Invariance of quantum rings under ordinary flops I:

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doi:10.14231/AG-2016-026

Invariance of quantum rings under ordinary flops I:

Quantum corrections and reduction to local models

Yuan-Pin Lee, Hui-Wen Lin and Chin-Lung Wang

Abstract

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to local models. In part II (“Invariance of quantum rings under ordinary flops II”, Algebraic Geometry, 2016), we develop a quantum Leray–Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuation in the K¨ahler moduli space for ordinary flops of split type. In part III (Lee, Lin, Qu and Wang, “Invariance of quantum rings under ordinary flops III”, Cambridge Journal of Mathematics, 2016), we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.

1. Introduction 1.1 Background review

Two complex manifolds X and X⁰ are K-equivalent, denoted by X =_K X⁰, if there are proper birational morphisms φ : Y → X and φ⁰: Y → X⁰ such that φ^∗KX = φ^0∗K_X⁰, where KX is the canonical divisor”. Major examples come from birational minimal models in Mori theory and especially from birational Calabi–Yau manifolds in the mathematical study of string theory. K- equivalent projective manifolds share the same Betti and Hodge numbers. It has been conjectured that a canonical correspondence T ∈ A(X ×X⁰) exists which induces isomorphisms of cohomology groups and preserves the Poincar´e pairing. For a survey, see [Wan04].

However, simple examples show that the classical cup product is generally not preserved under the above correspondence, and this leads to new directions of study in higher-dimensional birational geometry. On the other hand, according to the philosophy of the crepant transformation conjecture and string theory, the quantum product should be more natural and display a certain functoriality not available to the cup product among K-equivalent manifolds.

Received 30 July 2014, accepted in final form 01 February 2016.

2010 Mathematics Subject Classification 14N35, 14E30.

Keywords: quantum cohomology, ordinary flops, analytic continuations, degeneration formula, reconstructions.

This journal is c Foundation Compositio Mathematica2016. This article is distributed with Open Access under the terms of theCreative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact theFoundation Compositio Mathematica.

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Flops are typical examples of K-equivalent birational maps:

X

ψ

f //X⁰

ψ⁰

~~¯ X

In fact, they form the building blocks for connecting birational minimal models [Kaw08]. The simplest flop is the simple P¹ flop (Atiyah flop) in dimension 3. It is known that up to deformation, it generates, locally or symplectically, all K-equivalent maps for threefolds. The quantum corrections to the cup product by extremal ray invariants in the local 3-dimensional case were first observed by Aspinwall–Morrison and Witten [Wit93] and later globalized by Li–Ruan through the degeneration formula [LR01].

The higher-dimensional generalizations are known as ordinary P^r flops (also abbreviated as

“ordinary flops” or “P^r flops”). The local geometry is encoded in a triple (S, F, F⁰), where S is a smooth variety and F and F⁰ are two rank r + 1 vector bundles over S. If Z ⊂ X is the f -exceptional locus, then there exists a ¯ψ : Z ∼= P (F ) → S ⊂ ¯X with fibers spanned by the flopped curves C ∼= P¹ and N_Z/X = ¯ψ^∗F⁰⊗OZ(−1). A similar structure holds for Z⁰ ⊂ X⁰, with F and F⁰ exchanged. See Section 2.1 for details. (We note that the Atiyah flop corresponds to S = pt and r = 1.) Thus it is reasonable to expect that ordinary flops play a vital role in the study of K-equivalent maps. For example, up to complex cobordism, any K-equivalent map can be decomposed into P¹ flops [Wan03].

The study of the invariance of quantum products under ordinary flops in higher dimensions was started in [LLW10]. The canonical correspondence is given by the graph closure [¯Γ_f], and the quantum invariance under

F = [¯Γf]∗: QH(X) → QH(X⁰)

is proved for all simple P^r flops, that is, all P^r flops with S = pt. Here the quantum cohomology rings QH(X) and QH(X⁰) are defined below. The crucial idea is to interpret F -invariance in terms of analytic continuations in Gromov–Witten theory.

Let us explain this point in a little more detail. We use [CK99] as our general reference for early developments in Gromov–Witten (GW) invariants. Let M_g,n(X, β) be the moduli space of stable maps from genus g nodal curves with n marked points to X, and let e_i: M_g,n(X, β) → X be the evaluation maps. The Gromov–Witten potential

F_g^X(t) =X

n,β

q^β

n!htⁿi^X_g,n,β = X

n>0, β∈NE(X)

q^β n!

Z

[Mg,n(X,β)]^virt n

Y

i=1

e^∗_it

is a formal function in t ∈ H(X) and Novikov variables q^β, with β an element of NE(X), the Mori cone of effective classes of 1-cycles. Modulo convergence issues, it is a function on the complexified K¨ahler cone K^C^X := H^1,1

R + iKX via

ω 7→ q^β = e^2πi(β·ω).

Under the canonical correspondence F , the potentials Fg^X and F_g^X⁰ share the same variable t ∈ H ∼= H(X, C) ∼= H(X⁰, C). However, F does not identify NE(X) with NE(X⁰). Indeed, for the flopped curve classes ` = [C] and `⁰ = [C⁰], we have

F ` = −`⁰ ∈ NE(X/ ⁰) .

