GAMMA CONJECTURE AND TROPICAL GEOMETRY
HIROSHI IRITANI

Abstract. Hodge-theoretic mirror symmetry for a Calabi-Yau mirror pair says that thevariation of Hodge structure arising from quantum cohomology of a Calabi-Yau manifold
and that arising from deformation of complex structures on the dual Calabi-Yau manifoldbb
can be identiﬁed with each other, and it has been conjectured ( Γ-conjecture)that the Γintegral structure [10] in quantum cohomology corresponds to a natural integral structurebon the mirror side. Here the Γ-integral
structure is deﬁned via the topological K-group andbthe Γ-class, a characteristic class with transcendental coeﬃcients containing the Riemann ζbvalues. In this article, we explain an approach to the Γ-conjectureusing tropical geometry andobserve that the Riemann ζ-values arise as error terms of tropicalization in the computationof mirror periods. This is based on joint work [1] with Abouzaid, Ganatra and Sheridan.

1. Mirror symmetry
Mirror symmetry is a conjectural duality between symplectic and complex geometry. Itroughly speaking predicts that symplectic topology on a symplectic manifold Y “corresponds”(or equivalent) to complex geometry on another complex manifold Z. Then Z is called amirror of Y and vice versa. In this article, we consider two kinds of mirror correspondences:
Calabi-Yau mirror pairs and Fano/LG mirror pairs.
Calabi-Yau mirror pairs:(Y, (− log t)ω)←→(Zt , Ωt )t∈∆∗In the left-hand side, we consider a Calabi-Yau manifold Y equipped with a Kähler
form ω whose cohomology class is integral: they give a family (Y, (− log t)ω) of symplectic manifolds parametrized by small t > 0. On the right-hand side we consider afamily of Calabi-Yau manifolds Zt of the same dimension equipped with a holomorphic volume form Ωt . The family {Zt } maximally degenerates at t = 0 in a suitablen (Z ) is maximally unipotent (i.e. (M − id)n 6=
sense: the monodromy M on Hprimt
0, (M − id)n+1 = 0 for n = dim Zt ) and the limiting mixed Hodge structure is ofHodge-Tate type (see [2]). The limit point t = 0 is called the large complex structure
limit.
Fano/LG mirror pairs:F←→

W : (C× )n → C

In the left-hand side we consider a Fano manifold F (or a monotone symplectic manifold), i.e. c1 (F ) is represented by a positive (1, 1)-form, which gives a symplectic form.
In the right-hand side we consider a Landau-Ginzburg model, a Laurent polynomial
function W on the algebraic torus (C× )n with n = dim F .
Date: 19 December 2021.
1

2

HIROSHI IRITANIOn the complex geometry side, we consider (exponential) periods, namely, integrals of thefollowing formZΩt
for an n-cycle Ct (for Calabi-Yau mirrors)
Ct ⊂Zt
Z
dx1
dxn
e−tW
···
for a not necessarily compact n-cycle Γ (for Fano mirrors)x1xnΓUnder mirror symmetry, they should yield solutions to the quantum diﬀerential equation on
the symplectic side, which is deﬁned by counting rational curves (the genus-zero GromovWitten invariants). In the Calabi-Yau case, the mirror correspondence can be formulated as
an isomorphism of variation of Hodge structure (VHS) as given in Table 1.
Table 1. Mirror correspondence on the level of VHS (see e.g. [11]). Here {ϕi }P
1,1
is a basis of Halg
(X) and τ = ri=1 τ i ϕi .

bundleconnection
ﬁltration
polarization
Z-structure

L

symplectic side

p,p
1,1
1,1
H
(Y
)
× Halg(Y ) → Halg
(Y )
p alg

complex sideSn∗t Hprim (Zt ) → ∆rXquantum connection ∇ = d +(ϕi ⋆τ )dτ i
Gauss-Manin connection ∇GM
i=1
M
M n−k,k
k,k
Fp =
Halg
(Y )
Fp =
Hprim (Zt )
k≤n−p
k≥p
Z
Z
n(n−1)
deg /2n2((−1)α) ∪ β
α∪β
(2πi)
(−1)
Y
X
L
k,k
0 (X) →bΓ-integral
structure Kalg
H n (Zt , Z)
k Halg (X),b X (2πi)deg /2 ch(V )V 7→ Γ

Remark 1.1. In Table 1, we restrict our attention to the algebraic part of cohomology in thesymplectic side, and the primitive part of the middle cohomology in the complex side. Theb
Γ-integral
structure is also restricted to the K-group of algebraic vector bundles. We do notknow much about Hodge-theoretic mirror symmetry beyond these parts. On the symplecticbside, we can naturally deﬁne the Γ-integral
structure corresponding to the topolgoical K-group[10], so it might be possible to consider an integral structure on the complex side corresponding∗ (Y ) ∼ K ∗ (Z) has been
to the topological K-group as well. A conjectural isomorphism Ktop
= top
discussed in the literature [21].
b2. The Γ-class
Let X be an almost complex manifold. We write the total Chern class of the tangent bundle
T X asc(T X) = (1 + δ1 )(1 + δ2 ) · · · (1 + δn )where δ1 , . . . , δn are virtual cohomology classes called the Chern roots. Each δi may not exist
as a cohomology class, but any symmetric functions in δ1 , . . . , δn can be written as polynomialsb
in c1 (T X), . . . , cn (T X) and make sense as cohomology classes. The Γ-classof X is deﬁned tobeb X = Γ(1 + δ1 ) · · · Γ(1 + δn ) ∈ H ∗ (X, R)ΓGAMMA CONJECTURE AND TROPICAL GEOMETRY

