Geometric Monodromy around the Tropical Limit
Let {Vq}q be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of {Vq}q around q=∞ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin.
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irk-123456789-1477582019-02-16T01:26:16Z Geometric Monodromy around the Tropical Limit Yamamoto, Y. Let {Vq}q be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of {Vq}q around q=∞ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin. 2016 Article Geometric Monodromy around the Tropical Limit / Y. Yamamoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14T05; 14D05 DOI:10.3842/SIGMA.2016.061 http://dspace.nbuv.gov.ua/handle/123456789/147758 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Let {Vq}q be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of {Vq}q around q=∞ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin. |
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Yamamoto, Y. Geometric Monodromy around the Tropical Limit Symmetry, Integrability and Geometry: Methods and Applications |
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Yamamoto, Y. |
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Yamamoto, Y. |
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Geometric Monodromy around the Tropical Limit |
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Geometric Monodromy around the Tropical Limit |
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Geometric Monodromy around the Tropical Limit |
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Geometric Monodromy around the Tropical Limit |
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Geometric Monodromy around the Tropical Limit |
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geometric monodromy around the tropical limit |
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Інститут математики НАН України |
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2016 |
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Geometric Monodromy around the Tropical Limit / Y. Yamamoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 10 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT yamamotoy geometricmonodromyaroundthetropicallimit |
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2025-07-11T02:46:53Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 061, 23 pages
Geometric Monodromy around the Tropical Limit
Yuto YAMAMOTO
Graduate School of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
E-mail: yuto@ms.u-tokyo.ac.jp
Received September 02, 2015, in final form June 17, 2016; Published online June 24, 2016
http://dx.doi.org/10.3842/SIGMA.2016.061
Abstract. Let {Vq}q be a complex one-parameter family of smooth hypersurfaces in a toric
variety. In this paper, we give a concrete description of the monodromy transformation
of {Vq}q around q = ∞ in terms of tropical geometry. The main tool is the tropical locali-
zation introduced by Mikhalkin.
Key words: tropical geometry; monodromy
2010 Mathematics Subject Classification: 14T05; 14D05
1 Introduction
Let K := C{t} be the convergent Laurent series field, equipped with the standard non-archime-
dean valuation,
val : K −→ Z ∪ {−∞}, k =
∑
j∈Z
cjt
j 7→ −min{j ∈ Z | cj 6= 0}. (1.1)
Let n ∈ N be a natural number and M be a free Z-module of rank n+1. We write MR := M⊗ZR.
Let further ∆ ⊂MR be a convex lattice polytope, i.e., the convex hull of a finite subset of M . We
set A := ∆∩M . Let F =
∑
m∈A
kmx
m ∈ K
[
x±1 , . . . , x
±
n+1
]
be a Laurent polynomial over K in n+1
variables such that km 6= 0 for all m ∈ A. We fix a sufficiently large R ∈ R>0 such that 1/R is
smaller than the radius of convergence of km for all m ∈ A, and set S1
R := {z ∈ C | |z| = R}. For
q ∈ S1
R, let fq ∈ C[x±1 , . . . , x
±
n+1] denote the polynomial obtained by substituting 1/q to t in F .
Let F be the normal fan to ∆ and F ′ be a unimodular subdivision of F . Let XF ′(C) denote
the toric manifold over C associated with F ′. For each q ∈ S1
R, we define Vq ⊂ XF ′(C) as the
hypersurface defined by fq in XF ′(C). In this paper, we discuss the monodromy transformation
of {Vq}q∈S1
R
around q = ∞. The limit q → ∞ is called the tropical limit in this paper. The
motivation to address this problem comes from the calculation of monodromies of period maps.
Let trop(F ) : Rn+1 → R be the tropicalization of F defined by
trop(F )(X1, . . . , Xn+1) := max
m∈A
{
val(km) +m1X1 + · · ·+mn+1Xn+1
}
. (1.2)
The non-differentiable locus of trop(F ) is called the tropical hypersurface defined by trop(F )
and denoted by V (trop(F )). The tropical hypersurface V (trop(F )) is a rational polyhedral com-
plex of dimension n. The main theorem of this paper is Theorem 4.5, which gives a concrete
description of the monodromy transformation of {Vq}q∈S1
R
in terms of the tropical hypersurfa-
ce V (trop(F )) in the case where V (trop(F )) is smooth (see Definition 2.7). The monodromy
of {Vq}q∈S1
R
is also discussed in [2, Appendix B.2] and Theorem 4.5 is covered by [2, Proposi-
tion B.17]. However, this paper aims to make the relation of the monodromy of {Vq}q∈S1
R
to
tropical geometry clear. We give a self-contained proof and explicit examples.
