QUANTUM COHOMOLOGY OF BLOWUPS: A CONJECTURE
HIROSHI IRITANIAbstract. In this talk, I discuss a conjecture that a semiorthogonal decompositionof topological K-groups (or derived categories) due to Orlov should induce a relationship between quantum cohomology under blowups. The relationship between quantum
cohomology can be described in terms of solutions to a Riemann-Hilbert problem.

References: 1906.00801 (particularly the last section 8), discussion with Sergey Galkin
1. Introduction
Let X be a smooth projective variety and let QH(X) = (H ∗ (X), ⋆τ )τ ∈H ∗ (X) be the quantum cohomology. It is a family of (super)commutative product structures parametrizedby τ ∈ H ∗ (X). It is defined in terms of genus-zero Gromov-Witten invariants.
∑(α ⋆τ β, γ) =nz }| {1⟨α, β, γ, τ, . . . , τ ⟩0,n+3,dn!

n≥0,d∈H2 (X,Z)

where (·, ·) is the Poincaré pairing. We don’t know the convergence in general, but we willassume it.
Rem: We have⋆τ → ∪in the following large radius limit(∫ )2τ ∈ H (X), ℜτ → −∞for all eﬀective curve classes d ∈ H2 (X, Z) \ {0}de.g. τ = −rω with ω ample and r → ∞.
Crepant transformation conjecture (Y. Ruan)
A birational map ϕ : X1 99K X2 is crepant (or K-equivalent) if there exist a smoothprojective variety Z and a commutative diagram
ZAAAAAgAA}}
A}~}
ϕ_______/ X2X1}

f }}}with f, g birational morphisms, such that f ∗ KX1 = g ∗ KX2 .
Conjecture: Then, QH(X1 ) ∼= QH(X2 ) after analytic continuation in τ .
Rem: The isomorphism would depend on the choice of a path connecting the large radius
limit points.
Rem: ∃ an isomorphism as graded vector spaces: H ∗ (X1 ) ∼= H ∗ (X2 ). (Kontsevich,Batyrev, Yasuda, ...)
Talk at Online Algebraic Geometry Seminar at Nottingham (7 October 2020).
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Discrepant transformation conjecture (?) Suppose that f ∗ KX1 ≤ g ∗ KX2 , i.e. g ∗ KX2 −f ∗ KX1 is an eﬀective divisor.
=⇒ QH(X1 ) would be a direct summand of QH(X2 ) (after analytic continuation).
Goal: We want to understand a precise relationship in terms of quantum diﬀerential
equations and certain Betti data (coming from the topological K-group).
2. Structure of quantum connection
Fix τ ∈ H = H ∗ (X). Consider the meromorphic flat connection ∇(τ ) on the trivial
bundle H × Cz → Cz :∂11(τ )∇∂z =− E ⋆τ + µ∂z z 2zwhere∑(1 − 12 deg ϕi )τ i ϕi
E = c1 (X) +i

is the Euler vector field and1µ(ϕi ) = (deg ϕi − dimC X)ϕi2∑
is the grading operator. Here {ϕi } is a homogeneous basis of H = H ∗ (X) and τ = i τ i ϕi .
It is called quantum connection or Dubrovin connection.
µ ∈ End(H),(τ )Fact (Dubrovin): the family of flat connections {∇∂z }τ ∈H is isomonodromic, i.e. can beextended to a flat connection on the bundle H × (H × C) → H × C ∋ (τ, z).
A formula for the extended connection:∂1+ ϕi ⋆ τ .
∇ ∂ =∂τ i z∂τ i(τ )Rem: ∇∂z has regular singularity (or better, logarithmic singularity) at z = ∞;but has irregular singularities at z = 0 (in general) since it has order two poles at z = 0.
Rem: the quantum connection is self-dual with respect to the Poincaré pairing between
the fibers at z and −z.
(τ )Conjecture/Expectation: Write QC(X)τ := (H × Cz → Cz , ∇∂z ).
(1) (formal decomposition); this is expected from mirror symmetry and was introduced by Hertling-Sevenheck under the name require no ramifications; KatzarkovKontsevich-Pantev called it of exponential type.
Consider the restriction of QC(X)τ to the formal neighbourhood of z = 0QC(X)τ := QC(X) ⊗C[z] C[[z]]Then it should admit the following orthogonal decomposition:⊕QC(X)τ ∼(eu/z ⊗ Fu ) ⊗C{z} C[[z]]=u∈Spec(E⋆τ )where Spec(E⋆τ )) is the set of eigenvalues of E⋆τ and– eu/z denotes the rank one connection (C{z}, d + d(u/z));– Fu is a free C{z}-module with regular singular connection (and a pairing)
Rem: In general, the Hukuhara-Turrittin theorem says that we have a formaldecomposition after pulling back by a ramified covering w 7→ z = wr , r ∈ Z>0 .
But here we do ‘not require ramifications’.

