QUANTUM COHOMOLOGY AND BIRATIONAL TRANSFORMATIONHIROSHI IRITANIAbstract. A famous conjecture of Yongbin Ruan says that quantum cohomology ofbirational varieties becomes isomorphic after analytic continuation when the birationaltransformation preserves the canonical class (the so-called crepant transformation). Whenthe transformation is not crepant, the quantum cohomology becomes non-isomorphic,
but it is conjectured that one side is a direct summand of the other. In this talk, I
will explain a conjecture that a semiorthogonal decomposition of topological K-groups(or derived categories) should induce a relationship between quantum cohomology. The
relationship between quantum cohomology can be described in terms of solutions to aRiemann-Hilbert problem.

References: 1906.00801 (particularly the last section 8), discussion with Galkin
1. Introduction
X : smooth projective variety
QH(X) = (H ∗ (X), ⋆τ )τ ∈H ∗ (X) : quantum cohomology;
a family of (super)commutative product structures parametrized by τ ∈ H ∗ (X). It isdefined in terms of genus-zero Gromov-Witten invariants.
n∑z }| {1(α ⋆τ β, γ) =⟨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.
We have⋆τ → ∪
in the large radius limit(∫ )2
τ ∈ H (X), ℜ
τ → −∞ for all eﬀective curve classes d ∈ H2 (X, Z) \ {0}dCrepant resolution 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 diagramZAAAAAgAA}}A~}}
ϕX1 _ _ _ _ _ _ _/ X2}

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 China Geometry and Physics Seminar (9 June 2020).
12HIROSHI IRITANI

Discrepant case: 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: 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 :1∂1(τ )∇∂z =− 2 E ⋆τ + µ∂z zzwhere∑
E = c1 (X) +(1 − 12 deg ϕi )τ i ϕii

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 .
i∂τz∂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. First introduced
by Hertling-Sevenheck under the name require no ramifications; later it is calledof exponential type by Katzarkov-Kontsevich-Pantev.
Consider the restriction of QC(X)τ to the formal neighbourhood
QC(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, Hukuhara-Turrittin theorem says that we have a certain similar
decomposition over C((z)) after pulling back by a ramified covering z = wr , r ∈Z>0 .

QUANTUM COHOMOLOGY AND BIRATIONAL TRANSFORMATION

3(2) (analytic lift): this is a fact (Hukuhara-Turrittin theorem) when (1) holds. The
above formal decomposition lifts uniquely to an analytic decomposition over asector S = Sϕ centered at the angle ϕ and of angle > π⊕QC(X)τ ∼eu/z ⊗ Fu=SSu∈Spec(E⋆τ )withS = Sϕ = {z ∈ C× : | arg z − ϕ| < π2 + ϵ}
if 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, we have two analytic continuation maps t± :st+

V−S kVSt−given by ⟨t− (α), β⟩ = [α, β) and ⟨t+ (α), β⟩ = [β, α). These maps constitute theStokes data: the formal decomposition together with the Stokes data reconstructs
the analytic germ of QC(X)τ at z = 0.
+−SS

eiϕ-−(4) (Dubrovin/Gamma conjecture): This is due to Galkin-Golyshev-Iritani (in thesemisimple case) and Sanda-Shamoto (in general case). Under analytic continuation from z = 0 to z = ∞, the above SOD of the space of flat sections inducesban SOD of the topological K-group (or derived category) via the Γ-integralstructure. Roughly speaking, a flat section near z = ∞ corresponds one-to-one with acohomology class in H
H → {flat sections near z = ∞},1This is called mutation system by Sanda-Shamoto.

α 7→ sα (z) ∼ z −µ z c1 (X) α4HIROSHI IRITANI

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

E 7→

1b X ∪ (2πi)deg /2 ch(E)Γ(2π)dim X/2

This 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
the flat connections over S, −S and around z = ∞). This amounts to solving
a Riemann-Hilbert problem (which Dubrovin studied in the semisimple case indetails).
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)⟩. The category A corresponds to the eigenvalue 0.
3. Discrepant transformation
Conjecture: Suppose that we have a birational map ϕ : X1 99K X2 such that KX1 > KX2 .
Then(1) 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)|S
over a sector S corresponds to an SOD
K(X1 ) ∼= KL ⊕ K(X2 ) ⊕ KR
e be the blowup of X along a smooth subvariety Z ⊂ X of codimension
Example: Let X
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πZ

je/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 AND BIRATIONAL TRANSFORMATION

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 Euler eigenvalues align as inthe following picture:
Spec(E Z ⋆τ1 )

..
.

eiϕ−−→Spec(E Z ⋆τc−1 )XSpec(E ⋆σ )






e from the Stokes(2) Using Orlov’s SOD, we can reconstruct the Stokes data of QC(X)e at z = 0.
data of each piece. This recovers the analytic germ of QC(X)(3) we can glue it with the germ of the flat connection near z = ∞ given by
ec1 (X)µ̃∂−+∇τ,z=∞
∼
∂z
2
∂zzzb
e can bevia the Γ-integral structure. The parameter f (τ1 , . . . , τc−1 , σ) for QC(X)
determined. (Locally the parameter space should be the product of (c − 1) copiesof H ∗ (Z) and H ∗ (X).)
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
e is also weak Fano). See Example 7.34 (blowup of P4 along P1 ) where this picturethat Xis fully verified.