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By duality this implies K^C^X ∩ K^C^X0 = ∅ in H²

C. Hence F_g^X and F_g^X⁰ have different domains and comparison can only make sense after analytic continuation over a certain compactification of K^C^X ∪ K^C^X0 ⊂ H²

C. (Thus the naive K¨ahler moduli space K is usually regarded as the closure of the union of all K^C^X0 with X⁰ =_K X.) In other words, we set F q^β = q^{F β}. ThenF F_g^X cannot be a formal GW potential of X⁰.

In this paper, we will focus on genus zero theory, which carries a quantum product structure, or equivalently a Frobenius structure [Man99]. Let {T_µ} be a basis of H and {T^µ :=P g^µνT_ν} the dual basis with respect to the Poincar´e pairing, where gµν = (Tµ· T_ν) and (g^µν) = (gµν)⁻¹ is the inverse matrix. Denote by t = P t^µT_µ a general element of H. The big quantum ring (QH(X), ∗) uses only the genus zero potential with three or more marked points:

Tµ∗_tTν =X

κ

∂³F₀^X

∂t^µ∂t^ν∂t^κ(t)T^κ = X

κ, n>0, β∈NE(X)

q^β

n!hT_µ, Tν, Tκ, tⁿi^X_0,n+3,βT^κ.

The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation guarantees that ∗_t is a family of associative products on H parameterized by t ∈ H. Equivalently, it equips H with a structure of formal Frobenius manifold H_X with a family (in z ∈ C^×) of integrable (or, equivalently, flat) Dubrovin connections

∇^z = d − z⁻¹X

µ

dt^µ⊗ T_µ∗_t on the tangent bundle T H = H × H.

There is a natural embedding of K^C^X in H. With a suitable choice of coordinates we have q^` = e^2πit^` with K¨ahler constraint = t_` > 0. Since now F q^` = q^−`⁰, the pair {q^`, q^`⁰} serves as an atlas for P¹, the compactification of C/Z ∼= C^×. This gives the formal H an analytic P¹ direction. In [LLW10], for simple flops, the structural constants ∂_µνκ³ F₀^X(t) for the big quantum product are shown to be analytic (in fact algebraic) in q^`. Moreover, F identifies HX and HX⁰

through analytic continuations over this P¹. Based on this, in [ILLW12] the Frobenius structure is further exploited to deduce analytic continuation from F_g^X to F_g^X⁰ for all simple flops and for all g > 0.

1.2 Outline of the contents

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over a smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to local models.

In part II [LLW13], we show that the big quantum cohomology ring is invariant under analytic continuation in the K¨ahler moduli space for flops of split type. In part III [LLQW16], the final part of this series, we remove the splitting condition by developing a quantum splitting principle, hence solve the problem completely.

In particular, this is the first result on the K-equivalence (crepant transformation) conjecture where the local structure of the exceptional loci cannot be deformed to any explicit (for example, toric) geometry and the analytic continuation is non-trivial. As far as we know, this is also the first result for which the analytic continuation is established with non-trivial Birkhoff factorizations.

We give an outline of the contents of this paper below.

Conventions. Throughout this paper, we work on the even cohomology H = H^even to avoid complications with signs. In particular, by degree we always mean the Chow degree. Nevertheless,

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all our discussions and results work for the full cohomology spaces.

1.2.1 Defect of the cup product under the canonical correspondence. Let { ¯Ti} be a basis of H(S) with dual basis { ˇT¯i}. Let h = c₁(OZ(1)) and H_k = c_k(QF), where QF → Z = P (F ) is the universal quotient bundle. Similarly, we define h⁰ and H_k⁰ on the X⁰ side. The H_k are of fundamental importance since

F Hk= (−1)^r−kH_k⁰ and the dual basis of { ¯T_ih^j} in H(Z) is given by { ˇT¯_iH_r−j}.

Theorem (Theorem 2.8, topological defect). Let a1, a₂, a₃ ∈ H(X) with P deg a_i = dim X.

Then

(F a1·F a2·F a3)^X⁰ − (a₁· a₂· a₃)^X

= (−1)^rX

i∗,j∗

a1· ˇT¯i1Hr−j1

X

a2· ˇT¯i2Hr−j2

X

× a₃· ˇT¯i3Hr−j3

X

s_j₁_+j₂_+j₃−(2r+1)(F + F^0∗) ¯Ti1T¯i2T¯i3

S

, where s_i is the ith Segre class.

1.2.2 Quantum corrections attached to flopping extremal rays. We proceed to calculate the quantum corrections attached to the flopping extremal ray N`. Using the calculation, we demon- strate that the “quantum corrected product”, combining the classical product and the quantum deformation attached to the extremal ray, isF -invariant after analytic continuation.

The stable map moduli space for the extremal ray has a bundle structure over S:

M0,n(P^r, d`) ^//M0,n(Z, d`) ^eⁱ ^//

Ψn

Z

ψ¯

yyS

In this case, the GW invariants on X are reduced to twisted invariants on Z by certain obstruction bundles. We define the fiber integral (see Section3.1 for the details of the notation)

* _n Y

i=1

h^jⁱ +/S

d

:= Ψn∗

n

Y

i=1

e^∗_ih^jⁱ· e R¹f t∗e^∗_n+1N_Z/X

!

