X=PPr(O(−1)⊕(r+1)⊕O).
The cohomology (Chow ring) is given by
H∗(X) =A∗(X) =Z[h, ξ]/(hr+1,(ξ−h)r+1ξ).
Since c1(X) = (r+2)ξis semi-positive, X is a semi-Fano toric variety.
The toric fan4(X)of X is given by one dimensional edges w0, . . . , wr+1, v0, . . . , vr∈Zr+(r+1)
such that
w0+w1+ · · · +wr+1=0, v0+ · · · +vr=w0+ · · · +wr = −wr+1. Let{ei}i=0,...,r−1and{e0i}i=0,...,rbe the basis ofZr×Zr+1. Then we may pick
wi = e0i, 0≤i≤ r; wr+1= −e00− · · · −e0r; vi =ei+ei0, 0≤i≤r−1; vr= −e0− · · · −er−1+er0. This implies the following linear equivalence of toric divisors
Dv0 =Dv1 = · · · =Dvr =: h; ξ := Dwr+1 = Dwi+Dvi, i=0, . . . , r.
Thus Dwi =ξ−h for all i=0, . . . , r.
Remark 5.1. In terms of the homogeneous coordinate rings, X is defined by an embedding of(C∗)2 ,→ (C∗)2r+1, which is defined by the 2× (2r+1) matrix M : Lie(C∗)2r+1→Lie(C∗)2
M =1 . . . 1 −1 . . . −1 0 0 . . . 0 1 . . . 1 1
,
where on the first row, there are r 1’s, r(−1)’s. The K¨ahler cone is spanned by h and ξ on H2(X) ∼=C2.
We start with one-point descendent invariants. The toric data allows us to apply the known results of [4] [14] directly. Let
Pβ :=
∏
ρ∈41(X)∏
0m=−∞(Dρ+mz)∏
ρ∈41(X)∏
(mβ.D=−ρ)∞(Dρ+mz) . Lemma 5.2. For an effective curve class β= d1` +d2γ,JX(β, z−1) =
∏
0 m=−∞(ξ−h+mz)r+1
d1
m
∏
=1(h+mz)r+1
d2−d1
m
∏
=−∞(ξ−h+mz)r+1
d2
m
∏
=1(ξ+mz) .
Proof. Since(β.h) = d1 and(β.ξ) = d2, the right hand side is precisely Pβ. JX(β, z−1)is equal to Pβ without change of variables (“mirror transforma-tion”) due to the uniqueness theorem and the fact that Pβ =O(1/z2)in 1/z power series expansion. Indeed if d1≤d2,
Pβ = 1
(d1!)r+1((d2−d1)!)r+1d2! 1
zd2(r+2) + · · · ,
while if d1>d2(the key observation), Pβ = (ξ−h)r+1((d1−d2−1)!)r+1
(d1!)r+1d2! (−1)d1−d2−1 1
zd2(r+2)+r+1 + · · ·.
For more details see [4] [14].
It also follows that a presentation of the small quantum cohomology ring is given by Batyrev’s quantum ring (cf. [2], the proof of Proposition 11.2.17).
Namely for q1= q`and q2=qγ,
QH∗(X) =C[h, ξ][[q1, q2]]/(hr+1−q1(ξ−h)r+1,(ξ−h)r+1ξ−q2). Though the presentation does not provide enough information for our pur-pose, it does give a first test of the invariance property.
Proposition 5.3. The map FX : QH∗(X)[q−11] → QH∗(X0)[q10−1] defined by FXh = ξ0 −h0, FXξ = ξ0, FXq1 = q0−1 1 andFXq2 = q01q02 extends to a ring isomorphism.
Proof. SinceFX0◦FX =IdX, it is enough to check that the generators of the ideal are mapped into the corresponding ideal in the X0side:
FX(hr+1−q1(ξ−h)r+1) = (ξ0−h0)r+1−q0−1 1h0r+1
= −q0−1 1(h0r+1−q10(ξ0−h0)r+1); FX((ξ−h)r+1ξ−q2) =h0r+1ξ0−q01q02
= (h0r+1−q01(ξ0−h0)r+1)ξ0+q10((ξ0−h0)r+1ξ0−q02).