3R∞bwhere Γ(1 + x) = 0 e−t tx dt/t is the Euler Γ-function. By the Taylor expansion of the Γfunction, we can think of the right-hand side as a symmetric power series of δ1 , . . . , δn : thenb
the right-hand side makes sense as a cohomology class of X. The Γ-function
Γ(1 + x) has
simple poles at x = −1, −2, −3, . . . and it has the following inﬁnite product expansion:Γ(1 + x) = Q

e−γx
−x/n
n≥0 (1 + x/n)e

where γ = limn→∞ (1 + 21 + · · · + n1 − log n). This can be calculated asΓ(1 + x) = e−γx

=e

−γx

∞
Y
n=1
∞
Y

e− log(1+ n )+ n
x

exp

n=1

= exp −γx +∞X(−1)k xk
k=2
∞
X
k=2

where ζ(k) =written asP∞1
n=1 nk

x

k

!

nk

ζ(k) k
(−1)k
x
k

!

b
is the value of the Riemann ζ-function. Hence the Γ-classcan beb X = exp −γc1 (X) +
Γ

∞
X
k=2

ζ(k)
(−1)
chk (T X)(k − 1)!

!

kb
Remark 2.1. The Γ-class
has the following geometric interpretation. We consider a free loop
space LX equipped with the S 1 -action rotating loops. We also consider the set X of constantbloops in LX. Then the Γ-classcan be interpreted as a regularization of the S 1 -equivariant
Euler class of the positive normal bundle N+ of X in LX [15, 4]; here positive means the
positive weight part as an S 1 -representation. We haveb X ∼ eS 1 (N+ ) = Q Q(2π)n/2 z n− 2 z c1 (X) Γ
deg

i

1
.
k≥0 (δi + kz)b
3. Mirror symmetric Γ-conjecturebb
The mirror symmetric Γ-conjectureroughly speaking says that the Γ-integral
structure [10]on the symplectic side should correspond to a natural integral structure on the complex side.
In this article, we do not discuss Hodge-theoretic mirror symmetry with integral structurebanymore (we do not even give the deﬁnition of quantum cohomology or the Γ-integral
structure
on it): we refer the reader to e.g. [12, 3, 11]. Instead, we discuss a more concrete conjecture,b“mirror symmetric Γ-conjecture”
stated in terms of (exponential) periods. This problem ismore of a topological nature, and does not involve counting rational curves.
b
The mirror symmetric Γ-conjecture
originates from Hosono’s conjecture [9], which saysthat mirror periods equal the pairing of certain explicit hypergeometric series with the Chernclasses of vector bundles, for Batyrev mirror pairs of Calabi-Yau hypersurfaces. By taking
the asymptotics of Hosono’s conjecture at the large complex structure limit, we arrive at thefollowing conjecture.

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HIROSHI IRITANI

Calabi-Yau case: Let (Y, (− log t)ω) and (Zt , Ωt ) be a Calabi-Yau mirror pair as above.
For a certain family of n-cycles Ct ⊂ Zt , we have a K-theory class V on Y such thatZZ
b Y (2πi)deg /2 ch(V ) + O(tϵ )
(1)
Ωt =
t−ω Γ
Ct ⊂ZtY

as t → +0, where ϵ > 0 is a positive real number.
Fano/LG case: Let F and W : (C× )n → C be a Fano/LG mirror pair as above. For acertain (possibly noncompact) n-cycle Γ ⊂ (C× )n , we have a K-theory class V on F
such thatZZ
dx1 · · · dxn
b F (2πi)deg /2 ch(V ) + O(tϵ )
(2)
e−tW
=
t−c1 (F ) Γ
x
·
·
·
x
1
n
Γ
F
as t → +0, where ϵ > 0 is a positive real number.
This conjecture has been veriﬁed for (weak) Fano toric orbifolds and certain complete
intersections in them, for some choices of V and cycles Ct (or Γ), see [10, 11]. The paper [1]gives another proof for Batyrev mirror pairs based on the SYZ picture and tropical geometry.
Remark 3.1. (A) This conjecture is closely related to homological mirror symmetry (withsymplectic side and complex side interchanged). Homological mirror symmetry predicts thatthe derived category of coherent sheaf on one side should be equivalent to the Fukaya (orFukaya-Seidel) category of the other side:
Db Coh(Y ) ∼= Db Fuk(Zt )Db Coh(F ) ∼= Db FS(W )for Calabi-Yau mirror pairs
for Fano/LG mirror pairs

We expect that the K-classes V and the cycles Ct (or Γ) in the above conjecture should
correspond to each other under homological mirror symmetry. Namely, when V comes froma coherent sheaf V ∈ Db Coh(Y ) or Db Coh(F ), the cycle Ct (or Γ) should be the Lagrangiansubmanifold L mirror to V.
(B) The categorical equivalence in homological mirror symmetry is given only up to autoequivalences, and hence the correspondence in (A) is ambiguous. In this conjecture, more
precisely, we should consider the equivalence induced from Strominger-Yau-Zaslow (SYZ)
dual torus ﬁbrations (see [19, 6, 7]). The SYZ conjecture says that (in the Calabi-Yau case)
we have special Lagrangian torus ﬁbrations1 p1 : Y → B, p2 : Zt → B with singularities
Y?