∈ A^ν(S)

as a ¯ψ-relative invariant over S, a cycle of codimension ν :=P j_i− (2r + 1 + n − 3). The absolute invariant is obtained by the pairing on S: for ¯t_i ∈ H(S),

t¯1h^j¹, . . . , ¯tnh^jⁿX

d = h^j¹, . . . , h^jⁿ/S d ·

n

Y

i=1

¯ti

!S

.

If ν = 0, the invariant reduces to the simple case. This happens for n = 2, since then j1 = j2 = r. Thus we may calculate extremal functions based on the 2-point case by (divisorial) reconstruction. To state the result, let

f (q) := q

1 − (−1)^r+1q, which satisfies the functional equation f (q) + f (q⁻¹) = (−1)^r.

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For 3-point functions, we show that W_ν := P

d∈Nhh^j¹, h^j², h^j³i^/S_d q^d with 1 6 ji 6 r lies in A^ν(S)[f ] and is independent of the choices of the ji.

Theorem (Theorem 3.6, quantum corrections). The function Wν is the action on f by a polynomial in the operator δ = q d/dq with Chern classes as coefficients. (See Proposition 3.5.) It satisfies

W_ν− (−1)^ν+1W_ν⁰ = (−1)^rs_ν(F + F^0∗) .

This implies that the topological defect is corrected by the 3-point extremal functions. The analytic continuation for n > 4 points follows by reconstruction.

1.2.3 Degeneration analysis. The next step is to prove that the big quantum ring, involving all curve classes, isF -invariant. As a first step, this statement is reduced to a corresponding one on f -special descendent invariants on the projective local models

X_loc:= ˜E = P (N_Z/X⊕O)→ Z^p and

X_loc⁰ := ˜E⁰ = P (N_Z⁰_/X⁰ ⊕O)→ Z^p⁰ ⁰ by a degeneration analysis.

To compare GW invariants of non-extremal classes, application of the degeneration formula and deformation to the normal cone is well suited for ordinary flops with base S. This reduces the problem to local models with induced flop f : ˜E 99K ˜E⁰. The reduction has two steps. The first reduces the problem to relative local invariants hA | ε, µi^{( ˜}^E,E), where E ⊂ ˜E is the infinity divisor.

The second is a further reduction back to absolute local invariants, with possibly descendent insertions coupled to E, called of f -special type.

The local model ¯p := ¯ψ ◦ p : ˜E → S and the flop f are defined over S, with the simple case as fibers. In particular, the kernel of ¯p∗: N₁( ˜E) → N₁(S) is spanned by the p-fiber line class γ and ¯ψ-fiber line class `. The correspondenceF is compatible with ¯p. Namely,

N₁( ˜E) ^F ^//

¯

p∗⊕d2 %%

N₁( ˜E⁰)

¯ p⁰_∗⊕d⁰₂

xx

N1(S) ⊕ Z

is commutative. Here we write a class β in N1( ˜E) as βS+ d` + d2γ for some βS in N1(S) and d, d2 ∈ Z. Thus the functional equation of a generating series hAi is equivalent to the functional equations of its various subseries (fiber series) hAi_β_S_,d₂ labeled by NE(S) ⊕ Z.

Theorem 1.1 (Degeneration reduction). To prove F hαi^Xg ∼= hF αi^Xg ⁰ for all α ∈ H(X)^⊕n and g 6 g0, it is enough to prove the local case f : ˜E → ˜E⁰ for descendent invariants of f -special type:

F hA, τk1ε1, . . . , τkρερi^E_g,β^˜

S,d2

∼= hF A, τk1ε1, . . . , τkρερi^E_g,β^˜⁰

S,d2

for any A ∈ H( ˜E)^⊕n, k_j ∈ N ∪ {0}, εj ∈ H(E) ⊂ H( ˜E), g 6 g0, β_S∈ NE(S) and d₂> 0.

1.2.4 Further reduction to the big quantum ring, quasi-linearity on local models. While the degeneration reduction works for higher genera, for g = 0 more can be said. Using the topological recursion relation (TRR) and the divisor axiom (for descendent invariants), the F -invariance

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for f -special invariants can be completely reduced to theF -invariance of big quantum rings for local models (Theorem5.2).

We then employ the divisorial reconstruction [LP04] and the WDVV equation to make a further reduction to anF -invariance statement about elementary f-special invariants with at most one special insertion.

To state the result, we now assume X = X_loc = ˜E. Since X → S is a double projective bundle, H(X) is generated by H(S) and the relative hyperplane classes h for Z → S and ξ for X → Z. This leads to another useful reduction: by moving all the classes h, ξ and ψ into the last insertion (divisorial reconstruction), the problem is reduced to the case

¯t1, . . . , ¯tn−1, ¯tnτ_kh^jξⁱX βS,d2

with ¯t_l∈ H(S) and d₂ ∈ Z, where k 6= 0 only if i 6= 0.