Note that the virtual dimension of an n-point invariants in degree β = d1` +d2γis given by Dn,β = (r+2)d2+2r+n−2, so for a fixed set of cohomology insertions there could be at most one d2supporting non-trivial invariants and for the corresponding n-point function the summation over d2is unnecessary.
Lemma 5.4 (Quasi-linearity). Let JX := JX(q, z−1). For any α ∈ H∗(X), the one point functionhτkξαiXsatisfies the functional equation (without analytic continuation):
Fhτkξ.αiX =hτkF(ξ.α)iX0 =τkξ0.Fα X0. Equivalently,F is linear in Jξ:
F(JXξ.α) = JX0F(ξ.α) =JX0ξ0.Fα.
Proof. The key observation on Pβ is that if d2−d1 < 0 then the middle factor in the denominator of Pβ goes to the numerator instead which has a factor(ξ−h)r+1. Thus it vanishes after multiplication by ξ. Notice that the condition d2 ≥ d1 simply corresponds to the effectivity ofFβ = −d1`0 + d2(γ0+ `0) = (d2−d1)`0+d2γ0.
Since JX =∑β∈NE(X)qβPβ, by the above observation JXξ.α can be written
Notice that since the flop is an isomorphism outside Z = Pr ⊂ X, the cohomology correspondenceF is the “identity” one on classes ξ.α. Namely Fhi = (ξ0−h0)i for i ≤ r andF(ξ.α) =Fξ.Fα = ξ0.Fα for any α ∈ H∗(X), arrive at the corresponding expression of JX0ξ0.Fα.
Since for given insertion(s) there could be at most one d2 supporting non-trivial invariants, we find thathτkξαiis a finite sum andFhτkξ.αi = hτkξ0.Fαiholds without the need of analytic continuation. 5.2. The functional equations in general. Write β = d1` +d2γ. If d2 = 0, the whole setting on Gromov-Witten invariants goes back to quantum corrections attached to the extremal rayZ`. In §3 we had seen that while n-point functions with n ≥3 satisfy the functional equation underF up to analytic continuation, it is not the case for n = 2 or descendent invariants with n−3−k <0.
The results in §2 and the quasi-linearity lemma are the induction basis of our discussion on functional equations up to analytic continuation. In-troduce the notation A ∼= B for the two series A and B when they can be
The following lemma formalizes the argument in the proof of Corollary 3.2:
Lemma 5.5. The differential operator dH is F equivariant. That is, F◦dH = dFH◦F. In particular, if Fhαi ∼= hFαithenFdHhαi ∼=dFHhFαitoo.
Proof. This follows from the fact thatF preserves the Poincar´e pairing. In explicit terms, denote by (x, y) = (q`, qγ)and (x0, y0) = (q`0, qγ0)
Hence
F◦dH = (FH.F`)−x0 ∂
∂x0 +y0 ∂
∂y0
+ (FH.Fγ)y0 ∂
∂y0
= (FH.`0)x0 ∂
∂x0 + (FH, F(γ+ `))y0 ∂
∂y0 =dFH◦F.
IfFhαi ∼= hFαithenFdHhαi =dFHFhαi ∼=dFHhFαi. Theorem 5.6. Lethαi = hα1, . . . , αniwith αi ∈ H∗(X) ∪τ•H∗(E). If d2 6= 0 then
Fhαi ∼= hFαi.
Proof. We will prove the theorem by induction on d2 and then n. This is based on the following observations: (1) By the virtual dimension count, each set of insertions can support at most one d2. (2) Under divisor relations the degree β is either preserved or split into effective classes β= β1+β2, so d2is split accordingly as d2=dL2+dR2. (3) When summing over β∈ NE(X), the splitting terms can usually be written as the product of two generating series with no more marked points in a manner which will be clear in each context during the proof.
For d2 = 0, since ξ|Z =0 we get trivial invariant if one of the insertions involves ξ. Hence by §3 the statement in the theorem holds for d2 = 0 except for the unique casehhr, hri. In this case, by the divisor axiom
dhhhr, hri = hh, hr, hri,
which satisfies the functional equation up to analytic continuation, as we had shown before through explicit formulae incorporated with classical de-fect. Thus we may base our induction on d2 = 0 with special care on this case.
Let d2 ≥1. The case n =1 is contained in Theorem 5.4, so let n ≥2. We may and will make one more assumption that ξ appears in some αi. If not, then there will be no descendent insertions and we may write
hα1, . . . , αni = hα1, . . . , αn, ξi/d2
by the divisor axiom. In the following reduction steps, each term will either have smaller d2or with this condition being preserved.
By reordering we may assume that αn = τsξ a, s ≥ 0. Write α1 = τkhlξj. The induction procedure is to move divisors in α1 into αn in the order of ψ, h and ξ. That is we use induction on the following five numbers in the alphabetical order:
(d2, n, k, l, j).
For ψ we use equation ψ1 = −ψn+ [D1|n]virt. If k ≥1 then j6=0 and we get
hτkhlξj, . . . , τsξ ai = −hτk−1hlξj, . . . , τs+1ξ ai +
∑
i
hτk−1hlξj, . . . , TiihTi, . . . , τsξ ai.
For each i, if one of dL2 and dR2 is zero then since both terms contain ξ classes the splitting term must vanish. So we may assume that dL2 <d2and dR2 <d2 and these terms are done by the induction hypothesis. By performing this procedure to α1, . . . , αn−1 we may assume that the only descendent inser-tion is αn.
For h, if l ≥2 or l=1 but j6=0 we use the divisor equation to get hhlξj, . . . , τsξ ai = hhl−1ξj, . . . , τsξ ahi +dhhhl−1ξj, . . . , τs+1ξ ai
−
∑
i
dhhhl−1ξj, . . . , TiihTi, . . . , τsξ ai.
The only case for the splitting term to have one factor to have the same d2and n is of the form
dhhhl−1ξj, TiihTi, α2, . . . , αn−1, τsξ ai,
where the two-point invariant has dL2 = 0. But then l−1 < r forces it to vanish. We remark here that the case dL2 = 0 may still support nontrivial invariants with three or more points if j=0.
By induction (and Lemma 5.5) we are left with the case α1 = h. The divisor axiom implies that
hh, . . . , τsξ ai =dhh. . . , τsξ ai + h. . . , τs−1ξ ahi.
Since both terms have one less marked points, they are done by induction.
For ξ, the argument is entirely similar. For j≥2, the divisor relation says that
hξj, . . . , τsξ ai = hξj−1, . . . , τsξ2ai +dξhξj−1, . . . , τs+1ξ ai
−
∑
i
dξhξj−1, . . . , TiihTi, . . . , τsξ ai. We then have dL2 < d2and d2R<d2as before. If j=1 we get hξ, . . . , τsξ ai =dξh. . . , τsξ ai + h. . . , τs−1ξ2ai
and both terms have fewer marked points. The proof is complete. Practically the above inductive procedure leads to explicit determination of GW invariants, though the computations are somewhat tedious. For the interested readers, we list the results for the two typical series of examples of the local model of simpleP2flop.
Example 5.7. SimpleP2flop with d2 =1, n=3. The virtual dimension is 9.
Then on X (q1= q`, q2=qγ), hh2, h2, h2ξ3i = q
2 1
1+q1q2, hξ2, ξ2, h2ξ3i = (1+q1)q2,
hhξ, hξ, h2ξ3i = hhξ, ξ2, h2ξ3i = hhξ, h2, h2ξ3i = hξ2, h2, h2ξ3i =q1q2.
Similar formulae hold on X0. We compute (q01 =q`0, q02=qγ0) Fhh2, h2, h2ξ3i = q
0−2 1
1+q1−1q01q02= 1 1+q01q02;
hFh2,Fh2,Fh2ξ3i = h(ξ0−h0)2,(ξ0−h0)2,F[pt]i
= hξ02, ξ02,[pt]i +4hξ0h0, ξ0h0,[pt]i + hh02, h02,[pt]i
−4hξ0h0, h02,[pt]i −4hξ0h0, ξ02,[pt]i +2hξ02, h02,[pt]i
=(1+q10) +4q10 + q
02 1
1+q01 −4q01−4q01+2q01 q02
=1−q10 + q
02 1
1+q01
q02= 1 1+q01q02.
ThusFhh2, h2, h2ξ3i ∼= hFh2,Fh2,Fh2ξ3i. We leave the simpler verifications on the other five cases to the readers.
Example 5.8. Descendent invariants for simple P2 flop with d2 = 1 and n=3.
hh2, h2, τ4ξi =3q1q2−6 q1q2
1+q1, hξ2, ξ2, τ4ξi =9q2+9q1q2, hhξ, hξ, τ4ξi = hh2, ξ2, τ4ξi =3q2,
hhξ, h2, τ4ξi =0, hhξ, ξ2, τ4ξi =6q2+3q1q2. We omit the elementary verifications on functional equations.
6. MUKAI FLOPS
6.1. Generalized Mukai flops. Consider a flopping contraction of Mukai type. Namely ψ : (X, Z) → (X, S¯ ) with NZ/X = TZ/S∗ ⊗ψ¯∗L for some twisting line bundle L ∈ Pic(S). To construct the flop, as in the case of ordinary flops, it is natural to consider the blow-up φ : Y=BlZX→X and try to contract the exceptional set E in another fiber direction.
When S is a point, it is well known that E is the degree(1, 1) hypersur-face inPr×Pr. Though the general case is merely a fibered version of this simple model, it is even simpler to study it within the framework of ordi-nary flops. Indeed, we will construct the local model of it as a slice of the ordinary flop with F0 =F∗⊗L.
We start with an arbitrary pair (F, F0)of vector bundles of rank r and denote the corresponding maps in the ordinary Pr flop by Φ : Y → X, Φ0 : Y → X0, Ψ : X → X and Ψ¯ 0 : X0 → X. Also let g¯ = Ψ◦Φ = Ψ0◦Φ0. The restriction maps on the exceptional sets are denoted by ¯φ, ¯φ0, ¯ψ, ¯ψ0and
¯g respectively.
E=PS(F) ×SPS(F0) ⊂Y
Φ
ttjjjjjjjjjjjjjjjj
Φ0
**UU UU UU UU UU UU UU UU
g
Z=PS(F) ⊂X
ΨUUUUUUUUU **U UU
UU UU
UU Z0=PS(F0) ⊂X0
Ψ0
ttiiiiiiiiiiiiiiiiii S⊂X¯
First suppose that there exists a non-degenerate bilinear map F×SF0 →ηS
with ηS ∈Pic(S). (This happens precisely when F0 ∼= L∗⊗ηSfor some line bundle ηS.) The mapOP(F)(−1) →ψ¯∗F pulls back to ¯φ∗OP(F)(−1) → ¯g∗F, hence leads to a natural map
OE(−1,−1):=φ¯∗OZ(−1) ⊗Eφ¯0∗OZ0(−1) → ¯g∗(F⊗SF0) → ¯g∗ηS. Notice that the normal bundle NE/YequalsOE(−1,−1). That is,Y is the total space ofOE(−1,−1). Let p : NE/Y →E be the projection map.
We describe two equivalent ways to construct the space Y. The above linear map between line bundles induces a surjective map of invertible sheaves which fits into an exact sequence of the form
0→ NE/Y(−E) → NE/Y→ ¯g∗ηS →0
for an effective divisor E ⊂E. We then take Y = p−1(E) ⊂Y to be the col-lection of lines with origins in E. Alternatively Y is simply the irreducible component of the inverse image of the zero section of ¯g∗ηSinY other than the zero sectionE.
Let X = Φ(Y) ⊃ Z, X0 = Φ0(Y) ⊃ Z0, ¯X = g(Y) ⊃ S with restriction maps φ, φ0, ψ, ψ0. By tensoring the Euler sequence
0→OZ(−1) →ψ¯∗F→ Q →0
withS∗ =OZ(1)and noticing thatS∗⊗ Q ∼= TZ/S, we get by duality 0→ TZ/S∗ →OZ(−1) ⊗ψ¯∗F∗ →OZ →0.
The inclusion maps Z,→X,→X leads to
0→ NZ/X → NZ/X →NX/X|Z →0.
Here NX/X|Z = O(X)|Z = ψ¯∗O(X¯)|S. Denote O(X¯)|S by L. Recall that NZ/X ∼=OPS(F)(−1) ⊗ψ¯∗F0. By tensoring with ¯ψ∗L∗, we get
0→ NZ/X⊗ψ¯∗L∗ →OP
S(F)(−1) ⊗ψ¯∗(F0⊗L∗) →OZ →0.
So F0 = F∗⊗L if and only if NZ/X ∼= TZ/S∗ ⊗ψ¯∗L.
This is the case for generalized Mukai flops. We have then ηS∼= L.
6.2. Mukai flops as limits of isomorphisms. For Mukai flops, namely L∼= OS, we have F0 = F∗ with duality pairing F×SF∗ →OS.
Consider π :Y→C via
Y→ ¯g∗OS=OE∼=E×C−→π2 C.
We get a fibration withYt := π−1(t), which is smooth for t 6= 0 andY0 = Y∪E. The intersection E=Y∩E restricts to the degree(1, 1)hypersurface over each fiber ofE→S. Indeed the equation for π in coordinates reads as
t=
∑
r i=0xiyi.
From this, we also have thatYt ∼=E\E for all t6=0 under the projection p.
LetXt,X0tand ¯Xtbe the proper transforms ofYtinX, X0and ¯X. For t6=0, all maps in the diagram
Yt
~~}}}}}}}}
A A AA AA AA Xt
A A AA AA
AA X0t
~~}}}}}}} X¯t
are isomorphisms. For t=0 this is the Mukai flop. Thus local Mukai flops are limits of isomorphisms. More precisely, we have
Theorem 6.1. There is a projective compactification bY → P1which deforms the projectivized local model of Mukai flop
Xb0 =PZ(TZ/S∗ ⊕O) 99KPZ0(TZ∗0/S⊕O) =Xb00 into isomorphisms bXt ∼=Xb0t∼=E for all t 6=0.
Moreover, bY→P1is the blow-up ofE×P1along E× {0}, the deformation to normal cone of the pair(E, E)with E being the relative(1, 1)divisor of
E=PS(F) ×SPS(F∗)
over S. bX0, bX00and b¯X0 are the contractions ofE ⊂ bY0along the two rulings and the double ruling respectively.
Proof. We first consider the compactified normal bundle Y¯ =PE(O(−1,−1) ⊕O) 99KP1
which extends the map π by sending the infinity divisorE∞ ∼=E to ∞∈P1.
It is clear that E∞ := Y¯ ∩E∞is the “axis” where π is not defined. Indeed E∞ is the boundary divisor of everyYt. Thus the blow-up
bY :=BlE∞Y¯
ˆ π
J$$J JJ JJ JJ JJ
Y¯ _ _ _π_ //_P1
resolves the indeterminacy to get a morphism ˆπ : bY → P1. bYt is the com-pactification ofYtby adding E at infinity, hence bYt ∼=E for all t6=0.
We then have a compactified diagram as expected:
bYt
~~~~~~~
@ @
@@
@@
@
Xbt
? ?
??
??
?? Xb0t
b¯ Xt
For t = 0, by the very construction of Mukai flops from the ordinary flops, we havePZ(TZ/X∗ ) ∼= E. So the compactification bX0 coincides with PZ(TZ/S∗ ⊕O). Similarly bX00 ∼=PZ0(TZ∗0/S⊕O).
For the second statement, again by our construction bY0 = E∪Y with bb Y the total space of theP1bundle
PE(OE(−1,−1) ⊕O) ∼=PE(O⊕OE(1, 1)).
This is precisely the exceptional divisor coming from the blow-up eY=BlE×{0}E×P1.
The theorem follows from an easy comparison of eY withY.b In particular all interesting invariants which are continuous under de-formations are preserved. For example, the diffeomorphism type, Hodge type and quantum cohomology rings etc.. To be more precise, since the fiber product satisfies the base change property and for ordinary flops the fiber product equals the graph closure, the canonical isomorphism of Chow motives of projective local models of Mukai flops f : X 99KX0 is clearly to be induced by the correspondence[X×X¯ X0], which is the t=0 fiber of the graph ofX99KX0:
F := [X×X¯ X0] = [Γ¯f] + [E] ∈ A∗(X×X0)
where[Γ¯f] ≡ Y := BlZX = BlZ0X0. For global (generalized) Mukai flops we also consider the fiber product as the proposed correspondenceF.
The quantum cohomologies are not just isomorphic, in fact all quantum corrections attached to the extremal ray are zero: If not, then the deforma-tion invariance of Gromov-Witten invariants implies that some extremal curve class d` ∈ NE(X)survives as an effective curve in a nearby fiber as Ct ⊂Xt∼=X0t, then the class
[Ct0] =Ft[Ct] ∼Fd` = −d`0.
is both effective and anti-effective on X0, which is a contradiction. (For simple Mukai flops, the invariants on d`are zero have also been proved by Hu and Zhang [5] by direct computation via localizations.)
For a global Mukai flop, the local deformation equivalence may fail to extend to a global deformation equivalence since there are in general ob-structions to extend deformations from local to global. (For hyper-K¨ahler manifolds or more generally Calabi-Yau manifolds such global deforma-tions do exists.) Nevertheless, together with the degeneration analysis, the local deformation equivalence do lead to global results:
Theorem 6.2. For any Mukai flop f : X 99K X0 (not necessarily being simple), X is diffeomorphic to X0 and both have isomorphic Chow motives, Hodge struc-tures and full Gromov–Witten theory (in all genera) under the correspondenceF.
Moreover, all quantum corrections attached to the extremal ray vanish.
Proof. The diffeomorphism is obtained by patching the local deformation equivalence and the identity map on X\Z∼= X0\Z0.
For Chow motives, we investigate the induced mapping on Chow groups as in §2. For any T, idT× f : T×X 99KT×X0 is also a Mukai flop, with base S being replaced by T×S. Since the correspondence F is compati-ble with base change, to prove thatF∗◦F = ∆X, by the identity principle, we only need to show that F∗F = id on A∗(X)for any Mukai flop. Let p : X×X0 →X and p0 : X×X0 →X0be the projections. From
FW = p0∗(([Γ¯f] + [E]).p∗W)
and the property of intersection product we see that theF∗◦F= ∆X is re-ally a local statement which depends only on the normal bundles NZ/Xand NZ0/X. Thus the identity follows from the case of local models. Similarly F◦F∗ =∆X0. SoF induces an isomorphism on Chow motives of X and X0. The Hodge realizations leads to equivalence of Hodge structures.
Now we treat the Gromov-Witten invariants. As in the case of ordinary flops, we consider deformation to normal cone W →A1of X and W0 →A1 of X0respectively. W0 =Y∪Xlocwith Y=BlZX and Xloc =PZ(TZ/S∗ ⊕O). Similarly W00 = Y0∪Xloc0 with Y0 = BlZ0X0 and Xloc0 = PZ0(TZ∗0/S⊕O). By definition Y = Y0 and we have the induced Mukai flop for local models
f : Xloc 99KXloc0 .
By the degeneration formula, any Gromov-Witten invarianthαiXg,n,βsplits into sum of products of relative invariants of(Y, E)and(Xloc, E). Now we compare it with the similar splitting ofhFαiXg,n,βinto(Y, E)and(X0loc, E).
Notice that most of the setting on degeneration analysis in §4 is still valid in the Mukai case. In particular, the cohomology reduction lemma (Lemma 4.4) works in the Mukai case too.
In fact, the situation now is very simple. We match the relative invari-ants on(Y, E)from both sides and then we need to compare only the cases (Xloc, E)and(Xloc0 , E). But they are deformation equivalent while the de-formations leave the boundary divisor E unchanged. By the deformation invariance of (relative) Gromov-Witten theory and the fact thatF is induced from this deformation, we find that the relative invariants are also the same on this part. Hence we have proved
hαiXg,n,β= hFαiXg,n,Fβ0
for any g, n, β including descendent invariants. Namely the full GW theory on X and X0 are equivalent.
The statement on vanishing of GW invariants of extremal rays follows from the previous discussion. The proof is now complete. Remark 6.3. Instead of using deformation invariance of relative GW the-ory, we may also proceed in the same way as the case of ordinary flops, at least for simple Mukai flops. By Proposition 4.6, the equivalence problem is reduced to the case of absolute invariants and then we may use the de-formation invariance of absolute GW theory to conclude. Indeed the defor-mation invariance of relative GW theory can be deduced form the absolute case and the result in [17].