mailto:yuto@ms.u-tokyo.ac.jp
http://dx.doi.org/10.3842/SIGMA.2016.061
2 Y. Yamamoto
When ∆ is smooth and reflexive and the polynomial F gives a central subdivision of ∆,
Zharkov [10] also gave a concrete description of the monodromy transformation of {Vq}q∈S1
R
. The
idea of his description is the same as that of ours. By treating his construction systematically,
we generalize his result to the case where ∆ is any polytope and the subdivision of ∆ given by F
is not necessarily central.
Since the claim of Theorem 4.5 is technical and it is necessary to make preparations in order
to state it, we do not state it here and discuss its corollary in the following. Assume n = 1.
Let {ρi}i∈{1,...,d} be the set of all bounded edges of V (trop(F )). For each ρi, let νi1, νi2 ∈ Rn+1
be the endpoints of ρi. Let further V ∈ Zn+1 be the primitive vector such that νi1 − νi2 = lV
for some l ∈ R>0. We define the length L(ρi) of ρi as l ∈ R>0. Assume that the tropical
hypersurface V (trop(F )) is smooth, in the sense that for any vertex ν of V (trop(F )), there exists
a Z-affine transformation
(
(mij)1≤i,j≤2, (ri)i=1,2
)
∈ GL2(Z) n R2 such that in the coordinate
(Y1, Y2) on R2 defined by
Y1 = m11X1 +m12X2 + r1, Y2 = m21X1 +m22X2 + r2,
the tropical hypersurface V (trop(F )) coincides locally with the tropical hyperplane defined by
max{0, Y1, Y2} around ν. Then we have νi1, νi2 ∈ Zn+1. The amoeba of Vq converges to the
tropical hypersurface V (trop(F )) as q →∞ in the Hausdorff metric [8, 9] and the hypersurface Vq
is obtained by ‘thickening’ the amoeba of Vq. Let Ci (i = 1, . . . , d) be the simple closed curve
in Vq=R turning around ρi (see Fig. 1 for an example). Let further Ti : VR → VR be the Dehn
twist along Ci.
Corollary 1.1. If n = 1 and V (trop(F )) is smooth, then the monodromy transformation of
{Vq}q∈S1
R
around q =∞ is given by T
L(ρ1)
1 ◦ · · · ◦ TL(ρd)
d .
Corollary 1.1 is conjectured by Iwao [4]. Let us illustrate this claim with a simple example.
Consider the polynomial F given by
F (x1, x2) = x2
2 + x2
(
x3
1 + t−2x2
1 + t−2x1 + t−1
)
+ 1. (1.3)
Then we have
fq(x1, x2) = x2
2 + x2
(
x3
1 + q2x2
1 + q2x1 + q1
)
+ 1,
trop(F )(X1, X2) = max{2X2, 3X1 +X2, 2X1 +X2 + 2, X1 +X2 + 2, X2 + 1, 0}.
The tropical hypersurface V (trop(F )) and the hypersurface Vq in this case are shown in Fig. 1.
Let ρi and Ci (i = 1, . . . , 7) denote edges of V (trop(F )) and simple closed curves in Vq as
shown in Fig. 1. Then the edges ρ1, . . . , ρ7 correspond to the simple closed curves C1, . . . , C7,
respectively. By simple calculations, we have
L(ρ1) = 2, L(ρ2) = 4, L(ρ3) = 12, L(ρ4) = L(ρ5) = 1, L(ρ6) = L(ρ7) = 2.
It follows from Corollary 1.1 that the monodromy transformation of {Vq}q∈S1
R
is given by T 2
1 ◦
T 4
2 ◦ T 12
3 ◦ T4 ◦ T5 ◦ T 2
6 ◦ T 2
7 .
The organization of this paper is as follows: First, we set up the notation in Section 2. In
Section 3, we recall the notion of the tropical localization introduced by Mikhalkin [8]. This is
the main tool to construct the monodromy transformation of {Vq}q∈S1
R
. In Section 4, we give an
explicit description of the monodromy transformations in any dimension. In Section 5, we show
that Corollary 1.1 follows from Theorem 4.5. In Section 6, we give examples in dimension 1
and 2. In Section 7, we discuss the relation between Zharkov’s description and ours. This section
may also be useful for understanding this paper and a possible first step for getting our idea.
