗LQ≅ Rn−df∗′(Z ⊗Lf′∗Q) ≅ Rn−df∗′Q.
Then
IC(T, Rn−df∗Q) ≅ IC (T, Rn−df∗′Q) .
By the explicit decomposition theorem of [BM], Rf∗Q[n] ≅ ⊕
T∈T
IC(T, Rn−df∗Q) ≅ ⊕
T∈T
IC(T, Rn−df∗′Q) ≅ Rf∗′Q[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.