QUANTUM COHOMOLOGY OF BLOWUPS: A CONJECTURE

3(2) (analytic lift): this is a fact (the Hukuhara-Turrittin theorem) when (1) holds.
The above formal decomposition lifts uniquely to an analytic decomposition overa sector S = Sϕ centered at the angle ϕ and of angle > π⊕QC(X)τ ∼eu/z ⊗ Fu
=SSu∈Spec(E⋆τ )HereS = Sϕ = {z ∈ C× : | arg z − ϕ| < π2 + ϵ}
and we assume that the direction eiϕ is admissible (in the sense that u1 − u2 ∈/R>0 eiϕ for all u1 , u2 ∈ Spec(E⋆τ )).
(3) (SOD and Stokes data): this is also a general fact provided (1) holds. The above analytic decomposition induces a semiorthogonal decomposition 1 (SOD) of the space
VS of flat sections of the quantum connection over S:⊕VS =Vuu∈Spec(E⋆τ )If we equip VS with the pairing()[s1 , s2 ) = s1 (e−πi z), s2 (z) Poincaréthe decomposition is semiorthogonal in the sense that[Vu1 , Vu2 ) = 0 if ℑ(u1 /eiϕ ) < ℑ(u2 /eiϕ ).
The analytic decomposition associated with the opposite sector −S is dual to theabove decomposition⊕V−S =Vu′u∈Spec(E⋆τ )with respect to the natural pairing V−S × VS → C. Since the sectors S and −Soverlap in two connected components (see the figure below), we have two analyticcontinuation maps t± :st+

V−S kVSt−

These maps satisfy ⟨t− (α), β⟩ = [α, β) and ⟨t+ (α), β⟩ = [β, α). The maps t±
constitute the Stokes data: the formal decomposition together with the Stokes data
reconstructs the analytic germ of QC(X)τ at z = 0.
+

−S  Seiϕ-−(4) (Dubrovin/Gamma conjecture): See Galkin-Golyshev-Iritani (in the semisimplecase) and Sanda-Shamoto (in general case). Under analytic continuation fromz = 0 to z = ∞, the above SOD of the space of flat sections induces an SOD of theb
topological K-group (or derived category) via the Γ-integralstructure. Roughlyspeaking, a flat section near z = ∞ corresponds one-to-one with a cohomologyclass in H
H → {flat sections near z = ∞},α 7→ sα (z) = (1 + O(1/z))z −µ z c1 (X) α1This is called mutation system by Sanda-Shamoto.

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which is then related to a K-class by the map
Ktop (X) → H,

E 7→1(2π)dim X/2b X ∪ (2πi)deg /2 ch(E)ΓThis map intertwines the Euler pairing on the K-group with the above pairing[·, ·) on the space of flat sections.
Conclusion: if we know the formal decomposition of QC(X)τ and the corresponding SOD of the K-group, we can recover the quantum connections (by gluing theflat connections over S, −S and around z = ∞). This amounts to solving a
Riemann-Hilbert problem (recall the work of Dubrovin in the semisimple case).
Example (Sanda-Shamoto): Let X ⊂ Pn be a degree k < n Fano hypersurface. The
eigenvalues of the small quantum multiplication E⋆τ = c1 (X)⋆τ at τ = 0 are{0} ∪ {T ζ : ζ n+1−k = 1}
kwith T = (n+1−k)k n+1−k . After certain mutation, the corresponding SOD of the derived
category is given byDb (X) = ⟨A, O, O(1), . . . , O(n − k)⟩ .
where O, . . . , O(n − k) is an exceptional collection and A is the right orthogonal of⟨O, O(1), . . . , O(n − k)⟩ (Kuznetsov). The category A corresponds to the eigenvalue 0.
3. Discrepant transformation/Blowups
Conjecture: Suppose that we have a birational map ϕ : X1 99K X2 such that KX1 ≥ KX2 .
Then we have(1) an orthogonal decomposition QC(X1 )τ = L⊕QC(X2 )f (τ ) ⊕R for some connections
L, R.
(2) the analytic lift
QC(X1 )τ |S ∼= (L ⊕ QC(X2 )f (τ ) ⊕ R)|Sover a sector S corresponds to an SODK(X1 ) ∼= KL ⊕ K(X2 ) ⊕ KR
e be the blowup of X along a smooth subvariety Z ⊂ X ofThe case of blowups: Let X
codimension c. We have Orlov’s SOD:⟨⟩
e = Db (Z)−(c−1) , . . . , Db (Z)−1 , Db (X)
Db (X)
e given by α 7→
where Db (Z)k is the image of the fully faithful functor Db (Z) → Db (X)∗j∗ (O(k) ⊗ π (α)):
EπZje/X
/Xe should be recovered
where E is the exceptional divisor. The quantum connection QC(X)from those for Z and X as follows:

QUANTUM COHOMOLOGY OF BLOWUPS: A CONJECTURE

5e := QC(Z)τ ⊕ · · · ⊕ QC(Z)τ
(1) Put QC(X)
Here we assume that1c−1 ⊕ QC(X)σ .
∗∗τi ∈ H (Z) and σ ∈ H (X) are chosen so that the eigenvalues of the Euler vector
fields align as in the following picture:
Spec(E Z ⋆τ1 )

..
.

eiϕSpec(E Z ⋆τc−1 )XSpec(E ⋆σ )−−→


i.e. the collections Spec(E Z ⋆τ1 ), . . . , Spec(E Z ⋆τc−1 ), Spec(E X ⋆σ ) of eigenvalues are
separated by lines parallel to an admissible eiϕ and line up in this order.
e from those of(2) Using Orlov’s SOD, we reconstruct the Stokes data of QC(X)QC(Z)τi and QC(X)σ . Namely, the Stokes data for QC(Z)τi and QC(X)σ determine SODs of Ktop (Z) and Ktop (X) (for a given phase eiϕ ); they are glued
e via the Orlov decomposition
together to give an SOD of Ktop (X)e ∼Ktop (X)= Ktop (Z)−(c−1) ⊕ · · · ⊕ Ktop (Z)−1 ⊕ Ktop (X)e This recovers the analytic germ ofand give rise to the Stokes data for QC(X).
eQC(X) at z = 0.
(τ,z=∞)
b(3) we can glue it (via the Γ-integralstructure) with the germ ∇∂zof the flat
connection near z = ∞ given byec1 (X)∂µ̃(τ )
(τ,z=∞)−∇∂z gauge∼∇∂z:=+ .
2∂zzz
equivalentThen we obtain a global vector bundle over P1z which should be trivial. The gaugetransformation between ∇(τ ) and ∇(τ,z=∞) gives a fundamental solution (the soe can also becalled calibration). The parameter τ = f (τ1 , . . . , τc−1 , σ) for QC(X)
determined. (Locally the parameter space should be isomorphic to the product of(c − 1) copies of H ∗ (Z) and H ∗ (X) as an F -manifold.)
Rem: This picture has been partially verified in 1906.00801 when X is a weak Fano torice is a weighted blowup of X along Z (we assumeorbifold, Z is a toric suborbifold, and X
ethat X is also weak Fano).
e → X be the blowup
Example (Example 7.34 in 1906.00801) Let X = P4 and let φ : X12eealong a line P . We have a projection π : X → P and X is a P2 -bundle over P2 . Set
p1 = π ∗ H and p2 = φ∗ H and consider the parameterτ = p1 log q1 + p2 log q2
There are two limiting pictures:
fibration picture: |q1 | ≪ |q2 |, −τ approaches the ample class on P2
blow-down picture: |q2 | ≪ |q1 |, −τ approaches the ample class on X = P4
In these limits, the quantum cohomology decomposes diﬀerently (see the figure in Example7.34).
e τ ∼QH∗ (X)= QH(base P2 )t ⊕ QH(base P2 )t ⊕ QH(base P2 )t
12e τ ∼QH (X)= QH(P1 )τ1 ⊕ QH(P4 )σ ⊕ QH(P1 )τ2∗3