Remark 6.4. For generalized Mukai flops with non-trivial twisting line bun-dle L, we take F0 = F∗⊗L and ηS := L. The pairing F×SF0 →ηSis simply F×S(F∗⊗L) →L. Since
φ¯0∗OP(F∗⊗L)(−1) =φ¯0∗(OP(F∗)(−1) ⊗ψ¯0∗L) =φ¯0∗OP(F∗)(−1) ⊗ ¯g∗L, the linear mapY → ¯g∗L is obtained by tensoring the corresponding map for Mukai flops with ¯g∗L. The inverse image of the zero section gives Y∪E.
Again the proper transforms of Y in various spaces give rise to the gener-alized Mukai flop. The difference is that since ¯g∗L is not a trivial bundle, we do not have a fibration structureY → C as before. But we still get the equivalence of Chow motives via the fiber product.
Example 6.5. To see how the extra component corrects the graph closure, we shall carry out the detailed computations for the case of simple Mukai flops. So Z ∼= Pr, NZ/X = TZ∗ and E ⊂ Pr×Pr is the universal family of lines in Pr from both sides, namely, it is the hypersurface of bi-degree (1, 1). By weak Lefschetz, H2(E) =Pic E = Zx|E⊕Zy|E with x and y the generators of PicPr×Pras pull backs of h and h0. As in the ordinary case, NE/Y =OE(−1,−1):=φ¯∗OZ(−1) ⊗φ¯0∗OZ0(−1).
Let F0 = [Γ¯f]. The argument to compute F0 as in the ordinary case fails precisely when α ∈ Ar(X), so we would like to find F0[Z]. Since
φ∗[Z] = j∗(cr−1(E)), withE defined by 0→ NE/Y →φ¯∗NZ/X →E→0, we get
cr−1(E) = (1−x)r+1(1− (x+y))−1|E(r−1)
= (x+y)r−1−Cr1+1x(x+y)r−2+ · · · + (−1)r−1Crr+−11xr−1 E
= (yr−1−2yr−2x+3yr−3x2+ · · · + (−1)r−1rxr−1)|E. So
F0[Z] =φ0∗φ∗[Z] = (−1)r−1r[Z0],
which implies thatF0induces isomorphism on cohomologies overQ, but not overZ.
For 0< s≤r, since E∼x+y, we have φ∗hs= j∗(cr−1(E). ¯φ∗hs)
= (yr−1−2yr−2x+3yr−3x2+ · · · + (−1)r−1rxr−1)|E.xs
= (yr−yr−1H+ · · · + (−1)r−1yxr−1+ (−1)r(1−r)xr)xs
=xsyr−xs+1yr−1+ · · · + (−1)r−sxrys.
By symmetry this implies thatF0(hs) = (−1)r−sh0swhen s6=0.
LetF= [X×X¯ X0] =F0+F1withF1 = Z×SZ0 = [Pr×Pr]. We claim thatF1[Z] = (−1)r(r+1)[Z0]andF1hs=0 for s6=0. Indeed,
F1[Z] =p0∗(p−1[Z].[Z×Z0])
with p (resp. p0) the projection of X×X0 to X (resp. X0). Then Z2 =cr(NZ/X) =cr(TZ∗) = (−1)rχ(Pr) = (−1)r(r+1).
SoF1[Z] = p0∗([Z×X0].[Z×Z0]) = (−1)r(r+1)[Z0]. ForF1hs, notice that we may choose W ∼ Z with W∩hs = ∅. Hence F1hs= p0∗([hs×X0].[W× Z0]) =0.
ThusF(hs) = (−1)r−sh0s for 0 ≤ s ≤ r andF induces integral isomor-phisms.
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DEPARTMENT OFMATHEMATICS, UNIVERSITY OFUTAH, SALTLAKECITY, UTAH84103, U.S.A.
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DEPARTMENT OFMATHEMATICS, NATIONALCENTRALUNIVERSITY, CHUNG-LI, TAI
-WAN.
E-mail address: [email protected]
DEPARTMENT OFMATHEMATICS, NATIONALCENTRALUNIVERSITY, CHUNG-LI, TAI
-WAN., NATIONALCENTER FORTHEORETICSCIENCES, HSINCHU, TAIWAN. E-mail address: [email protected]
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