??
??p1
??
?

p2 ~~~

B

Zt

~~
~~~

that are dual to each other, where B is a real n-dimensional manifold homeomorphic to asphere. Here Y and Zt are equipped with Ricci-ﬂat Kähler metrics. It is expected that Zt
converges2, as t → 0, to the base B in the sense of Gromov-Hausdorﬀ topology (where wenormalize the metric so that the diameter is constant). The SYZ ﬁbrations should induce
1For tropical computation of periods, we do not need Ricci-ﬂat metrics or special Lagrangian ﬁbrations: weonly need a weaker version as in the Gross-Siebert program [8].
2Likewise, we can consider a maximal degeneration of complex strcutures on Y , which corresponds under
mirror symmetry to the large-radius limit for the Kähler (symplectic) structure on Zt , and the complex degeneration induces the collapse Y → B. When we take into account both the symplectic and complex structures,
we should consider a mirror pair (Ys , (− log t)ωY ) ↔ (Zt , (− log s)ωZ ) of maximally degenerating families.

GAMMA CONJECTURE AND TROPICAL GEOMETRY

5

categorical equivalences as above3, and it is expected that a Lagrangian section of p2 : Zt → B
corresponds to a line bundle on Y . In particular, if we take a Lagrangian section C0,t (or Γ0 )corresponding to the structure sheaf, we haveZZ
b Y + O(tϵ )
Ωt =
t−ω Γ
(in the Calabi-Yau case)C0,t ⊂Zt

Z

(3)

e−tW
Γ0

dx1 · · · dxn=
x1 · · · xnZYb F + O(tϵ )t−c1 (F ) Γ(in the Fano case)F

as t → +0. In examples such as Batyrev mirrors, C0,t arises as a “positive real locus”. In theFano case, we expect that Γ0 = (R>0 )n .
When V is the class of the structure sheaf of a point, the corresponding cycle should be aﬁbre p−1
2 (b) of the SYZ ﬁbration p2 : Zt → B in the Calabi-Yau case, and the compact torus(S 1 )n ⊂ (C× )n in the Fano case. Since ch(V ) = [pt] in this case, we do not see nontrivialbcomponents of the Γ-classin the asymptotics of Rthe corresponding (exponential) periods.
1 nn
Tonkonog [20] showed that the exponential period (S 1 )n e−tW dxx11 ···dx···xn of (S ) is a generating
series of gravitational descendants of F (the pairing of the J-function and [pt]), when one
chooses a monotone Lagrangian submanifold L in F and deﬁnes W by counting holomorphic
discs with boundaries in L. See also (C) below.
b(C) The above conjectures (1), (2), (3) say that the Γ-classappears in the asymptotics of(exponential) periods of the mirror in the large complex structure limit t → 0. The leading
asymptotics are polynomials in log t. The information of curve counting is contained in thehigher order terms in t, which are exponentially small compared to the asymptotic part. If
we include all the higher-order terms, the right-hand side in the conjecture should becomeZb Y (2πi)deg /2 ch(V )JY (ω log t, −1) ∪ Γ(in the Calabi-Yau case)YZ
b F (2πi)deg /2 ch(V )JF (c1 (F ) log t, −1) ∪ Γ(in the Fano case)F

where JX (τ, z) is the small J-function
JX (τ, z) = eτ /z 1 +X
iXd∈H2 (X,Z)

ϕiz(z − ψ)e⟨τ,d⟩ ϕi 

deﬁned in terms of gravitational Gromov-Witten invariants (see [4, 3, 12] for the notation). In
the Calabi-Yau case, we need to normalize the volume form Ωt by a Hodge-theoretic condition,
as discussed in [2]; we also need to assume that the parameter t of the mirror family {Zt } isnormalized so that the mirror map is trivial τ = ω log t.
Example 3.2. Let Y be a Calabi-Yau 3-fold. The asymptotic part of the ﬁrst equation of(3) takes the form:
Z
Z
Z
Z
ω3
b = (− log t)3t−ω Γ−(− log t)ζ(2)ω ∪ c2 (Y ) − ζ(3)c3 (Y )Y 3!
Y
YY
| {z }| {z }
volume of Y

Euler number

Note that the leading term is the symplectic volume of (Y, (− log t)ω).
3More precisely, the categorical equivalence would be given up to the twist by a line bundle, and theambiguity would be ﬁxed by choosing a Lagrangian section that corresponds to the structure sheaf.

6

HIROSHI IRITANIExample 3.3. Let F be the projective space Pn and let Y be a degree n + 1 Calabi-Yau