Geometric Monodromy around the Tropical Limit 3
Figure 1. The tropical hypersurface V (trop(F )) and the hypersurface Vq for (1.3).
2 Preliminaries
2.1 Tropical toric varieties
Let M be a free Z-module of rank n + 1 and N := HomZ(M,Z) be the dual lattice of M . We
set MR := M ⊗Z R and NR := N ⊗Z R = HomZ(M,R). We have a canonical R-bilinear pairing
〈−,−〉 : MR ×NR → R.
Let F be a fan in NR. We write the toric variety associated with F over C as XF (C). For each
cone σ ∈ F , we set
σ∨ := {m ∈MR | 〈m,n〉 ≥ 0 for all n ∈ σ},
σ⊥ := {m ∈MR | 〈m,n〉 = 0 for all n ∈ σ}.
Let Uσ(C) := Hom(σ∨ ∩M,C) denote the affine toric variety and Oσ(C) := Hom(σ⊥ ∩M,C∗)
denote the torus orbit corresponding to σ. We write the closure of Oσ(C) in XF (C) as XF ,σ(C).
Let T := R∪ {−∞} be the tropical semi-ring, equipped with the following arithmetic opera-
tions for any a, b ∈ T;
a⊕ b := max{a, b}, a� b := a+ b.
We can also define the toric variety over T as follows. For each cone σ ∈ F , we define Uσ(T) as
the set of monoid homomorphisms σ∨ ∩M → (T,�),
Uσ(T) := Hom
(
σ∨ ∩M,T
)
with the compact open topology. For cones σ, τ ∈ F such that σ ≺ τ , we have a natural
immersion,
Uσ(T)→ Uτ (T),
(
v : σ∨ ∩M → T
)
7→
(
τ∨ ∩M ⊂ σ∨ ∩M v−→ T
)
,
where σ ≺ τ means that σ is a face of τ . By gluing {Uσ(T)}σ∈F with each other, we have the
tropical toric variety XF (T) associated with F ,
XF (T) :=
( ∐
σ∈F
Uσ(T)
)/
∼.
Tropical toric varieties are first introduced by Kajiwara [5], see [5] or [6] for details. For a projec-
tive toric variety, the associated tropical toric variety is homeomorphic to the moment polytope
of it [6, Remark 1.3].
4 Y. Yamamoto
Example 2.1. The tropical projective space of n-dimension is homeomorphic to the n-dimen-
sional simplex.
We define the torus orbit Oσ(T) over T corresponding to σ by
Oσ(T) := Hom
(
σ⊥ ∩M,R
)
,
and write the closure of Oσ(T) in XF (T) as XF ,σ(T). Let R ∈ R>0 be a positive real number
and LogR : C→ T denote the map defined by
c 7→
{
logR |c|, c 6= 0,
−∞, c = 0.
We have a canonical map LogR : XF (C)→ XF (T) defined by
Uσ(C) = Hom
(
σ∨ ∩M,C
)
→ Uσ(T) = Hom
(
σ∨ ∩M,T
)
, v 7→ LogR ◦v. (2.1)
2.2 Polyhedral complex
We define the product R≥0 × T→ T by
r · t :=
r × t, t 6= −∞,
−∞, r 6= 0, t = −∞,
0, r = 0, t = −∞,
for r ∈ R≥0 and t ∈ T. Here, × denotes the ordinary multiplication of R. We also define the
product
(
R≥0
)n+1 × Tn+1 → T by
a · b :=
n+1∑
i=1
ai · bi,
for a = (a1, . . . , an+1) ∈
(
R≥0
)n+1
and b = (b1, . . . , bn+1) ∈ Tn+1. For each subset I ⊂ {1, . . . ,
n+ 1}, we set
Tn+1
I :=
{
X ∈ Tn+1 |Xi = −∞ for any i ∈ I
}
.
Definition 2.2. A subset ρ of Tn+1 is a convex polyhedron if there exist a finite collection
{Hj}j∈J of half-spaces of the form
Hj =
{
X ∈ Tn+1 | cj ·X ≤ dj
}
, cj ∈
(
R≥0
)n+1
, dj ∈ R,
and a subset I ⊂ {1, . . . , n+ 1} such that
ρ = ∩j∈JHj ∩ Tn+1
I .