By a further application of WDVV equations, the F -invariance can always be reduced to the case i 6= 0 even if k = 0. Since ξ is the class of the infinity divisor, which is within the isomorphism locus of the flop, such an F -invariance statement is intuitively plausible. We call it the type I quasi-linearity property (cf. Theorem5.5).

The above steps furnish a complete reduction to projective local models X_loc, which works for any F and F⁰.

To proceed, notice that these descendent invariants are encoded by their generating function, that is, the so-called (big) J function: for τ ∈ H(X),

J^X τ, z⁻¹ := 1 + τ

z + X

β,n,µ

q^β n!T_µ

T^µ

z(z − ψ), τ, . . . , τ

X 0,n+1,β

.

The determination of J usually relies on the existence of C^×-action. Certain localization data I_β coming from the stable map moduli are of hypergeometric type. For “good” cases, say c₁(X) is semipositive and H(X) is generated by H², the sum I(t) = P I_βq^β determines J (τ ) on the small parameter space H⁰⊕ H² through the “classical” mirror transform τ = τ (t). For a simple flop, X = X_loc is indeed semi-Fano toric and the classical mirror theorem (of Lian–Liu–Yau and Givental) is sufficient [LLW10]. (It turns out that τ = t and I = J on H⁰⊕ H².)

For a general base S with given QH(S), the determination of QH(P ) for a projective bundle P → S is far more involved. To allow fiberwise localization to determine the structure of the GW invariants of Xloc, the bundles F and F⁰ are then assumed to be split bundles. This is the main subject to be studied in part II of this series [LLW13].

Remark 1.2. Results in this paper have been announced by the authors, in increasing degree of generality, at various conferences during 2008–2010; see, for example, [Lin10, Wan11, LLW12], where more examples are studied.

2. Defect of the classical product 2.1 Cohomology correspondence for P^r flops

We recall the construction of ordinary flops in [LLW10] to fix the notation.

Let X be a smooth complex projective manifold and ψ : X → ¯X a flopping contraction in the sense of minimal model theory, with ¯ψ : Z → S the restriction map on the exceptional loci.

Assume that

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(i) ¯ψ equips Z with a P^r-bundle structure ¯ψ : Z = P (F ) → S for some rank r + 1 vector bundle F over a smooth base S;

(ii) N_Z/X|_Z_s ∼=OP^r(−1)^⊕(r+1) for each ¯ψ-fiber Zs for s ∈ S.

Then there is another rank r + 1 vector bundle F⁰ over S such that N_Z/X ∼=OP (F )(−1) ⊗ ¯ψ^∗F⁰.

We may blow up X along Z to get φ : Y → X. The exceptional divisor E = P (N_Z/X) ∼= P ( ¯ψ^∗F⁰) = ¯ψ^∗P (F⁰) = P (F ) ×_SP (F⁰)

is a P^r × P^r-bundle over S. We may then blow down E along another fiber direction to get φ⁰: Y → X⁰. This induces another contraction ψ⁰: X⁰ → ¯X, with exceptional loci ¯ψ⁰: Z⁰ = P (F⁰) → S and N_Z⁰_/X⁰|_ψ_¯0−fiber∼=OP^r(−1)^⊕(r+1).

We call f : X 99K X⁰ an ordinary P^r flop. The various sets and maps are summarized in the following commutative diagram:

E

φ¯

}}

φ¯⁰

!!

^j // Y

φ

}} ^φ

0

!!

Z

ψ¯

ⁱ // X

ψ

!!

Z⁰

ψ¯⁰

}}

ⁱ⁰ // X⁰

ψ⁰

}}S ^j⁰ // X where the normal bundle of E in Y is

N_E/Y = ¯φ^∗OP (F )(−1) ⊗ ¯φ^0∗OP (F⁰)(−1) .

First of all, we have found a canonical correspondence between the cohomology groups of X and X⁰.

Theorem 2.1 ([LLW10]). For an ordinary P^r flop f : X 99K X⁰, the graph closure T := [¯Γf] ∈ A(X × X⁰) identifies the Chow motives ˆX of X and ˆX⁰ of X⁰; that is, ˆX ∼= ˆX⁰ via T^t◦ T = ∆_X and T ◦ T^t = ∆_X⁰. In particular, F := T∗: H(X) → H(X⁰) preserves the Poincar´e pairing on cohomology groups.

In practice, the correspondence T induces a map on Chow groups:

F : A(X) → A(X⁰) , W 7→ p⁰_∗(¯Γ_f · p^∗W ) = φ⁰_∗φ^∗W , where p and p⁰ are the projection maps from X × X⁰ to X and X⁰, respectively.

Second, parallel to the procedure in [LLW10], we need to determine the explicit formulae for the associated map F restricted to A(Z). The Leray–Hirsch theorem says that

A(Z) = ¯ψ^∗A(S)[h]/f_F(h) ,

where fF(λ) = λ^r+1 + ¯ψ^∗c1(F )λ^r + · · · + ¯ψ^∗cr+1(F ) is the Chern polynomial of F and h = c1(OP (F )(1)). Thus a class α ∈ A(Z) is of the form α =Pr

i=0hⁱψ¯^∗ai for some ai ∈ A(S).