hypersurface in F = Pn . The mirror of F is given by the Laurent polynomial
W = x1 + · · · + xn +1
x1 · · · xn

and the mirror of Y is given by a Calabi-Yau compactiﬁcation Zt of the ﬁbre W −1 (1/t)
equipped with the holomorphic volume form
Ωt =

dx1
dxn
x1 ∧ · · · ∧ xn

t · dW.
W −1 (1/t)

b
The mirror symmetric Γ-conjecture
for these mirror pairs holds when V is the structure sheaf
and the cycle is the positive real locus [10, 11]. We haveZZ
b Y + O(tϵ )Ωt =t−c1 (F ) ΓnZt ∩(R>0 )
Y
Z
Z
dx
·
·
·
dx
1
n
b F + O(tϵ )t−c1 (F ) Γ
=
e−tW
x
·
·
·
dx
n
1
n
F
(R>0 )
for any ϵ with 0 < ϵ < n + 1. The higher order term can be also given explicitly as hypergeometric series. We remark that the exponential period in the second line can be written as
a Fourier transform of the period in the ﬁrst line. (The ﬁrst identity was obtained from thesecond one by inverse Fourier transformation in [11].)bRemark 3.4. The Γ-conjecture
for Fano manifolds [4] proposed by Galkin, Golyshev andthe author does not rely on mirror symmetry (although it sometimes follows from the mirrorbsymmetric Γ-conjecture).
It is formulated purely in terms of quantum cohomology of a Fanobmanifold. The Γ-conjecture
in [4] is related to the Stokes structure of the quantum connection.
4. Periods via tropical geometry
We explain how to compute the asymptotics (3) of periods using tropical geometry. Firstly
we see that the termZωn(− log t)n
Y n!
Rshould appear as the leading term of the period C0,t Ωt using the SYZ picture. Away fromthe descriminant locus (the singular locus of the ﬁbration), the base B is equipped with aZ-aﬃne structure (i.e. an atlas as a topological manifold such that every coordinate change
is Z-aﬃne linear, that is, belongs to GL(n; Z) ⋉ Rn ) and the SYZ dual ﬁbrations are locallymodelled on
T ∗ B/Λ∗

HH
HHp1
HH
HH
H#

T B/Λ

p2 yyy

B

yy
yy
|yy

where Λ ⊂ T B is the lattice deﬁned by the Z-aﬃne structure. Let x1 , . . . , xn be Z-aﬃne
local
on B. Then the symplectic form on T ∗ B/Λ∗ is given by (− log t)ω =Pn coordinates∗
j=1 dxj ∧ dyj and the holomorphic volume form on T B/Λ is given by Ω = dz1 ∧ · · · ∧ dzn

GAMMA CONJECTURE AND TROPICAL GEOMETRY

7

with zj = xj + iyj , where yj∗ are ﬁbre coordinates on T ∗ B dual to dxj and yj are ﬁbre
coordinates on T B dual to ∂/∂xj . Hence we have((− log t)ω)np1∗= aﬃne volume form dx1 ∧ · · · ∧ dxn on B = s∗ Ωn!
where s is a section of p2 . From this we expect the leading asymptotics:
ZZωn
as t → +0.
Ωt ∼ (− log t)n
C0,t
Y n!
The error term of this approximation arises from the discriminant locus of the SYZ ﬁbration.
We explain that tropicalization gives an approximate SYZ(-like) ﬁbration and how we cancompute periods tropically by means of example.
3

2

0

1

-3

-2

-1

1

2

3

x

x0

x1

-1

y
-2

-3

Figure 1. Amoeba

Figure 2. Tropical amoeba

Figure 3. Pair of pants (P1 minus three points, on the left) and stretching
the neck (on the right)
Example 4.1. We start with a simple example of tropicalization. Consider the variety
P = {(X, Y ) ∈ (C× )2 : X + Y + 1 = 0}.
This is P1 minus three points, and is called a ‘pair of pants’. We consider the family of maps
for t > 0:
Logt : P → R2 , (X, Y ) 7→ (x, y) = (logt |x|, logt |y|).
The image of the map is called amoeba. In this example, the image is determined by thetriangle inequality ||X| − 1| ≤ |Y | = |X + 1| ≤ |X| + 1 ⇔ |tx − 1| ≤ ty ≤ |tx + 1| ⇔logt (1 + tx ) ≤ y ≤ logt |1 − tx |. See Figure 1. If we take the limit t → 0, this approaches to thetropical amoeba (tropical variety) in Figure 2. It is given by the singular locus of the piecewise
linear function min(x, y, 0). We can think of Logt as an approximate torus ﬁbration of P over

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HIROSHI IRITANI

the tropical amoeba. For example, the ﬁbre at (x0 , 0) (with x0 > 0) is approximately thecircle |X| = tx0 , Y = −1. Most of the points on P go very close to the origin, and three ‘neck’
neighbourhoods of the missing points X = 0, X = −1, X = ∞ are stretched to semi-inﬁnite
cylinders (see Figure 3). The integral of a holomorphic 1-form
Ω=

dX
dY
X ∧ Y

d(X + Y + 1)

X+Y +1=0

on a section over the interval [x0 , x1 ] in the x-axis is approximately the length of the interval
multiplied by (− log t).
Z
Z
dX
∗
s Ω≈
= (− log t)(x1 − x0 ).
XY[x0 ,x1 ][x0 ,x1 ]X=tx ,Y =−1
Example 4.2. Next we consider a tropicalization of an elliptic curve. Let Et be the aﬃneelliptic curve:
1
Et = {(X, Y ) ∈ (C× )2 : t(X + Y +) = 1}
XY
This is a compact elliptic curve minus 3 points. The limit of Log t (Et ) as t → 0 (tropicalamoeba) and the approximate torus ﬁbration are depicted in Figure 4. The tropical amoeba
is the singular locus of min(0, x + 1, y + 1, 1 − x − y). We see from this picture that E is
composed of three ‘pairs of pants’ in the previous example. This is an instance of pairs-ofpants decomposition by Mikhalkin [16]. The approximate ﬁbration given by Logt is “singular”
at three vertices, but we can make it a smooth ﬁbration by adding three missing points (atinﬁnity) to Et and contracting the unbounded edges by a linear projection around each vertex.
For example, around the vertex (−1, −1) of the tropical amoeba, we can use the projection
x − y = logt |X| − logt |Y | along the ray R≥0 (−1, −1) to deﬁne the ﬁbration. Then we get asmooth S 1 -ﬁbration over the boundary ∂∆2 ∼= S 1 of the 2-simplex (shown in blue colour).

Figure 4. Tropical elliptic curve
The integral of the holomorphic 1-formΩt =d log X ∧ d log Yd(t(X + Y + 1/(XY ))) Et

over the compact cycle C0,t (shown in Figure 4) is asymptotically the same as the aﬃne length(= 9 = 3 × 3) of the cycle in the tropical base (shown in blue colour) multiplied by (− log t).
This elliptic curve Et is mirror to a cubic curve Y in P2 . A precise choice of the cubic curve
Y does not matter, but Y can be for example:
Ys = {[z0 , z1 , z2 ] ∈ P2 : s(z03 + z13 + z23 ) + z0 z1 z2 = 0}.

GAMMA CONJECTURE AND TROPICAL GEOMETRY

9

The dual SYZ ﬁbration for Ys is approximately given by the restriction of the moment mappingµ : P2 → R2 (with respect to the anticanonical class −KP2 on P2 ) to Ys .

3|z2 |2
3|z1 |2µ([z0 , z1 , z2 ]) =,|z0 |2 + |z1 |2 + |z2 |2 |z0 |2 + |z1 |2 + |z2 |2
Note that the image of µ is the 2-simplex {(x, y) : x ≥ 0, y ≥ 0, x+y ≤ 3} and µ(Ys ) approaches
its
as s → 0. The aﬃne length of the tropical cycle equals the symplectic volume
R boundary2 ) = 9.
c
(P
1
Ys
Example 4.3. Now we consider the case where the SYZ ﬁbration has singularities. In
dimension two, the following local model of singularities appears in the Gross-Siebert program(see [5, 14]).
Z = {(X1 , X2 , Y ) ∈ C2 × C× : X1 X2 = 1 + Y } = C2x \ {X1 X2 = 1} = P2 \ (line ∪ conic)
d log X1 ∧ d log X2 ∧ d log YdX1 dX2dX1 dYdX2 dY


Ω=
=
=
∧
=
∧
1 − X1 X2
X1
Y
X2
Y
d 1+Y
X1 X2

i
(dX1 ∧ dX1 + dX2 ∧ dX2 )2This is a log CY variety, in the sense that Ω has log poles along the boundary divisor P2 \ Z =line ∪ conic. The Lagrangian4 torus ﬁbration on Z is given as follows. We ﬁrst consider the
sympletic reduction by the S 1 -action (X1 , X2 , Y ) 7→ (eiθ X1 , e−iθ X2 , Y ) at the level λ. The
symplectic reduction is identiﬁed with the Y -plane C× .
{(X1 , X2 , Y ) ∈ Z : |X1 |2 − |X2 |2 = λ}/S 1 ∼= {Y ∈ C× }ω=

By pulling back the standard Lagrangian torus ﬁbration {(|Y | = r)}r>0 on C× , we see that
Z is foliated by the family of Lagrangian tori
Tλ,r = {(X1 , X2 , Y ) ∈ Z : |X1 |2 − |X2 |2 = λ, |Y | = r}parametrized by λ ∈ R and r > 0. This gives a torus ﬁbration on Z. Here T0,1 is a unique
singular ﬁbre (pinched torus) with singularity at (0, 0, −1) ∈ T0,1 .
We compare this ﬁbration with the tropicalization map
