UNDER
STRATIFIED MUKAI FLOPS
BAOHUA FU AND CHIN-LUNG WANG
ABSTRACT. For stratified Mukai flops of type An,k, D2k+1and E6,I, it is shown the fiber product induces isomorphisms on Chow motives.
In contrast to (standard) Mukai flops, the cup product is generally not preserved. For An,2, D5 and E6,I flops, quantum corrections are found through degeneration/deformation to ordinary flops.
1. INTRODUCTION
1.1. Backgroumd. Two smooth projective varieties overC are K equivalent if there are birational morphisms φ : Y → X and φ0 : Y → X0 such that φ∗KX = φ0∗KX0. This basic equivalence relation had caught considerable
attention in recent years through its appearance in minimal model theory, crepant resolutions, as well as other related fields.
The conjectural behavior of K equivalence has been formulated in [W]. A canonical correspondenceF ∈ A∗(X×X0)should exist and gives an iso-morphism of Chow motives[X] ∼= [X0]. Under F, X and X0 should have isomorphic B-models (complex moduli with Hodge theory on it) as well as A-models (quantum cohomology ring up to analytic continuations over the extended K¨ahler moduli space).
Basic examples of K equivalence are flops (with exceptional loci Z, Z0, S)
(X, Z) ψ $$I I I I I I I I I f // (X0, Z0) ψ0 zztttttt ttt (X, S¯ ) .
Among them the ordinary flops had been studied in [LLW1] where the equiv-alence of motives and A-models was proved. In that caseF is the graph closureF0 := Γ¯f. In general,F must contain degenerate correspondences.
The typical examples are Mukai flops. They had been extensively studied in the literature in hyper-K¨ahler geometry. Over a general base S, they had also been studied in [LLW1], where the invariance of Gromov-Witten theory was proved. In that caseF=X×X¯ X0 =F0+F1withF1= Z×SZ0.
We expect that for flopsF should be basically X×X¯ X0. 1
To understand the general picture we are led to study flops withF con-sisting of many components. The stratified Mukai flops provide such ex-amples. They appear naturally in the study of symplectic resolutions [Fu] [Na2] and they should play important roles in higher dimensional bira-tional geometry. For hyper-K¨ahler manifolds, see for example [Mar].
In this paper, we study general stratified Mukai flops without any as-sumptions on the global structure of X and X0. By way of it, we hope to develop tools with perspective on future studies.
1.2. Stratified Mukai flops. Fix two natural numbers n, k such that 2k <
n+1. Consider two smooth projective varieties X and X0. Let Fk ⊂ Fk−1 ⊂ · · ·F1 ⊂X and Fk0 ⊂ Fk0−1 ⊂ · · ·F10 ⊂ X0 be two collections of closed
subva-rieties. Assume that there exist two birational morphisms X −→ψ X¯ ψ0
←− X0. The induced birational map f : X99KX0 is called a (stratified) Mukai flop of type An,k over ¯X if the following conditions are satisfied:
(i) The map f induces an isomorphism X\F1
∼
−→X0\F10; (ii) ψ(Fj) =ψ0(Fj0) =: Sjfor 1≤ j≤ k;
(iii) Sk is smooth and there exists a vector bundle V of rank n+1 over
it such that Fk is isomorphic to the relative Grassmanian GSk(k, V) of k-planes over Sk and the restriction ψ|Fk : Fk → Sk is the natural projection. Furthermore, the normal bundle NFk/X is isomorphic to the relative cotangent bundle TF∗
k/Sk. The analogue property holds for Fk0and ψ0 with V replaced by its dual V∗;
(iv) If k = 1, we require that f is a usual Mukai flop along Fk. When k ≥ 2, let Y (resp. Y0, ¯Y) be the blow-up of X (resp. X0, ¯X) along Fk
(resp. Fk0, Sk). By the universal property of the blow-ups, we obtain
morphisms Y → Y¯ ← Y0. The proper transforms of Fj, Fj0 give
collections of subvarieties on Y, Y0. We require that the birational map Y99KY0is a Mukai flop of type An−2,k−1.
We define a Mukai flop of type D2k+1 in a similar way with the following
changes: (1) one requires that Skis simply connected; (2) the vector bundle V is of rank 4k+2 with a fiber-wise non-degenerate symmetric 2-form. Then the relative Grassmanians of k-dimensional isotropic subspaces of V over Sk has two components G+iso and Giso−. We require that Fk (resp. Fk0) is
isomorphic to Giso+ (resp. Giso−); (3) when k=1, f is a usual Mukai flop. Similarly one can define a Mukai flop of type E6,Iby taking k = 2 with V
being an E6-vector bundle of rank 27 over S2 and F2 is the relative E6/P1
-bundle over S2inP(V). The dual variety F20 is given by the relative E6/P6
-bundle inP(V∗), where P1, P6are maximal standard parabolic subgroup in
E6corresponding to the simple roots α1, α6respectively. By [CF], when we
blow up the smallest strata of the flop, we obtain a usual Mukai flop. 1.3. Main results. Our main objective of this work is to prove the follow-ing theorems.
Theorem 1.1. Let f : X99KX0 be a Mukai flop of type An,k, D2k+1or E6,I over
¯
X. LetF be the correspondence X×X¯ X0. Then X and X0 have isomorphic Chow
motives underF. Moreover F preserves the Poincar´e pairing of cohomology. Note that the flops of type An,1 and D3, i.e. k = 1, are the usual Mukai
flops, and in these cases the theorem has been proven in [LLW1]. Our proof uses an induction on k (for all n) via (iv). We shall give details of the proof for An,k flops, while omitting the proof of the other two types, since the
argument is essentially the same.