By the pull-back formula from intersection theory, it is easy to see that for a ∈ A_k(Z) we have

φ^∗(i∗a) = j∗ c_r(E ) · ¯φ^∗a ∈ A_k(Y ) ,

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whereE is the excess normal bundle defined by

0 → N_E/Y → φ^∗N_Z/X →E → 0 .

By the functoriality of the pull-back and push-forward, together with the above formula, we can conclude from F (i∗(P hⁱψ¯^∗ai)) =P i⁰_∗(F (i∗(hⁱ)) · i⁰_∗ψ¯^0∗ai)^Z⁰ that F restricted to A(Z) is A(S)-linear. Here we identify the ring A(S) with its isomorphic images in A(Z) and A(Z⁰) via ¯ψ^∗ and ¯ψ^0∗, respectively.

Under such an identification, we will abuse the notation to denote c_i(F ), ¯ψ^∗c_i(F ) and ¯ψ^0∗c_i(F ) by the same symbol c_i. Similarly, we denote c_i(F⁰), ¯ψ^∗c_i(F⁰) and ¯ψ^0∗c_i(F⁰) by c⁰_i. We use this abbreviation for any class in A(S). And for α ∈ A(Z) we often omit i∗from i∗α when α is regarded as a class in A(X), unless possible confusion should arise. We do likewise for α⁰ ∈ A(Z⁰) ,→ A(X⁰).

The A(S)-linearity of F restricted to A(Z) allows us to focus on the study of a basis for A(Z) over A(S). Recall that for a simple P^r flop we have the basic transformation formula F (h^k) = (−1)^r−kh^0k. Unfortunately, for a general P^rflop, this does not hold anymore, so a better candidate has to be sought.

Note that the key ingredient in the pull-back formula is c_r(E ). From the Euler sequence 0 →OZ⁰(−1) → ¯ψ^0∗F⁰ → Q_F⁰ → 0

and the short exact sequence defining the excess normal bundleE , we get E = ¯φ^∗OP (F )(−1) ⊗ φ¯^0∗Q_F⁰. A simple computation leads to

cr(E ) = (−1)^r φ¯^∗h^r− ¯φ^0∗H₁⁰φ¯^∗h^r−1+ ¯φ^0∗H₂⁰φ¯^∗h^r−2+ · · · + (−1)^rφ¯^0∗H_r⁰ , where H_k⁰ = ck(QF⁰). Explicitly,

H_k⁰ = h^0k+ c⁰₁h^0k−1+ · · · + c⁰_k, where h⁰ = c₁(OP (F⁰)(1)). Similarly, we write

H_k= c_k(Q_F) = h^k+ c₁h^k−1+ · · · + c_k.

Notice that H_k= 0 = H_k⁰ for k > r. Finally, we find that H_k and H_k⁰ are the correct choices.

Proposition 2.2. For all positive integers k 6 r, we have F (Hk) = (−1)^r−kH_k⁰.

Proof. First of all, we have the basic identities h^r+1+ c₁h^r+ · · · + c_r+1 = 0, ¯φ⁰_∗φ¯^∗hⁱ = 0 for all i < r and ¯φ⁰_∗φ¯^∗h^r = [Z⁰]. The latter two follow from the definitions and dimension considerations.

In order to determine F (Hk) = ¯φ⁰_∗(c_r(E ) · ¯φ^∗H_k), we need to take care of the class φ¯⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗hⁱ· ¯φ^∗H_k

with 0 6 i 6 r; here H0⁰ := 1.

If i > r − k, then

φ¯⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗hⁱ· ¯φ^∗H_k = ¯φ⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗ h^k+i+ c₁h^k+i−1+ · · · + c_khⁱ

= − ¯φ⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗ c_k+1hⁱ⁻¹+ c_k+2hⁱ⁻²+ · · · + c_r+1h^i+k−r−1 = 0 , since the power in h is at most i − 1 < r.

If i < r − k, then again ¯φ⁰_∗( ¯φ^0∗H_r−i⁰ φ¯^∗hⁱ· ¯φ^∗H_k) = 0, since the power in h is at most i + k < r.

For the remaining case i = r − k,

φ¯⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗hⁱ· ¯φ^∗H_k = ¯φ⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗h^r = H_r−i⁰ = H_k⁰ .

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We conclude that

F (Hk) = (−1)^r

r

X

i=0

(−1)^r−iφ¯⁰_∗ φ¯^0∗H_r−i⁰ φ¯^∗hⁱ· ¯φ^∗Hk = (−1)^r−kH_k⁰ .

Remark 2.3. The image class of h^k under F looks more complicated here than in the case of simple P^r flops. As a simple corollary of Proposition 2.2, we may show, by induction on k, that for all k ∈ N,

F (h^k) = (−1)^r−k a₀h^0k+ a₁h^0k−1+ · · · + a_k ∈ A(Z⁰) , where a₀ = 1 and the a_k∈ A(S) are determined by the recursive relations

c⁰_k= a_k− c₁a_k−1+ c₂a_k−2+ · · · + (−1)^kc_k. Symmetrically,

F^∗(h^0k) = (−1)^r−k a⁰₀h^k+ a⁰₁h^k−1+ · · · + a⁰_k ∈ A(Z) with a⁰₀ = 1 and a⁰_k= c⁰₁a⁰_k−1− c⁰₂a⁰_k−2+ · · · + (−1)^k−1c⁰_k+ c_k.

To put these formulae into perspective, we consider the virtual bundles A := F⁰− F^∗, A⁰ := F − F^0∗.

Then a_k = c_k(A) and a⁰_k = c_k(A⁰). Notice that since a_k and a⁰_k are Chern classes of virtual bundles, they may survive even for k > r + 1.

It is also interesting to notice that the explicit formula reduces to F (h^k) = (−1)^r−kh^0k,

without lower-order terms precisely when F⁰ equals F^∗, the dual of F . 2.2 Triple product

Let { ¯T_i^k} be a basis of H^2k(S) and { ˇT¯_i^k} ⊂ H^2(s−k)(S) its dual basis, where s = dim S. It is an easy but quite crucial discovery that the dual basis of the canonical basis { ¯T_i^kh^j} in H(Z) can be expressed in terms of {Hk}_k>0.

Lemma 2.4. The dual basis of { ¯T_i^k−jh^j}_j6min{k,r} in H^2k(Z) is { ˇT¯_i^k−jH_r−j}_j6min{k,r} in H^2(r+s−k)(Z).

Proof. We have to check that ( ¯T_i^k−jh^j· ˇT¯_i^k−jHr−j) = 1 and ( ¯T_i^k−jh^j · ˇT¯_i^k−j⁰H_r−j⁰) = 0 for any j 6= j⁰. Indeed,

T¯_i^k−jh^j· ˇT¯_i^k−jH_r−j = ¯T^s h^r+ c₁h^r−1+ · · · = ¯T^sh^r= 1 , since ¯T^sc_i= 0 for all i > 1 by degree considerations.

Notation 2.5. When X is a bundle over S, classes in H(S) may be considered as classes in H(X) by the obvious pull-back, which we often omit in the notation. To avoid confusion, we consistently employ the notation ˇT¯i as the dual class of ¯Ti∈ H(S) with respect to the Poincar´e pairing in S.

The “raised” index form, for example T^µ as the dual of T_µ∈ H(X), is reserved for duality with respect to the Poincar´e pairing in X.

Now if j⁰> j, then

k − j + (s − (k − j⁰)) = s + (j⁰− j) > s ,

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which implies ¯T_i^k−jTˇ¯_i^k−j⁰ = 0. Conversely, if j⁰ < j, then ¯T_i^k−jTˇ¯_i^k−j⁰ ∈ H^2(s−(j−j⁰⁾⁾(S) and h^jH_r−j⁰ = h^r+(j−j⁰⁾+ c₁h^r+(j−j⁰⁾⁻¹+ · · · + c_r−j⁰h^j

= −c_r−j⁰₊₁h^j−1− · · · − c_r+1h^j−j⁰⁻¹. Again, since

(s − (j − j⁰)) + (r − j⁰+ z) = s + (r + z − j) > s for z > 1, we have ¯T_i^k−jTˇ¯_i^k−j⁰c_r−j⁰_+zh^j−z−1= 0. The result follows.

Now we can determine the difference of the pull-backs of the classes a andF a as follows.

Proposition 2.6. For a class a ∈ H^2k(X), let a⁰ =F a in X⁰. Then φ^0∗a⁰ = φ^∗a + j∗

X

i

X

16j6min{k,r}

a · ˇT¯_i^k−jH_r−jT¯_i^k−jx^j− (−y)^j x + y , where x = ¯φ^∗h and y = ¯φ^0∗h⁰.

Proof. Recall that

N_E/Y = ¯φ^∗OZ(−1) ⊗ ¯φ^0∗OZ⁰(−1) ,

and hence c1(N_E/Y) = −(x + y). Since the difference φ^0∗a⁰− φ^∗a has support in E, we may write φ^0∗a⁰− φ^∗a = j∗λ for some λ ∈ H^2(k−1)(E). Then

(φ^0∗a⁰− φ^∗a)|_E = j^∗j∗λ = c₁(N_E/Y)λ = −(x + y)λ .

Notice that while the inclusion-restriction map j^∗j∗ on H(E) may have non-trivial kernel, elements in the kernel never occur in φ^0∗a⁰− φ^∗a, by the Chow moving lemma. Indeed, if j^∗j∗λ ≡ j∗λ|E = 0, then j∗λ is rationally equivalent to a cycle λ⁰ disjoint from E. Applying φ⁰_∗ to the equation

φ^0∗a⁰− φ^∗a = j∗λ ∼ λ⁰ gives rise to

φ⁰_∗λ⁰ ∼ φ⁰_∗φ^0∗a⁰− φ⁰_∗φ^∗a = a⁰− a⁰ = 0 . This leads to λ⁰∼ 0 on Y .

Hence

λ = − 1

x + y (φ^0∗a⁰)|_E− (φ^∗a)|_E = − 1

x + y φ¯^0∗(a⁰|_Z⁰) − ¯φ^∗(a|_Z) . By Lemma2.4, we get

φ¯^∗(a|_Z) = ¯φ^∗



 X

i

X

j6min{k,r}

a · ˇT¯_i^k−jH_r−jT¯_i^k−jh^j





=X

i

X

j6min{k,r}

a · ˇT¯_i^k−jHr−j

T¯_i^k−jx^j.

Similarly, we have

φ¯^0∗(a⁰|_Z⁰) =X

i

X

j6min{k,r}

a⁰· ˇT¯_i^k−jH_r−j⁰ T¯_i^k−jy^j. SinceF preserves the Poincar´e pairing, we have

a⁰· ˇT¯_i^k−jH_r−j⁰ = F a · F (−1)^r−(r−j)Tˇ¯_i^k−jH_r−j = (−1)^j a · ˇT¯_i^k−jH_r−j.

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Putting these together, we obtain λ =X

i

X

16j6min{k,r}

a · ˇT¯_i^k−jH_r−jT¯_i^k−jx^j − (−y)^j x + y .

Remark 2.7. Notice that since the power in x (and in y) is at most r − 1, the class λ clearly contains non-trivial fiber directions of both morphisms ¯φ and ¯φ⁰. Thus this proposition in particular gives rise to an alternative proof of the equivalence of Chow motives under ordinary flops (Theorem3.6). Indeed, this is precisely the quantitative version of the original proof in [LLW10].

Now, we may compare the triple products of classes in X and X⁰.

Theorem 2.8. Let ai ∈ H^2kⁱ(X) for i = 1, 2, 3 with k1+ k2+ k3 = dim X = s + 2r + 1. Then (F a1·F a2·F a3) = (a1· a₂· a₃) + (−1)^rX

a1· ˇT¯_i^k₁¹^−j¹Hr−j1

a2· ˇT¯_i^k₂²^−j²Hr−j2

× a₃· ˇT¯_i^k³^−j³

3 H_r−j₃

˜

s_j₁_+j₂_+j₃−2r−1T¯_i^k¹^−j¹

1

T¯_i^k²^−j²

2

T¯_i^k³^−j³

3 ,

where the sum is over all possible i1, i2, i3 and j1, j2, j3 subject to the constraints 1 6 jp 6 min{r, k_p} for p = 1, 2, 3 and j₁+ j₂+ j₃ > 2r + 1. Here

˜

s_i := s_i(F + F^0∗) is the ith Segre class of F + F^0∗.

Proof. First of all, φ^0∗F ai = φ^∗a_i+ j∗λ_i for some λ_i ∈ H^2(kⁱ⁻¹⁾(E) which contains both fiber directions ¯φ and ¯φ⁰. Hence

(F a1·F a2·F a3) = (φ^0∗F a1· φ^0∗F a2· (φ^∗a3+ j∗λ3)) = (φ^0∗F a1· φ^0∗F a2· φ^∗a3)

= ((φ^∗a1+ j∗λ1) · (φ^∗a2+ j∗λ2) · φ^∗a3) . When we expand this, the first term is clearly equal to (a1· a₂· a₃).

For those terms with two pull-backs like φ^∗a₁· φ^∗a₃, the intersection values are zero since the remaining part necessarily contains a non-trivial fiber direction of ¯φ.

The terms with φ^∗a3 and two exceptional parts contribute sums over i1, i2, j1 and j2 of products of

φ^∗a3· j_∗T¯_i^k¹^−j¹

1

x^j¹ − (−y)^j¹ x + y

· j_∗T¯_i^k²^−j²

2

x^j² − (−y)^j² x + y

= −φ^∗a3· j∗ T¯_i^k₁¹^−j¹T¯_i^k₂²^−j² x^j¹ − (−y)^j¹

x^j²⁻¹+ x^j²⁻²(−y) + · · · + (−y)^j²⁻¹

(2.1) and (a₁· ˇT¯_i^k¹^−j¹

1 H_r−j₁)(a₂· ˇT¯_i^k²^−j²

2 H_r−j₂). The terms with non-trivial contribution must contain y^q with q > r, which implies j1+ j2− 1 > r; hence these terms are

−(−y)^j¹ x^j²^{−1−(r−j}¹⁾(−y)^r−j¹ + x^j²^{−1−(r−j}¹⁾⁻¹(−y)^r−j¹⁺¹+ · · · + (−y)^j²⁻¹, and their contribution after taking φ∗ is

(−1)^r+1 h^j¹^+j²^−r−1− h^j¹^+j²^−r−2s⁰₁+ · · · + (−1)^j¹^+j²^−r−1s⁰_j₁_+j₂_−r−1 ,

where s⁰_i := s_i(F⁰) is the ith Segre class of F⁰. Here we use the property of Segre classes to obtain φ∗y^q= s⁰_q−r for q > r + 1.

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In a bundle-theoretic formulation,

h^j¹^+j²^−r−1− h^j¹^+j²^−r−2s⁰₁+ · · · + (−1)^j¹^+j²^−r−1s⁰_j₁_+j₂_−r−1

= (1 − s⁰₁+ s⁰₂+ · · · ) 1 + h + h²+ · · ·

j1+j2−r−1

=

s(F^0∗) 1 (1 − h)

j1+j2−r−1

=

c(F )

(1 − h)s(F )s(F^0∗)

j1+j2−r−1

= c(QF) · s(F + F^0∗)

j1+j2−r−1

= Hj1+j2−r−1+ Hj1+j2−r−2˜s1+ · · · + ˜sj1+j2−r−1. With respect to the basis { ˇT¯_i^k}, the term ˜spT¯_i^k₁¹^−j¹T¯_i^k₂²^−j² is of the form

X

i3

˜

s_pT¯_i^k¹^−j¹

1

T¯_i^k²^−j²

2

T¯_i^k³^{−(2r+1+p−j}¹^−j²⁾

3

Tˇ¯_i^k³^{−(2r+1+p−j}¹^−j²⁾

3 .