Logt : Z → R3 ,(X1 , X2 , Y ) 7→ (logt |X1 |, logt |X2 |, logt |Y |)for 0 < t  1. The last coordinate logt |Y | is constant on Tλ,r , but the ﬁrst two logt |X1 |,logt |X2 | are not. We see however that the map Logt approximates the torus ﬁbration awayfrom the singularity X1 = X2 = 0, i.e. away from the region where both |X1 | and |X2 | aresmall. When we set x1 = logt |X1 |, x2 = logt |X2 |, y = logt |Y |, for positive ϵ > 0,


√
2
2
|y| > ϵ =⇒ xi ≈ 21 logt ±λ+ 2λ +4r on Tλ,rx1 < x 2 − ϵ=⇒|X1 |  |X2 |=⇒x1 ≈ 21 logt |λ| on Tλ,rx2 < x 1 − ϵ=⇒|X2 |  |X1 |=⇒x2 ≈ 21 logt |λ| on Tλ,r .

The image Logt (Z) is approximated by the tropical amoeba, which is the singular locus ofmin(0, x1 + x2 , y), as in Figure 5. From this picture we can see that the above three regions
4This is not special Lagrangian since Ω ∧ Ω is not a constant multiple of ω 2 . But we have ω|Tr,λ = 0 andIm Ω|Tr,λ = 0. Therefore the base of the torus ﬁbration has a Z-aﬃne structure (complex aﬃne structure)deﬁned by ﬂuxes of Im Ω.

10

HIROSHI IRITANI(|y| > ϵ), (x1 < x2 − ϵ), (x2 < x1 − ϵ) cover the region away from the singularity or the origin(x1 , x2 , y) = (0, 0, 0). Therefore
the map (x1 , y) approximates the torus ﬁbration in the region (|y| > ϵ) ∪ (x1 < x2 − ϵ);the map (x2 , y) approximates the torus ﬁbration in the region (|y| > ϵ) ∪ (x2 < x1 − ϵ).
Moreover these maps deﬁne (approximately) aﬃne-linear coordinates in the respective regions.
The map (xi , y) contracts the face (x1 + x2 ≥ 0 = y) to the line y = 0 and send the union(x1 + x2 = min(0, y)) of the other two faces isomorphically to R2 . These coordinates (x1 , y),(x2 , y) are glued together as in Figure 6: the base of the Lagrangian ﬁbration has the focusfocus singularity with monodromy

1
0

1
1

(see [14, 22]).

Figure 5. Tropical amoeba of Z
y

y

6

6

- x1

x2 

Figure 6. Two approximately aﬃne-linear charts (x1 , y) and (x2 , y). Each
chart (xi , y) is aﬃne-linear away from the positive xi -axis. The coordinatechange between the charts is given by x1 + x2 = min(0, y). The green line
shows the polytope corresponding to R.
Let C ⊂ Z be the positive-real cycle given by X1 > 0, X2 > 0, Y > 0. The cycle C is
homeomorphic to the image Logt (C) ⊂ R3 under the map Logt :Logt (C) = {(x1 , x2 , y) ∈ R3 : x1 + x2 = logt (1 + ty )}GAMMA CONJECTURE AND TROPICAL GEOMETRY

11

The image Logt (C) is very close to the union (x1 + x2 = min(0, y)) of two faces of the tropicalamoeba. We now compute the period of C with respect to the holomorphic volume formΩ. The volume form Ω restricted to C can be identiﬁed with the aﬃne area form in thecoordinates (xi , y):Ω|C = (− log t)2 dx1 ∧ dy = (− log t)2 dx2 ∧ dy.
To make the computation ﬁnite, we consider the integral over the ﬁnite region R delimitedby the aﬃne-linear equations (with a1 , a2 , b positive constant):
R = {(X1 , X2 , Y ) ∈ C : x1 ≤ −a1 , x2 ≤ −a2 , −b ≤ y ≤ b}
The corresponding polytope in the aﬃne chart is shown in green colour in Figure 6. The aﬃnearea of this polytope is 2(a1 + a2 )b − b2 /2. On the other hand, the actual shape of R in thecoordinate (x1 , y) slightly diﬀers from this polytope because of the error in tropicalization,
see Figure 7. We haveZZ
Ω = x +x =log (1+ty ) (− log t)2 dx1 dy
R⊂C

1

2

t

xi ≥−ai ,|y|≤bZ b2= (− log t)−b(a2 + logt (1 + ty ) − (−a1 ))dyZ b
= (− log t)2−b(a1 + a2 + min(0, y))dy −



Z b
y

−b

(− logt (1 + t ) + min(0, y))dy= (− log t)2 (area of the polytope) − ζ(2) + O(tb ).
where ζ(2) arises from the ‘error in tropicalization’ integral (area of the blue region):
Z ∞
2
(4)
ζ(2) = (− log t)(− logt (1 + ty ) + min(0, y))dy.
−∞

This means that the singularity contributes −ζ(2) to the period integral.
y
2

1

-4

-2

2

4

x1

-1

-2

Figure 7. The region R in coordinates (x1 , y). It is given by −a1 ≤ x1 ≤a2 + logt (1 + ty ), |y| ≤ b. This diﬀers from the green polytope in Figure 6 by
the blue region.

12

HIROSHI IRITANIExample 4.4. Let (Zt , Ωt ) be the family of aﬃne K3 surfaces