For k=1 (i.e. the usual Mukai flops), the cohomology ring as well as the Gromov-Witten theory are also invariant under F [LLW1]. However, the general situation is more subtle:
Theorem 1.2. When k ≥ 2, the cup product is generally not preserved underF. For An,2, D5and E6,Iflops the defect is corrected by the genus zero Gromov-Witten
invariants attached to the extremal ray, up to analytic continuations.
While Theorem 1.1 is as expected, Theorem 1.2 is somehow surprising, since stratified Mukai flops are in some sense locally (holomorphically) symplectic and it is somehow expected that there is no quantum correc-tions for flops of these types. Indeed stratified Mukai flops among hyper-K¨ahler manifolds can always be deformed into isomorphisms [Huy] hence there is no quantum correction. As it turns out, the key point is that for the projective local models of general stratified Mukai flops, in contrast to the case k=1, we cannot deform them into isomorphisms!
1.4. Outline of the contents. In Section 2, the existence of An,kflops in the
projective category is proved via the cone theorem. In Section 3, a general criterion on equivalence of Chow motives via graph closure is established for strictly semi-small flops. While a given flop may not be so, generic deforma-tions of it may sometimes do. When this works, we then restrict the graph closure of the one parameter deformation back to the central fiber to get the correspondence, which is necessarily the fiber product.
It is thus crucial to study deformations of flops. Global deformations are usually obstructed, so instead we study in Section 4 the deformations of pro-jective local models of An,k flops. While open local models can be deformed into isomorphisms, the projective local models cannot be deformed into isomorphisms in general but only be deformed into certain A∗n−2,k−1flops. These flops, which we called stratified ordinary flops, do not seem to be stud-ied before in the literature. Nevertheless this deformation is good enough for applying the equivalence criterion of motives.
To handle global situations, we consider degenerations to the normal cone to reduce problems on An,kflops to problems on An−2,k−1flops and on local
models of An,k flops. This is carried out for correspondences in Section 5.
This makes inductive argument work since local models are already well handled. We also carry out this for cup product by proving an orthogonal
decomposition under degenerations to the normal cone. This in particular applies to the Poincar´e pairing and completes the proof of Theorem 1.1.
In section 6 we prove Theorem 1.2 for An,2flops. We apply the
degenera-tion formula for GW invariants [LR] [Li] to splits the absolute GW invariants into the relative ones. After degenerations, the flop is split into two simpler flops, one is a Mukai flop and another one can be deformed into ordinary Pn−2 flop. It turns out that each GW invariant attached to the extremal ray
must go to one of these two factors completely. For the former the extremal invariants indeed vanish. For the latter we use a recent result on ordinary flop with general base [LLW2] to achieve the quantum corrections up to analytic continuations. This then completes the proof.
At the end we compare Theorem 1.2 with the hyper-K¨ahler case, where the ring structure is preserved and there are no non-trivial Gromov-Witten invariants. When f is not standard Mukai, all these may fail without the global hyper-K¨ahler condition. A careful comparison of the degeneration analysis in this case with the local model case leads to some new topological constraint on hyper-K¨ahler manifolds (c.f. Proposition 6.4).
1.5. Acknowledgements. B. Fu is grateful to the Department of Mathe-matics, National Central University (Jhongli, Taiwan) for providing excel-lent environment which makes the collaboration possible.
C. L. Wang would like to thank the MATHPYL program of the F´ed´eration de Math´ematiques des Pays de Loire for the invitation to Nantes.
2. EXISTENCE OF(TWISTED) An,kFLOPS
Given k, n ∈ N, 2k < n+1, a flopping contraction ψ : (X, F) → (X, S¯ )
is of type An,k if it admits the following inductive structure: There is a
fil-tration F = F1 ⊃ · · · ⊃ Fk with induced filtration S = S1 ⊃ · · · ⊃ Sk,
Sj := ψ(Fj)such that ψSk : Fk
∼
= GSk(k, V) → Sk is a G(k, n+1)bundle for some vector bundle V →Sk of rank n+1 with
NFk/X
∼
= TF∗k/Sk⊗ψS∗kLk for some Lk ∈ Pic Sk.
Moreover, the blow-up maps φ, ¯φfit into a cartesian diagram Y=BlFkX⊃ E φ ¯ ψ ((R R R R R R R R R R R R R X ψ ))R R R R R R R R R R R R R R R R R R R Y¯ =BlS kX¯ ⊃ E¯ ¯ φ ¯ X
such that the induced contractions ¯ψ: (Y, ˜F) → (Y, ˜¯ S)with filtrations ˜F = ˜
F1 ⊃ · · · ⊃ F˜k−1, ˜Fj := φ∗−1(Fj), ˜S = S˜1 ⊃ · · · ⊃ S˜k−1, ˜Sj = φ¯−∗1(Sj),
contraction is an isomorphism. By definition, An,1contractions are twisted
Mukai contractions.
The main results of this paper are all concerned with the (untwisted) stratified Mukai flops, namely Lk ∼= OSk. The starting basic existence theo-rem of flops does however hold for the twisted case too.
Proposition 2.1. Given any An,k contraction ψ, the corresponding An,kflop (X, F) ψHHH$$H H H H H H f // (X0, F0) ψ0 zzuuuuuu uuu (X, S¯ )
exists with ψ0being an An,kcontraction.
Proof. We construct the flop by induction on k. The case k = 1 has been done in [LLW1], section 6, so we let k≥2 and n+1>2k. By induction we have a diagram Y φ ¯ ψ ? ? ? ? ? ? ? ? g // Y0 ¯ ψ0 ~~~~~~ ~~ ? φ0 X ψ ?? ? ? ? ? ? ? Y¯ ¯ φ X0 ¯ X
where g :(Y, ˜F) 99K (Y0, ˜F0)is an An−2,k−1flop and ¯ψ0 : (Y0, ˜F0) → (Y, ˜¯ S)is
an An−2,k−1contraction.
Let C⊂E be a φ-exceptional curve and C0 = g∗C be its proper transform
in E0 = g∗E. We shall construct a blow-down map φ0 : Y0 →X0 for C0. Let γ(resp. γ0) be the flopping curve for ¯ψ(resp. ¯ψ0).
Since the Poincar´e pairing is trivially preserved by the graph correspon-dence F0 of g in the divisor/curve level, and F0C = C0 +aγ0 for some
a ∈N (in fact a=1), we compute
(KY0.C0) = (KY0.F0C) = (KY.C) <0.
To show that C0is an Mori (negative) extremal curve, it is thus sufficient to find a supporting divisor for it.
Let ¯L be a supporting divisor for ¯C = ψ¯(C) in ¯Y. Then ¯ψ0∗¯L is a sup-porting divisor for the extremal face spanned by C0 and γ0. The idea is to perturb it to make it positive along γ0 while keeping it vanishing along C0. Let D be a supporting divisor for C in Y with λ := (D.γ) > 0. Let D0 =g∗D=F0D. SinceF0γ= −γ0, we compute
(D0.γ0) = −(D, γ) = −λ<0,
Let H0be a supporting divisor for γ0in Y0with c0 := (H0.C0) >0. Then W := aλH0−c0D0
has the property that(W.γ0) >0 and(W.C0) =0. Now for m large enough, the perturbation
L0 :=m ¯ψ0∗¯L+W
is a supporting divisor for C0. Indeed, L0takes the same values as W on γ0 and C0, while(L0.β0) > 0 for other curve classes β0 in Y0. That is, L0 is big and nef which vanishes precisely on the rayZ+[C0].
By the (relative) cone theorem applying to ¯φ◦ψ¯0 : Y0 →X, we complete¯ the diagram and achieve the flop f : X99KX0:
Y φ ¯ ψ ? ? ? ? ? ? ? ? g // Y0 ¯ ψ0 ~~~~~~ ~~ φ0 X ψ ?? ? ? ? ? ? ? Y¯ // ¯ φ X0 ψ0 ~~~~~~ ~~ ¯ X
It remains to show that the contraction ψ0 : X0 → X is of type A¯ n,k. By
construction, it amounts to analyze the local structure of Fk0 := φ0(E0). Since the flop f is unique and local with respect to ¯X, it is enough to determine its structure in a neighborhood of Sk. This can be achieved by explicit
con-structions.
Suppose that Fk = GSk(k, V). We consider the pair of spaces (X˜ 0, ˜F0
k)
defined by duality. Namely ˜Fk0 :=GSk(k, V
∗)and
˜
X0is the total space of TF∗˜
k/Sk⊗ψ ∗
SkLk.
It is well-known that, in a neighborhood of Sk, X 99K X00 is an An,k flop.
Thus the local structure of (X0, Fk0) must agree with(X˜0, ˜Fk0). The proof is
complete.
Remark 2.2. In the definition of An,k contractions, the restriction to
excep-tional divisors ¯ψ|E : (E, ˜F|E) → (E, ˜¯ S|E¯) is also an An−2,k−1 contraction.
Moreover, in the proposition the restriction
(E, ˜F|E) g|E // ¯ ψ|E %%K K K K K K K K K (E0, ˜F0|E0) ¯ ψ0|E0 yyrrrrrr rrrr (E, ˜¯ S|E¯) is also an An−2,k−1flop.
3. EQUIVALENCE CRITERIA OF MOTIVES
Let X −→ψ X¯ ψ0
←− X0 be two projective resolutions of a quasi-projective normal variety ¯X, and f : X 99KX0 the induced birational map. Consider the graph closureΓ of f and X←−φ Γ φ
0
−→X0the two graph projections. Then we obtain a morphism between Chow groups:
F := φ∗0φ∗: A∗(X) →A∗(X0). For any i, we will consider the closed subvariety
Ei = {x∈ X|dimxψ−1(ψ(x)) ≥i}.
In a similar way we define the subvariety E0ion X0. By Zariski’s main theo-rem, ψ is an isomorphism over X\E1, thus ψ(E1) =ψ0(E01) =X¯sing.
The following criterion generalizes the one for ordinary flops in [LLW1]: Proposition 3.1. If for any irreducible component D, D0of Eiand E0irespectively,
we have
2 codim D >codim ψ(D), and 2 codim D0 >codim ψ0(D0),
thenF is an isomorphism on Chow groups which preserves the Poincar´e pairing on cohomology groups.
Moreover, the correspondence[Γ]induces an isomorphism between Chow mo-tives:[X] ' [X0].
Proof. For any smooth T, f ×idT : X×T 99K X0×T is also a birational
map with the same condition. Thus by the identity principle we only need to prove the equivalence of Chow groups underF.
For any α ∈ Ak(X), up to replacing by an equivalent cycle, we may assume that α intersects E := ∑i≥1Ei properly. Then we have Fα = α0, where α0is the proper transform of α under f . If we denote by ˜α the proper transform of α in A∗(Γ), then we have
φ0∗α0 = ˜α+
∑
C
aCFC,
where FC are some irreducible k-dimensional subvariety inΓ and aC∈Z.
For any C, note that φ0(FC) is contained in the support of α0∩E10. As
ψ0(α0∩E01) = ψ(α) ∩X¯sing = ψ(α∩E1), FC is contained in φ−1ψ−1(BC),
where BC := ψφ(FC) ⊂ ψ(α∩E1). Take the largest i such that there exists
an irreducible component D of Ei with BC⊂ ψ(α∩D). For a general point
s ∈BC, we denote by FC,sits fiber by the map ψ◦φ. Then we have dim FC,s≥dim FC−dim BC ≥dim FC−dim(α∩D) =codim D. By our assumption, we have codim D > dim D−dim ψ(D), the latter being the dimension of a general fiber of ψ−1(BC) → BC. Thus the general
fiber of the map φ|FC has positive dimension, which gives that φ∗(FC) = 0.
This gives thatF0◦F = Id, where F0 = φ∗φ0∗. A similar argument then shows thatF◦F0 =Id, thusF and F0are isomorphisms.
Since FC has positive fiber dimension in both φ and φ0 directions, the
statement on Poincar´e pairing follows easily as in [LLW1], Corollary 2.3.
Now consider two (holomorphic) symplectic resolutions: X−→ψ X¯ ψ0
←−X0. A conjecture in [FN] asserts that ψ and ψ0 are deformation equivalent, i.e. there exist deformations of ψ and ψ0 over C: X −→Ψ X¯ Ψ
0
←− X0, such that for any t 6= 0, the morphisms Ψt,Ψ0t are isomorphisms. This conjecture
has been proved in various situations, such as nilpotent orbit closures of classical type [FN] [Na2], or when W is projective [Na1].
Assume this conjecture and consider the birational map F : X 99K X0. Recall that every symplectic resolution is automatically semi-small by the work of Kaledin [Ka] and Namikawa [Na1]. We obtain that the deformed resolutionsΨ and Ψ0 satisfy the condition of the precedent proposition. As a consequence, we obtain:
Theorem 3.2. Consider two symplectic resolutions X−→ψ X¯ ψ0
←−X0. Suppose that they are deformation equivalent (say given by F : X 99K X0). If we denote by Γ the graph of F and Γ0 its central fiber. Then the correspondence [Γ0]induces an
isomorphism of motives[X] ' [X0]which preserves also the Poincar´e pairing. 4. DEFORMATIONS OF LOCAL MODELS
From now on all the stratified Mukai flops are untwisted.
4.1. Deformations of open local models. Let S be a smooth variety and V → S a vector bundle of rank n+1. The relative Grassmanian bundle of k-planes in V is denoted by ψ : F := GS(k, V) → S. Let T be the universal
sub-bundle of rank k on F and Q the universal quotient bundle of rank n+
1−k. As is well-known, the relative cotangent bundle TF/S∗ is isomorphic to HomF(Q, T). Thus it is natural to construct deformations of TF/S∗ inside
the endomorphism bundle EndFψ∗V =ψ∗EndSV.
Consider the vector bundleE over F defined as follows: For x∈ F, Ex:= {(p, t) ∈End Vψ(x)×C|Im p⊂Tx, p|Tx =t IdTx}.
We have an inclusion TF/S,x∗ = Hom(Qx, Tx) → Ex which sends q ∈
Hom(Qx, Tx)to(˜q, 0) ∈Ex, where ˜q is the composition
˜q : Vψ(x) →Qx
q
−→Tx ,→Vψ(x).
The projection to the second factor π : E → C is then an one-dimensional
deformation of π−1(0) =TF/S∗ .
Equivalently, the Euler sequence 0→T →ψ∗V→Q→0 leads to 0→ TF/S∗ =HomF(Q, T) →HomF(ψ∗V, T) →EndFT→0.
The projection to the first factor, followed by ψ∗ E $$I I I I I I I I I I I //ψ∗EndSV ψ∗ // EndSV F ψ //S
gives rise to a map E → EndSV, which is a birational morphism onto its
image ¯E. Indeed, Ψ : E→E is isomorphic over the loci with rank p¯ =k. In particular, it is isomorphic outside π−1(0). For any s∈S,
¯
Es:= {p∈End Vs|rank p=k and p2 =tp for some t∈C}
is the cone of scaled projectors with rank at most k. Thus π=π¯ ◦Ψ, where ¯
π : ¯E→C via φ7→ 1 kTr φ.
For t6=0,Ψt :Et→∼ E¯t. For t=0, ψ := Ψ0 : TF/S∗ =E0 → E¯0is the open
local model of an An,kcontraction.
We do a similar construction for the dual bundle V∗ → S. Under the canonical isomorphism EndSV'EndSV∗, we see that ¯E is identified with
¯
E0 =E¯(V∗). Thus we get a birational map F :E99KE0over ¯E. This proves Proposition 4.1. The birational map F overC:
E Ψ > > > > > > > > π . .. .. .. .. .. .. .. F // E0 Ψ0 π0 ¯ E C
deforms the birational map (An,kflop) f : TF/S∗ 99KTF∗0/Sinto isomorphisms.
LetΓ be the graph closure of F : E99K E0 andΓ0be its central fiber. By
Proposition 3.1, the map Γ∗ : A∗(E) → A∗(E0)is an isomorphism. Since
Γ→EׯEE0 is birational,(E×
¯EE0)∗ : A∗(E) → A∗(E0)is again an
isomor-phism. It follows that its central fiberFopen := TF/S∗ ׯE0 T ∗
F0/S induces an
isomorphism A∗(TF/S∗ ) →A∗(TF∗0/S).
Consider the fiber product
Floc:=PF(TF/S∗ ⊕O) ×PS(¯E0×C)PF0(T ∗ F0/S⊕O) andF∞ := PF(TF/S∗ ) ×P S(¯E0)PF0(T ∗
F0/S). Note that the push-forward map
Proposition 4.2. We have the following commutative diagrams with exact hori-zontal rows (induced by the localization formula in Chow groups):
0 −−−−→ A∗(P(TF/S∗ )) −−−−→ A∗(P(TF/S∗ ⊕O)) −−−−→ A∗(TF/S∗ ) −−−−→ 0 F∞ y Floc y Fopen y 0 −−−−→ A∗(P(TF∗0/S)) −−−−→ A∗(P(TF∗0/S⊕O)) −−−−→ A∗(TF∗0/S) −−−−→ 0
Thus ifF∞is an isomorphism, so isFloc. Note that PF(TF/S∗ ) 99K PF0(T∗
F0/S) is a stratified Mukai flop of type
An−2,k−1. This allows us to perform inductive argument later.
4.2. Deformations of projective local models. Consider the rational map π : P(E⊕O) 99KP1 which extends the map π :E →C and maps P(E)to ∞. The map π is undefined exactly along E := P(E0). Blow upP(E⊕O)
along E resolves the map π, thus we obtain ˆπ: X :=BlEP(E⊕O) →P1.
Since E is a divisor of the central fiber P(E⊕O)0 = P(TF/S∗ ⊕O), we have ˆπ−1(0) ' P(TF/S∗ ⊕O). When t 6= 0, Xt ' P(E)which is the
com-pactification ofEt by E ∼= P(E0) = P(TF/S∗ ). This gives a deformation of
P(TF/S∗ ⊕O)overP1with other fibers isomorphic toP(E).
We do a similar construction on the dual side, which gives a deformation of P(TF∗0/S⊕O)by ˆπ0 : X0 → P1. We get also an induced birational map F: X99KX0over ¯Xextending F :E99KE0over ¯E.
The flop Ft : Xt 99KX0t for t 6=0 has the property that there are smooth
divisors E ⊂ Xt and E0 ⊂ X0t such that (i) the exceptional loci Z ⊂ Xt (resp. Z0 ⊂ X0t) is contained in E (resp. E0), (ii) Ft|E : E99K E0 is a stratified
Mukai flop. We call such flops stratified ordinary flops if furthermore (iii) NE/Xt|P1 ∼=O and NE0/X0t|P1
∼
=O along the flopping extremal rays.
If Ft|E is of type A, D or E, then we say Ft is of type A∗, D∗ or E∗
re-spectively. Notice that stratified ordinary flops of type A∗m,1 are precisely ordinaryPmflops, which explains the choice of terminology.
Proposition 4.3. The birational map F overP1: X Ψ @ @ @ @ @ @ @ @ ˆ π 0 00 00 00 00 00 00 00 F // X0 Ψ0 ~~}}}}}} }} ˆ π0 ¯ X P1
deforms the An,k flop f = F0 : P(TF/S∗ ⊕O) 99K P(TF∗0/S⊕O)into A∗n−2,k−1
Proof. It remains to check condition (iii), which is equivalent to(E.C) = 0 for any flopping curve C ∼= P1. Since(E.C)is independent of t ∈ P1 we
may compute it at t=0. As a projective bundle ρ : X0 →F it is clear that
KX0 = −(2 dim F/S+2)E+ρ ∗K
X0|F.
Since(KX0.C) =0 by the definition of flops, we get(E.C) =0 as well. Clearly for t 6= 0, the map Ft : P(E) 99K P(E0)is an isomorphism only
for the case of ordinary Mukai flop, i.e. k =1.
Remark 4.4. For An,2flops, the deformed flop Ftis a family of ordinary flop,
which has defect of cup product by [LLW1]. As the classical cohomology ring is invariant under deformations, the fiber product of f does not pre-serve the ring structure. This implies that we cannot deform a projective local stratified Mukai flop of type An,2 into isomorphisms, which is a
cru-cial difference to usual Mukai flops. Thus there exist quantum corrections even in this local case.
Corollary 4.5. For projective local model of An,kflops X
ψ
−→ X¯ ψ0
←− X0, the corre-spondence defined by fiber productF = X×X¯ X0 induces isomorphism of Chow
motives[X] ∼= [X0]which preserves also the Poincar´e pairing. Proof. By definition, the An,kcontraction satisfies
2 codim D=codim ψ(D)
for each irreducible component D of Ei. The deformation
X−→Ψ X¯ ←Ψ−0 X0
constructed by Proposition 4.3 is not isomorphic on general fibers, instead it gives A∗n−2,k−1flops. Thus the additional deformation dimension makes it satisfying the assumption of Proposition 3.1. The result follows by notic-ing that the graph closure restricts toF on the central fiber.
Remark 4.6. Proposition 4.3 suggests certain inductive structure on An,k
flops. It will become more useful (e.g. for the discussion of global An,k
flops or Gromov-Witten theory) after we develop detailed analysis on cor-respondences.
5. DEGENERATION OF CORRESPONDENCES
5.1. Setup of degeneration. Let f : X 99K X0 be a stratified Mukai flop, say of type An,k with 2k < n+1. The aim of the following theorem is to
show that the degeneration to normal cone for(X, Fk)and(X0, Fk0)splits the
correspondenceFf defined by X×X¯ X0 into the oneFgdefined by Y×Y¯Y0
to Fkand Fk0. Conversely we may defineF inductively by gluing these two
parts. Here is the blow-up diagram Y=BlFkX φ g // &&M M M M M M M M M M M Y 0 =Bl Fk0X0 φ0 wwppppppp pppp X // ψ ''N N N N N N N N N N N N N N N Y¯ =BlSkX¯ X0 ψ0 wwoooooo oooooo ooo ¯ X
with g : Y99KY0being the induced An−2,k−1flop. To save the notations we
use the same symbolF for Ff and its local models as well if no confusion is likely to arise.
We consider degenerations to the normal cone W → A1of X, where W
is the blow-up of X×A1along F
k× {0}. Similarly we get W0 →A1for X0.
Note that the central fiber
W0 :=Y1∪Y2 =Y∪Xloc, W00 :=Y10∪Y20 =Y0∪X0loc,
where Xloc = P(TF∗k/S⊕O) and X0loc = P(TF∗0
k/S
⊕O). The intersections E := Y∩Xlocand E0 := Y0∩Xloc0 are isomorphic respectively toP(TF∗k/S) andP(TF∗0
k/S
). The map f : X 99K X0 induces the Mukai flop of the same type for local models: f : Xloc99KXloc0 and Mukai flop g : Y99KY0of type
An−2,k−1. Let p : Xloc→ Fkbe the projection and similarly we get p0.
5.2. Correspondences. A lifting of an element a∈ A∗(X)is a couple(a1, a2)
with a1 ∈ A∗(Y)and a2 ∈ A∗(Xloc)such that φ∗a1+p∗a2 = a and a1|E =
a2|E. Similarly one defines the lifting of an element in A∗(X0).
Theorem 5.1. Let a ∈ A∗(X)with (a1, a2)and(a01, a02)being liftings of a and
Fa respectively. Then
Fa1 =a01⇐⇒Fa2 =a02.
Moreover it is always possible to pick such liftings.
It is instructive to re-examine the Mukai case (k = 1) first. In this case Y = Y0 and f is an isomorphism outside the blow-up loci Z = F1 and
Z0 = F10. Let us denote F = F0+F1 withF0 = [Γ¯f] = φ∗0φ∗ andF1 the
degenerate correspondence[Z×SZ0].
By Lemma 4.2 in [LLW1], it is enough to prove the result for any single choice of a1 =a10. Consider the standard liftings
a(0) = (a1, a2) = (φ∗a, p∗(a|Z)), (Fa)(0) = (φ0∗Fa, p0∗(Fa|Z0)).
Since φ0∗Fa = φ∗a+λwith λ supported on E0 = E, we may select lifting
(a01, a20)with a01=a1. In that case,
(Fa2−a02)|E0 =a2|E−a02|E0 =a1|E−a01|E =0
by the compatibility constraint on E and the fact thatF restricts to an iso-morphism on E. HenceFa2−a20 = ι∗(z0), for some z0 ∈ A∗(Z0), where
ι: Z0 → X0locis the natural inclusion. To prove that z0 =0, consider
z0 = p0∗ι∗z0 = p0∗Fa2−p0∗a02.
By substituting φ0∗a01+p0∗a20 =Fa, a01= φ∗a and a2 = p∗(a|Z), we get
p0∗a02=Fa−φ∗0φ∗a=F1a.
Let q, q0 be the projections of Z×SZ0to the two factors and j : Xloc×X¯locX0loc→Z×SZ0
the natural morphism. Then
z0 = p0∗Flocp∗(a|Z) −F1a=q0∗j∗j∗q∗(a|Z) −q0∗q∗(a|Z).
Note that there exists a unique irreducible component in Xloc×X¯loc X 0
loc
birational to Z×SZ0via j, so j∗j∗ =Id, which gives z0 =0.
Now we proceed for general An,k flops. It is enough to prove the result
for any single choice of a1and a01, since other choices differ from this one by
elements supported on E and E0where the theorem holds by induction on k. To make a01= Fa1, notice that g is an isomorphism outside ˜F1 =φ−∗1(F1)
and ˜F10 = φ∗0−1(F10)but we may adjust the standard lifting φ0∗Ff a only by elements lying over Fk0, namely classes in E0 =P(TF∗0
k/Sk).
The following simple observation resolves this as well as later difficul-ties. Recall thatF=∑jFjwithFj = [Fj×Sj F
0
j].
Lemma 5.2. We have decomposition of correspondences: Ff =
φ0∗Fgφ∗+Fkf. In particular, φ0∗Ff = Fgφ∗modulo A∗(E0).
Proof. This follows from the definition and the base change property of
fiber product.
Thus we may pick
a1=φ∗a, a01=Fg(φ∗a) =Fa1.
Then
(Fa2−a02)|E0 =F(a2|E) −a02|E0
and soFa2−a20 = ι∗(z0), for some z0 ∈ A∗(Fk0), where ι : Z0 → Xloc0 is the
natural inclusion.
To prove that z0 =0, consider
z0 = p0∗ι∗z0 = p0∗Fa2−p0∗a02.
By substituting φ0∗a01+p0∗a20 =Fa, a10 =Fφ∗a and a2= p∗(a|Fk), we get p0∗a02=Fa−φ0∗Fφ∗a=Fka
by the above lemma.
Let q, q0 be the projections of Fk×SkFk0to the two factors and j : Xloc×X¯loc X0loc→Fk×Sk F
0
k
the natural morphism. Then
z0 = p0∗Flocp∗(a|Fk) −Fka =q 0
∗j∗j∗q∗(a|Fk) −q 0
∗q∗(a|Fk).
Note that there exists a unique irreducible component in Xloc×X¯loc X 0
loc
birational to Fk×Sk F 0
k via j, so j∗j∗ = Id, which gives z0 = 0. The proof is
complete.
5.3. Cup product and the Poincar´e pairing. Besides correspondences, we also need to understand the effect on the Poincar´e pairing under degener-ation. We will in fact degenerate classical cup product and this works for any degenerations to normal cones W → X×A1 with respect to Z ⊂ X.
Let W0 = Y1∪Y2, where φ : Y1 = Y → X is the blow up along Z,
p : Y2 = E˜ = PZ(NZ/X⊕O) → Z is the local model and Y1∩Y2 = E
is the φ exceptional divisor. Let i1: E,→Y1, i2 : E,→Y2.
Lemma 5.3. Let a, b ∈ H∗(X). Then for any lifting(a1, a2)of a and any lifting (b1, b2)of b, the pair(a1b1, a2b2)is a lifting of ab.
In particular, if a, b are of complementary degree, then we have an orthogonal splitting of the Poincar´e pairing: (a.b)X= (a1.b1)Y1+ (a2.b2)Y2.
Proof. We compute
a.b= φ∗a1.b+p∗a2.b= φ∗(a1.φ∗b) +p∗(a2.p∗(b|Z)).
Since a1.φ∗b|E = a2.p∗(b|Z)|E,(a1b1, a2b2)is a lifting of ab for the special
lifting (b1, b2) = (φ∗b, p∗(b|Z)) of b. By [LLW1], Lemma 4.2, any other
lifting of b is of the form(b1+i1∗e, b2−i2∗e)for some class e in E.
Since i∗1a1.e=i2∗a2.e is a class e0 ∈ H∗(E). The correction terms are
a1.i1∗e=i1∗(i∗1a1.e) =i1∗e0, −a2.i2∗e= −i2∗(i∗2a2.e) = −i2∗e0.
The lemma then follows from
i∗1(i1∗e0) =e0.c1(NE/Y1) = −e 0 .c1(NE/Y2) =i ∗ 2(−i2∗e0) and φ∗i1∗e0−p∗i2∗e0 = (φ|E)∗e0− (φ|E)∗e0 =0. Theorem 5.4 (= Theorem 1.1). For An,k flops, F induces an isomorphism on
Proof. If f : X 99K X0 is an An,k flop, f ×id : X×T 99K X0×T is also an
An,k flop. Thus by the identity principle, to prove[X] ∼= [X0]we only need
to prove the equivalence of Chow groups underF for any An,kflop.
We prove this for all n with n+1 >2k by induction on k. We start with k =0, which is trivial.
Given k≥1, by Theorem 5.1, the equivalence of Chow groups is reduced to the An−2,k−1case and the local An,kcase. The former is true by induction.
The later follows from Corollary 4.5 directly.
The same procedure proves the isomorphism of Poincare pairings by us-ing Theorem 5.1, Lemma 5.3 and Corollary 4.5.
For cohomology rings we need to proceed carefully. In order to run in-duction on k, using Theorem 5.1, Lemma 5.3 we must first consider the local An,k case. By remark 4.4, for k = 2, the classical cup product is not
preserved by the correspondenceF! This is analyzed in the next section. 6. QUANTUM CORRECTIONS
6.1. The proof of Theorem 1.2. We now prove the invariance of big quan-tum product attached to the extremal rays, up to analytic continuations, under An,2flops.
As in the precedent section, we consider degenerations to the normal cone W → A1of X and W0 → A1 of X0. Note that the map f : X 99K X0
induces the Mukai flop of the same type for local models: f : Xloc99K X0loc
and Mukai flop g : Y99KY0of type An−2,1.
By the degeneration formula (for the algebraic version used here, c.f. [Li]), any Gromov-Witten invariant on X splits into products of relative invariants of(Y, E)and(Xloc, E). Let a∈ H∗(X)⊕m with lifting(a1, a2):
haiX g,m,β=
∑
η∈Ω(g,β) Cη ha1i•(Γ1Y,E).ha2i•(Γ2Xloc,E) Eρ .Here ρ is the number of gluing points (in E) andΓ1∪Γ2forms a connected
graph. Thus ρ=0 if and only if one of theΓi is empty.
The relative invariants take values in H∗(Eρ)and the formula is in terms of the Poincar´e pairing of Eρ.
We apply it to X0as well and get:
hFaiX0 g,m,Fβ=
∑
η0∈Ω(g,Fβ) Cη0 ha01i•(Γ0Y0,E0) 1 . ha02i•(X0loc,E0) Γ0 2 E0ρ .There is a one to one correspondence between admissible triples η = (Γ1,Γ2, Iρ)and η0 = (Γ10,Γ02, Iρ0)via η0 := Fη. The combinatorial structure is kept the same, while the curve classes are related byF. We do still need the cohomology class splitting on X and X0be to compatible.
By Theorem 5.1 we may split the cohomology classes a∈ H∗(X)⊕minto
H∗(Yi0)⊕m, such that
Fa1 =a10, Fa2= a02.
By Theorem 5.4, the Poincar´e pairing is preserved byF under stratified Mukai flops E 99K E0, the same holds true for Eρ 99K E0ρ by Fρ, which for simplicity is still denoted byF. Thus by the degeneration formula the problem is reduced to showing thatF maps the relative invariants of(Y, E)
and(Xloc, E)to the corresponding ones of(Y0, E0)and(X0loc, E0).
Since we are only interested in invariants attached to the extremal ray β = d`, for any splitting β = (β1, β2) = (d1`, d2`), we must have ρ = (E.β2) = d2(E.`) = 0 (since ` can be represented by a curve in F2). But
this implies that β is not split at all and in the degeneration formula the invarianthaiX
g,m,d`goes to Y or Xloccompletely: haiXg,m,d` = ha1ig,m,dY `+ ha2iXg,m,dloc `.
Lemma 6.1. F maps isomorphically the cup product and full Gromov-Witten the-ory of Y to those of Y0. Moreoverha1iYg,m,d` =0 for all d∈N.
Proof. The birational map g : Y99KY0is a Mukai flop of type An−2,1. Hence
this follows from [LLW1], Theorem 6.3.
Indeed this follows from Lemma 5.3 and the above degeneration formula by applying it to the Mukai flop Y 99K Y0. Here we use the facts that projective local models of Mukai flops gloc : Yloc 99KYloc0 can be deformed
into isomorphisms gt : Yt→∼ Yt0 and that the cup product as well as the
Gromov-Witten theory are both invariant under deformations. For`being the extremal ray of gloc, if d` ∼Ctfor t6=0 then C0t=∼ gt(Ct) ∼Fd` = −d`0,
which is impossible. Thusha1iYg,m,d` =0 for all d∈N.
Denote by haif = ∑∞d=0hai0,m,d`qd`, the generating function of g = 0
Gromov-Witten invariants attached to the extremal ray. Then the degener-ation formula and Lemma 6.1 lead to
haiXf =δn3ha1iY0,3,0+ ha2iXfloc.
The correspondence F acts on qβ by Fqβ = qFβ. In particular for the extremal rays`and`0 we haveFqd` =q−d`0. If we regard q` =e−2π(ω.`) as an analytic function on ω ∈ HR1,1(X), then it is known thathaiX
f converges
in the K¨ahler coneKXof X. Under the identification HR1,1(X) ∼=FHR1,1(X) =
HR1,1(X0), it makes sense to compareFhaiX
f withhFaiX 0
f as analytic functions
onKX∪ KX0 ⊂ HR1,1up to analytic continuations.
Lemma 6.2. For m≥3,Fha2iXfloc ∼= hFa2iX 0
loc
f up to analytic continuations.
Proof. By Proposition 4.3, the An,2 flop f : Xloc 99K X0loc can be
Gromov-Witten theory, the lemma is reduced to the case of ordinary flops (with non-trivial base). For simple ordinary flops the invariance
Fha2iXfloc ∼= hFa2iX 0
loc
f
up to analytic continuations is proved in [LLW1]. It has been extended to general ordinary flops with base in [LLW2]. Hence the lemma follows.
Notice that the g =0, d= 0 invariants are non-zero if and only if m= 3 and they are given by the cubic product. By Lemma 6.1,ha11, a12, a13iY0,3,0 = hFa11,Fa12,Fa13iY
0
0,3,0. From Lemma 6.2 we getFhaiXf = hFaiX 0
f for m≥3.
Together with the F invariance of Poncar´e pairing, the big quantum product attached to the extremal ray is invariant underF. This completes the proof of Theorem 1.2 for type An,2. The cases of type D5 and E6,I are
completely similar, since the geometric picture in Proposition 4.3 is the same by [CF]. The proof is complete.
Remark 6.3. The degeneration formula is in terms of the Poincar´e pairing of relative GW invariants. Thus invariance of the Poincar´e pairing is crucial in our study. Indeed, the Poincar´e pairing together with 3-point functions determine the (small) quantum product. So far this is the only constraint we have found for the correspondenceF under K equivalence to be canonical. 6.2. A new topological constraint. Consider a stratified Mukai flops of type An,2, f : X 99K X0 with i : F2 ,→ X, such that F preserves the cup
product (e.g. for X and X0 being hyper-K¨ahler manifolds). By Proposition 4.3 and [LLW2], there exists defect of cup product on Xloc. A priori there
seems to be a contradiction. A closer look at them leads to
Proposition 6.4. For a Mukai flop f : X 99K X0 of type An,2, D5or E6,I, if the
restriction map i∗ : H∗(X,Q) →H∗(F2,Q)is surjective thenF does not preserve
the cup product. In particular, if X is hyper-K¨ahler then i∗ is not surjective. Proof. We shall investigate the degeneration analysis on cup product for an arbitrary An,2flop f as presented above. The other cases are similar.
Let a = (a1, a2)and b = (b1, b2) be two elements in H∗(X)with their
lifting. By Lemma 5.3, ab = (a1b1, a2b2). Then Theorem 5.1 implies that
F(ab) = (F(a1b1),F(a2b2)). By Lemma 5.3 again
F(a)F(b) = (F(a1)F(b1),F(a2)F(b2)) = (F(a1b1),F(a2)F(b2)),
where the last equality follows from Lemma 6.1 applied to g : Y 99KY0. So F(ab) =F(a)F(b) ⇐⇒ F(a2b2) =F(a2)F(b2).
That is, the invariance of cup product on H∗(X)is equivalent to the in-variance on elements in H∗(Xloc) which come from lifting of elements in
H∗(X). Indeed let i : F2 ,→ X and p : Xloc → F2 being the projection, we
may choose standard lifting a2 = p∗i∗a. Such elements form a subring
By applying this analysis to the case X = Xloc = PF2(T ∗
F2/S⊕O)where the cup product is not preserved under F, we find that the defect of cup product is completely realized in the subring p∗H∗(F2)since
i∗H∗(Xloc) = H∗(F2).
For general X, if i∗ : H∗(X,Q) → H∗(F2,Q)is surjective, then∆f ⊗Q =
p∗H∗(F2) ⊗Q must contain the defect on Xloc, hence the ring structure on
H∗(X)is not preserved underF. This completes the proof.
Example 6.5. Consider a simple A4,2 flop on hyper-K¨ahler manifold X of
dimension 12, where F2 = G(2, 5)is of dimension 6. The divisor c1(G) =
i∗H for some H ∈ H2(X,Q). But c2(G) 6∈i∗H4(X,Q).
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B. FU; C.N.R.S., LABO. J. LERAY, FACULTE DES SCIENCES, UNIVERSITE DE´ NANTES, 2, RUE DE LAHOUSSINIERE` , BP 92208, F-44322 NANTESCEDEX03 FRANCE.
E-mail address: [email protected]
C.-L. WANG; DEPT.OFMATH., NATIONALCENTRALUNIVERSITY, JHONGLI, TAIWAN. NATIONALCENTER FORTHEORETICSCIENCES, HSINCHU, TAIWAN.