We define the new index j3 = 2r + 1 + p − j1 − j₂; thus j1 + j2 + j3 > 2r + 1 and p = j₁+ j₂+ j₃− 2r − 1.

By summing all together, we get the result.

There is a particularly simple case where no Hi or Segre classes ˜si are needed in the defect formula, namely the P¹ flops.

Corollary 2.9. For P¹ flops over any smooth base S of dimension s, consider a_i ∈ H^2kⁱ(X) for i = 1, 2, 3 with k1+ k2+ k3= dim X = s + 3. Then

(F a1·F a2·F a3) = (a₁· a₂· a₃) −X

a₁· ˇT¯₁

a₂· ˇT¯₂

a₃· ˇT¯₃( ¯T₁T¯₂T¯₃) with the ¯Ti running over all classes in the chosen basis of H^2(kⁱ⁻¹⁾(S).

There is a trivial but useful observation on when the product is preserved.

Corollary 2.10. For a P^r flop f : X 99K X⁰ and classes a1 ∈ H^2k¹(X) and a2 ∈ H^2k²(X) with k₁+ k₂6 r, we have F (a1· a₂) =F a1·F a2.

This follows from Theorem 2.8, since all correction terms vanish for any a₃. In fact, it is a consequence of a dimension count.

3. Quantum corrections attached to the extremal ray 3.1 The set-up with non-trivial base

Let a_i∈ H^2kⁱ(X) for i = 1, . . . , n with

n

X

i=1

ki= 2r + 1 + s + (n − 3) .

Since

a_i|_Z=X

si

X

ji6min{ki,r}

a_i· ˇT¯_s^k_iⁱ^−jⁱH_r−j_iT¯_s^k_iⁱ^−jⁱh^jⁱ,

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we compute

ha₁, . . . , ani^X_0,n,d`=X

~s,~j

Z

M0,n(Z,d`) n

Y

i=1

ai· ˇT¯_s^k_iⁱ^−jⁱHr−jie^∗_i ψ¯^∗T¯_s^k_iⁱ^−jⁱ.h^jⁱ

· e R¹f t∗e^∗_n+1N

=X

~s,~j n

Y

i=1

(ai· ˇT¯_s^k_iⁱ^−jⁱHr−ji)

" _n Y

i=1

T¯_s^k_iⁱ^−jⁱ· Ψ_n∗

n

Y

i=1

e^∗_ih^jⁱ· e R¹f t∗e^∗_n+1N

!#S

,

where the sum is taken over all ~s = (s₁, . . . , s_n) and all admissible ~j = (j₁, . . . , j_n). By the fundamental class axiom, we must have j_i > 1 for all i.

Here we make use of

[M0,n(X, d`)]^virt= [M0,n(Z, d`)] ∩ e(R¹f t∗e^∗_n+1N ) and the fiber bundle diagram over S

M_0,n+1(Z, d`)

en+1

''

f t

N = N_Z/X

M_0,n(P^r, d`) ^//M_0,n(Z, d`) ^eⁱ ^//

Ψn

Z

ψ¯

vvS

as well as the fact that classes in S are constants among the bundle morphisms (by the projection formula applied to Ψ_n= ¯ψ ◦ e_i for each i).

We must haveP(k_i− j_i) 6 s to get non-trivial invariants. That is,

n

X

i=1

j_i> 2r + 1 + n − 3 .

If the equality holds, then Qn

i=1T¯s^kiⁱ^−jⁱ is a zero-dimensional cycle in S and the invariant readily reduces to the corresponding one on any fiber, namely the simple case, which is completely determined in [LLW10]:

T¯_s^k₁¹^−j¹· · · ¯T_s^k_nⁿ^−jⁿS

hh^j¹, . . . , h^jⁿi^simple_0,n,d` = Y ¯T_s^k_i¹^−j¹

S

N_~jdⁿ⁻³(−1)^(d−1)(n+1).

If, on the contrary, the strict inequality holds, then by dimension counting in the simple case, the restriction of the fiber integral Ψ_n∗(·) to points in S vanishes. In fact, the fiber integral is represented by a cycle S_~j ⊂ S with codimension

ν :=X

j_i− (2r + 1 + n − 3) . The structure of S_~j necessarily depends on the bundles F and F⁰.

One would expect the end formula for Ψn∗(·) to be s_ν(F + F^0∗)N_~jdⁿ⁻³

with N_~j = 1 for n 6 3, so that the difference of the corresponding generating functions on X and X⁰ cancels out with the classical defect on the cup product. Unfortunately, the actual behavior of these Gromov–Witten invariants with base dimension s > 0 is more delicate than this.