A subset µ of ρ is a face of ρ if there exist subsets J ′ ⊂ J and I ′ ⊂ {1, . . . , n + 1} such that
I ′ ⊃ I and
µ =
{
X ∈ ρ | cj ·X = dj for all j ∈ J ′, Xi = −∞ for all i ∈ I ′
}
.
We write µ ≺ ρ when µ is a face of ρ.
Definition 2.3. A fan F in NR is called unimodular if every cone in F can be generated by
a subset of a basis for N .
Geometric Monodromy around the Tropical Limit 5
Figure 2. Tropical hyperplanes of dimensions 1 and 2 in tropical projective spaces.
Let F be a complete and unimodular fan in NR in the following.
Definition 2.4. A subset ρ of XF (T) is a convex polyhedron ρ if ρ∩Uσ(T) is a convex polyhedron
in Uσ(T) ∼= Tn+1 for any (n+ 1)-dimensional cone σ ∈ F . A subset µ in ρ is called a face of ρ
when µ ∩ Uσ(T) is a face of ρ ∩ Uσ(T) for any (n+ 1)-dimensional cone σ ∈ F . We write µ ≺ ρ
when µ is a face of ρ.
Definition 2.5. A finite set P of convex polyhedra in XF (T) is a polyhedral complex if it
satisfies the following conditions:
• For any convex polyhedron ρ ∈ P , all faces of ρ are elements of P .
• For any two convex polyhedra ρ1, ρ2 ∈ P , ρ1 ∩ ρ2 is a face of ρ1 and ρ2.
Each element ρ ∈ P is called a cell. In particular, we call ρ a k-cell when ρ is k-dimensional.
Let P be a polyhedral complex in XF (T). For each σ ∈ F , we define
Pσ := {ρ ∈ P | relint(ρ) ⊂ Oσ(T)},
where relint(ρ) denotes the relative interior of ρ.
2.3 Hypersurfaces in toric varieties
Let K := C{t} be the convergent Laurent series field, equipped with the standard non-archime-
dean valuation (1.1). Let further ∆ ⊂ MR be a convex lattice polytope. We set A := ∆ ∩M .
Let F =
∑
m∈A
kmx
m ∈ K
[
x±1 , . . . , x
±
n+1
]
be a Laurent polynomial over K in n+ 1 variables such
that km 6= 0 for all m ∈ A. Let F denote the normal fan to ∆. We choose a unimodular
subdivision F ′ of F .
The tropicalization of F is the piecewise-linear map trop(F ) : O{0}(T) ∼= Rn+1 → R given
by (1.2). Let V{0}(trop(F )) denote the non-differentiable locus of trop(F ) in O{0}(T) ∼= Rn+1.
Let further V (trop(F )) denote the closure of V{0}(trop(F )) in XF ′(T). The tropical hypersur-
face V (trop(F )) has a structure of a polyhedral complex in XF ′(T). Let P denote the polyhedral
complex given by V (trop(F )) in the following.
Example 2.6. Let F , G be polynomials defined by F = 1 + x1 + x2 and G = 1 + x1 +
x2 + x3. Then the tropicalizations of F and G are trop(F ) = max{0, X1, X2} and trop(G) =
max{0, X1, X2, X3}. The tropical hypersurfaces V (trop(F )) and V (trop(G)) are shown in Fig. 2.
The polyhedral complex given by V (trop(F )) consists of four 0-cells and three 1-cells. The
polyhedral complex given by V (trop(G)) consists of eleven 0-cells, sixteen 1-cells, and six 2-cells.
Let v : A → Z be the function defined by v(m) := val(km). Let further Γv be the subset in
MR × R defined by
Γv := {(m, r) ∈ A× R | r ≤ v(m)},
6 Y. Yamamoto
Figure 3. The set conv(Γv) and the polyhedral subdivision Dv for F = t−1 + x1 + x2 + x−11 x−12 .
and conv(Γv) be the convex hull of Γv in MR × R. We write the polyhedral subdivision of ∆
given by the projections of all bounded faces of conv(Γv) to MR as Dv. Note that all vertices
of any polyhedron in Dv are contained in M . It is well known that the tropical hypersur-
face V{0}(trop(F )) is dual to the polyhedral subdivision Dv [7, