Decomposition of quantum cohomology underblowups∗
Hiroshi Iritani

Abstract
Quantum cohomology is a deformation of the cohomology ring defined bycounting rational curves. A close relationship between quantum cohomology
and birational geometry has been expected. For example, when the quantum
parameter Q approaches an “extremal ray”, the spectrum of the quantum
cohomology ring clusters in a certain way (predicted by the corresponding
extremal contraction), inducing a decomposition of the quantum cohomology.
In this talk, I will discuss such a decomposition for blowups: quantum cohomology of the blowup of X along a smooth center Z will decompose intoQH(X) and (codim Z-1) copies of QH(Z). The proof relies on Fourier analysis and shift operators for equivariant quantum cohomology. We can describe
blowups as a variation of GIT of a certain space W with C∗ action. The equivariant quantum cohomology of W acts as a “global” mirror family connecting
X and its blowup.

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Introduction

First we explain how a decomposition of quantum cohomology naturally arises frombirational geometry.
Let X be a smooth projective variety. The quantum cohomology QH(X) is asupercommutative algebra over the Novikov ring
C[[Q]] = C[[NEN (X)]]
which is the completed group ring of the monoid NE
P N (X) ⊂ H2 (X, Z) generated byeffective curve classes. It consists of formal sums d∈NEN (X) cd Qd with cd ∈ C.
Consider the Mori cone NE(X) ⊂ H2 (X, R) spanned by NEN (X). An extremal
ray R = R≥0 d0 ⊂ NE(X) is a 1-dimensional face of the Mori cone generated by aclass d0 satisfying c1 (X) · d0 > 0. By the Contraction Theorem, we have an extremal
contraction f : X → Y to a normal projective variety Y such that a curve C ⊂ X
contracts to a point if and only if the homology class [C] lies in the ray R.
Associated to the extremal contraction f , we have the f -exceptional quantum cohomology defined by counting only f -exceptional curves1 . The f -exceptional (small)∗Talk at M-seminar at Kansas State University, 18 April 2024 (online)
A similar quantum ring has been introduced by Yongbin Ruan in his study of crepant resolution
conjecture.
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quantum product is defined by(α ⋆f β, γ) =X⟨α, β, γ⟩0,3,nd0 Qnd0n≥0where (·, ·) is the Poincaré pairing and ⟨α, β, γ⟩0,3,d is the 3-point Gromov-Witten
invariant. An important thing is that the left-hand side is a finite sum since c1 (X) ·d0 > 0. If we write QHexc (X) for the exceptional quantum cohomology, it is therefore
defined over the polynomial ring C[q] with q := Qd0 .
The exceptional quantum cohomology QHexc (X) arises as a limit of QH(X).
Conversely we may regard QH(X) as a deformation2 of QHexc (X). Therefore, wehave a natural ring decomposition
MQH(X) ∼Aα=α

indexed by points α of the zero-dimensional scheme Spec(QHevexc (X)|q=1 ). A naturalquestion here is:
Does each summand Aα arise from (big) quantum cohomology of a space?
There are several examples where the answer is affirmative.
Example 1.1 ([I19]). Let X be a smooth toric DM stack. Let X 99K X + be atoric flip (or more generally a discrepant transformation arising from a toric VGIT).
Then QH(X) contains (big) quantum cohomology of X + as a direct summand.
Example 1.2 ([Koto-I]). Let V → Y be a vector bundle of rank r ≥ 2. Let
X = P(V ) → Y be the projective bundle over Y . In this case the exceptional
quantum cohomology associated with the extremal contraction f : X → Y isQHexc (X) ∼= H ∗ (Y )[p, q]/(pr + c1 (V )pr−1 + · · · + cr (V ) − q)rM∼
H ∗ (Y ) as a ring at q ̸= 0.
=i=1We have the corresponding decomposition (“quantum Leray-Hirsch”)
QH(X)τ ∼=r
M

QH(Y )σi (τ )i=1where the subscripts τ ∈ H ∗ (X), σi (τ ) ∈ H ∗ (Y ) are big quantum cohomology
parameters (bulk deformation parameters): the map τ 7→ (σ1 (τ ), . . . , σr (τ )) defines
a formal isomorphism. [Draw a picture of the deformation.]
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More precisely, we take the graded completion when defining the Novikov ring (with respect
to the grading deg Qd = 2c1 (X) · d). Therefore, the Novikov ring C[[Q]] can be presented as aring of the form C[q][[Q′ ]] consisting of formal power series in the other Novikov variables Q′ withcoefficients in the polynomial ring in q = Qd0 .