Zt = {(W1 , W2 , W3 ) ∈ (C× )3 : t(W1 + W2 + W3 +Ωt =d log W1 ∧ d log W2 ∧ d log W3d(t(W1 + W2 + W3 + W1 W12 W3 ))1) = 1}W1 W2 W3This can be compactiﬁed to a K3 surface Zt so that Ωt extends to a nowhere vanishing
holomorphic 2-form. This family is mirror to a quartic K3 surface Y ⊂ P3 equipped with asymplectic form in the class (− log t)c1 (P3 ). The tropical amoeba of Zt is given by the singularlocus of min(w1 + 1, w2 + 1, w3 + 1, 1 − w1 − w2 − w3 , 0) as in Figure 8. Observe that thecompact chamber∆ = {(w1 , w2 , w3 ) ∈ R3 : w1 ≥ −1, w2 ≥ −1, w3 ≥ −1, w1 + w2 + w3 ≤ 1}
bounded by the tropical amoeba is the same as the moment polytope of P3 with respect to
c1 (P3 ). By collapsing5 all the unbounded faces of the tropical amoeba (not contained in ∆),we should get a torus ﬁbration on Zt over the sphere ∂∆ ∼= S 2 . This ﬁbration has singularities,
since there is no uniform direction to collapse the unbounded faces, unlike the case of Example
4.2. (In fact, we have many ways to collapse, yielding diﬀerent ﬁbration structures on Zt .)Figure 8. Tropical K3 surface. This is composed of the boundary of the 3simplex ∆ and 6 unbounded faces. The green regions are the Log t -images ofneighbourhoods of singularities (as in Figure 5) and the red crosses represent
singular points of the aﬃne structure on ∂∆.
As before, we consider the period of the positive real cycle Ct = {(W1 , W2 , W3 ) ∈ Zt : W1 >0, W2 > 0, W3 > 0}. It is homeomorphic to its image under Logt :Logt (Ct ) = {(w1 , w2 , w3 ) ∈ R3 : tw1 +1 + tw2 +1 + tw3 +1 + t1−w1 −w2 −w3 = 1}.
As t → +0, Logt (Ct ) rapidly approaches ∂∆. We compute the period integral locally over
the tropical base ∂∆, reducing the computation to the local model from Example 4.3. Since
5See [22] for collapsing tropical hypersurfaces.

GAMMA CONJECTURE AND TROPICAL GEOMETRY

13

the ﬁbration has 24 singular points (see below), we expect to have the following asymptoticsZ(5)Ωt = (− log t)2 (aﬃne area of ∂∆) − 24ζ(2) + O(tϵ ).
Ct

R
R
b
b Y = −ζ(2) c2 (Y ).
The number −24ζ(2) also arises from the Γ-classof a quartic Y : Y ΓYWe shall explain the details below. We introduce a Z-aﬃne structure (with singularities) on∂∆ as follows (see [8]).
• on the interior of 2-dimensional faces of ∂∆, we consider the subspace aﬃne structure;• on a neighbourhood of a lattice point v on edges (1-dimensional face) of ∂∆, weconsider the aﬃne structure induced from the projection R3 → R3 /hvi and;• we need to have a singularity (shown in red cross) somewhere between adjacent lattice
points v1 , v2 on an edge because the aﬃne structures induced by the projections R3 →R3 /hvi i, i = 1, 2 are not compatible with each other; we have 4 singularities on eachedge, and thus 24 = 4 × 6 singularities in total.
The reason for this aﬃne structure is as follows. Near the interior of the 2-dimensional facegiven by (w1 = −1) for example, the cycle Ct ∼= Logt (Ct ) is approximated by the aﬃne-linear
subspace
tw1 +1 ≈ 1since the other terms tw2 +1 , tw3 +1 , t1−w1 −w2 −w3 are much smaller than tw1 +1 , and the volume
form Ωt |Ct is approximated by the aﬃne volume form (mulitplied by (− log t)2 ):d log tw1 +1 ∧ d log tw2 +1 ∧ d log tw3 +1
d(tw1 +1 + tw2 +1 + tw3 +1 + t1−w1 −w2 −w3 )(− log t)3 dw1 ∧ dw2 ∧ dw3≈(near the face (w1 = −1) ∩ ∆)d(tw1 +1 )

Ωt | Ct == (− log t)2 dw2 ∧ dw3 .
Next, around the interior of the edge given by (w1 = w2 = −1) for example, the cycle Ct isapproximated by
tw1 +1 + tw2 +1 ≈ 1(6)since the other terms tw3 +1 , t1−w1 −w2 −w3 are exponentially small. The volume form Ωt |Ct issimilarly approximated by the aﬃne volume form on R2 /hvi for a lattice point v on the edge.
Ω t | Ct ≈(− log t)3 dw1 ∧ dw2 ∧ dw3= (− log t)2 (aﬃne volume form on R3 /hvi).
d(tw1 +1 + tw2 +1 )In fact, if we complete v to a Z-basis v = v1 , v2 , v3 of Z3 and write (a1 , a2 , a3 ) for the linear∂w2
1
coordinates on R3 dual to (v1 , v2 , v3 ), we have wi (a1 v) = −a1 ; thus ∂w∂a1 = ∂a1 = −1 and(− log t)3 dw1 ∧ dw2 ∧ dw3(− log t)3 da1 ∧ da2 ∧ da3=
d(tw1 +1 + tw2 +1 )d(tw1 +1 + tw2 +1 )(− log t)3 da2 ∧ da3= (− log t)2 da2 ∧ da3 .
=∂(tw1 +1 + tw2 +1 )/∂a1
Note that the aﬃne structure depends on the choice of v, but the volume form does not.
Finally, around the vertex v = (−1, −1, −1) of ∂∆, Ct is approximated by
tw1 +1 + tw2 +1 + tw3 +1 ≈ 1
and Ωt |Ct is also approximated by the aﬃne volume form on R3 /hvi.

14

HIROSHI IRITANIWe next examine the cycle Ct near the singularity. Let v1 , v2 be adjacent lattice points on
the edge w1 = w2 = −1. Choose an integral vector v3 such that (v1 , v2 , v3 ) is a Z-basis of Z3
and let (a1 , a2 , a3 ) be the linear coordinates on R3 dual to (v1 , v2 , v3 ). We may assume thatv3 is parallel to the face w1 = −1 (i.e. w1 (v3 ) = 0) by adding a linear combination of v1 andv2 if necessary. The coordinates w1 , w2 , w3 can then be written as a Z-linear combination of
a1 , a2 , a3 ; examining the values at v1 , v2 , v3 we ﬁnd that the coordinate change is of the form:
w1 = −a1 − a2
w2 = −a1 − a2 + ma3
w3 = ja1 + ka2 + la3
for some j, k, l, m ∈ Z. Moreover, since w1 − w2 is a non-zero primitive covector, we havem = ±1. By ﬂipping the sign of a3 (or equivalently v3 ), we may assume that m = 1. Recall
from (6) that Ct is given by tw1 +1 + tw2 +1 ≈ 1 around the edge. This can be rewritten as:
1 + ta3 ≈ ta1 +a2 −1 = ta1 − 2 · ta2 − 2 .
1

1

Setting X1 = ta1 − 2 , X2 = ta2 − 2 , Y = ta3 , this equation can be identiﬁed with the local modelfrom Example 4.3. The holomorphic volume form (− log t)2 da1 ∧ da2 = d log X1 ∧ d log Y
is also the same as in Example 4.3. If we introduce a Lagrangian ﬁbration near {v1 , v2 } bylocally identifying Zt with the local model in Example 4.3 by this identiﬁcation (i.e. W1 =(tX1 X2 )−1 , W2 = Y (tX1 X2 )−1 , W3 = X1j X2k Y l t(j+k)/2 ), we see that the singularity appearsat (a1 , a2 , a3 ) = ( 12 , 12 , 0) on the base, which corresponds to the mid-point of v1 and v2 . At
each singular point, we have the error term −ζ(2) as we saw in Example 4.3 and arrive at theasymptotics (5).
We note that the choice of an (approximate) torus ﬁbration, in particular, the choiceof positions of singularities on ∂∆ has certain arbitrariness in the above discussion. We
introduced singularities on ∂∆ for the purpose of computing periods (and we had to since wecannot cover ∂∆ by compatible aﬃne charts) and the error terms in tropical approximation
occurred from those singularities.
1

1

Example 4.5. We have seen that ζ(2) arises from 2-dimensional singularities through the‘error in tropicalization’ integral (4). In dimension three, ζ(3) arises from the following ‘error
in tropicalization’ integrals (see [1, Eqn (22), Proposition 4.5]):
Z ∞

1
3
ζ(3) = (− log t)(logt (1 + ty ))2 − (min(0, y))2 dy
2
−∞
and

Z
(− log t)(− logt (1 + ty1 + ty2 ) + min(0, y1 , y2 ))dy1 dy2

3U= (− log t)ℓζ(2) + χU (0)ζ(3) + O(1/(− log t))
where U is a bounded domain in R2 such that ∂U intersects every stratum of the tropicalcurve L = Sing(min(0, y1 , y2 )) transversally, ℓ is the total aﬃne length of U ∩ L and(
1 if 0 ∈ U ,χU (0) =0 otherwise.
Remark 4.6. In dimension one, the relationship between periods and aﬃne-length of tropicalcurves was studied in [17, 13]. For toric Calabi-Yau hypersurfaces, the relationship between
periods and the radiance obstruction was studied in [22]. Ruddat and Siebert [18] computed

GAMMA CONJECTURE AND TROPICAL GEOMETRY

15

periods of cycles ﬁbering over 1-dimensional curves on the SYZ base, in the framework ofGross-Siebert program.
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Email address: iritani@math.kyoto-u.ac.jp
Department of Mathematics, Graduate School of Science, Kyoto University, KitashirakawaOiwake-cho, Sakyo-ku, Kyoto, 606-8502, JAPAN