2e = BlZ X be theExample 1.3 ([I23]). This is the main theme of this talk. Let X
blowup of a smooth projective variety X along a smooth subvariety Z ⊂ X. Let r bee → X is an extremal contraction
the codimension of Z. The blowdown morphism X
and the associated decomposition ise τ̃ ∼QH(X)= QH(X)τ ⊕r−1MQH(Z)σii=1e τ = τ (τ̃ ) ∈ H ∗ (X), σi = σi (τ̃ ) ∈ H ∗ (Z) are big quantum cohowhere τ̃ ∈ H ∗ (X),mology parameters. Again τ̃ 7→ (τ (τ̃ ), σ1 (τ̃ ), . . . , σr−1 (τ̃ )) is a formal isomorphism.
[Draw a picture of the deformation.]e = OPr−1 (−1) andFor example, if X = Cn , Z = pt, then X
e = {(p, q) : p(pr−1 + q) = 0}.
Spec QH(X)The configuration of the eigenvalues of the Euler vector field (or the quantum
multiplication by c1 (X) over the small quantum cohomology locus) is similar to1{0} ∪ {(−1) r−1 } near the “exceptional quantum cohomology limit”.

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More precise statement

We give a slightly more precise statement of the result of [I23]. We discuss the bigquantum cohomology and a D-module structure on it. The big quantum cohomology
is a supercommutative algebra(H ∗ (X) ⊗ C[[Q, τ ]], ⋆τ )parametrized (formally) by τ ∈ H ∗ (X) (sometimes calledP
bulk deformation parameter). Let {ϕi } be a basis of H ∗ (X) and we expand τ = i τ i ϕi . The ring C[[Q, τ ]]is defined3 to be C[[Q]][[τ 0 , . . . , τ s ]]. The quantum product ⋆τ is given by(α ⋆τ β, γ) =X⟨α, β, γ, τ, . . . , τ ⟩0,n+3,dn≥0

Qd.
n!

We introduce a new variable z and write
QDM(X) = H ∗ (X) ⊗ C[z][[Q, τ ]]for the quantum D-module. It is equipped with the following quantum connection(also known as Dubrovin connection):∂1+ (ϕi ⋆τ )i∂τz∂
1∇z∂z = z− (E⋆τ ) + µ∂z z∂1∇ξQ∂Q = ξQ+ (ξ⋆τ ) with ξ ∈ H 2 (X)∂Q z∇τ i =3We treat τ i as supercommuting variables.

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P
where E = c1 (X)+ i (1− 12 deg ϕi )τ i ϕi is the Euler vector field and µ ∈ End(H ∗ (X))
is the grading operator defined by µ(ϕi ) = ( 21 deg ϕi − n2 )ϕi (with n = dimC X). They
are supercommuting operators∇τ i , ∇z∂z , ∇ξQ∂Q : QDM(X) → z −1 QDM(X).
The quantum D-module is also equipped with the pairingZP (f, g) =f (−z) ∪ g(z),f, g ∈ QDM(X)X

given by the Poincaré pairing. It is flat with respect to the connection.
e →Theorem 2.1 ([I23]). There exists a formal invertible map (τ, σ1 , . . . , σr−1 ) : H ∗ (X)H ∗ (X) ⊕ H ∗ (Z)⊕(r−1) and an isomorphisme ∼QDM(X)= τ ∗ QDM(X) ⊕r−1
M

σi∗ QDM(Z)i=1

preserving the connection and the pairing.
e QDM(X), QDM(Z) are origRemark 2.2. (1) The quantum D-modules QDM(X),e C[[NEN (X)]], C[[NEN (Z)]].
inally defined over the different Novikov rings C[[NEN (X)]],
We need an extension of the Novikov rings to a common one. The common ringarises from the equivariant second homology group of a space W with C× -action (seebelow).
(2) We need to invert the variable q associated with the extremal ray. It corresponds to the class of a line in the exceptional divisor P(NZ/X ) that contracts to ae → X. The isomorphism is defined over the ring of the form
point in Z under X
C[z]((q −1/s ))[[Q, τ̃ ]]where s = r − 1 when r is even and s = 2(r − 1) when r is odd.
The above result has the following corollary:
Corollary 2.3. The derivative of τ̃ 7→ (τ, σ1 , . . . , σr−1 ) defines a ring isomorphisme τ̃ ∼QH(X)= QH∗ (X)τ ⊕r−1
M

QH(Z)σii=1

Remark 2.4. Katzarkov-Kontsevich-Pantev-Yu have announced application of thisresult to birational geometry, e.g. irrationality of generic cubic 4-fold. They alsoe from QH(X) and QH(Z) together withgive a reconstruction algorithm of QH(X)
certain classical topological data (Chern classes of normal bundles NZ/X etc).

43Idea of Proof

The basic idea is to describe blowups as a variation of GIT quotient. We also use
Teleman’s conjecture for quantum cohomology of reduction, adapted to quantum
D-modules. We prove Teleman’s conjecture in a special case, and the blowup result
follows from it.
Let W = BlZ×{0} (X × P1 ) be the blowup of X × P1 along Z × {0}. Consider
the C× action on W induced by the C× -action on P1 . We have two smooth GITquotients:eW//C× = X or X
depending on the stability conditions. [Draw a picture of W ]. The C× -fixed locusof W is×e ⊔ Z.
WC = X ⊔ X
Roughly speaking, a quantum D-module version of Teleman’s conjecture saysthat we have an isomorphism:(3.1)

QDMC× (W ) ∼= QDM(W//C× ).

This isomorphism should intertwine(λ: equivariant parameterwith
S: shift operator(z∇q∂q : quantum connection in the direction κ(λ)
qwhere κ(λ) ∈ H 2 (W//C× ) is the image of λ under the Kirwan map. The isomorphism(3.1) should be viewed as a Fourier transformation between difference modules andD-modules.
Remark 3.1. The C× -equivariant quantum D-module is equipped with the actionof a shift operator S : QDMC× (W ) → QDMC× (W ) of equivariant parameters, andwe have the commutation relation [λ, S] = zS, i.e. S ◦ λ = (λ − z) ◦ S.
The isomorphism (3.1) yields an obvious contradiction QDM(X) ∼= QDMC× (W ) ∼=eQDM(X), so it should be interpreted with care.
We explain how the equivariant quantum D-module QDMC× (W ), after Fouriertransformation, plays a role of a ‘global mirror’.
×The equivariant second homology H2C (W, Z) ∼= H2 (W, Z)⊕Z acts on QDMC× (W )[Q−1 ]with Novikov variables inverted: the H2 (W, Z) part acts as ordinary Novikov vari×ables and the Z = H2C (pt) part acts as shift operators. We can introduce anequivariant Mori monoid
××
NECN (W ) ⊂ H2C (W, Z)
that preserves the lattice QDMC× (W ) ⊂ QDMC× (W )[Q−1 ]. The equivariant Morimonoid is an extension of NEN (W ) ⊂ H2 (W, Z).
×The dual cone of the equivariant Mori cone NEC (W ) is the C× -ample cone4
CC× (W ) ⊂ HC2 × (W, R) consisting of equivariant ample classes [L] such that the4Introduced by Dolgachev-Hu and Thaddeus

5stable locus Wst (L) is nonempty. In the case at hand, there are two GIT chambersof this cone:
CC× (W ) = C X ∪ C Xee This gives a “fan” as in the following
corresponding to the GIT quotients X and X.
picture.
µX-cuspX-chambereX-cuspe
X-chamber0

H 2 (W, R)

ωAmp(X)

toric variety

By the Fourier transformation, we can regard QDMC× (W ) as a sheaf over thetoric variety associated with this fan. The quantum D-modules QDM(W//C× ) ofthe GIT quotients arise as certain completions of QDMC× (W ) at the corresponding
cusps.
We construct Fourier transformations:
QDMC× (W )FXewe
QDM(X)FZi

QDM(Z)FX'QDM(X)and show thate• FXe is an isomorphism after completion at the X-cups;• FX ⊕ FZ1 ⊕ · · · ⊕ FZr−1 extends to the completion and is also an isomorphism.

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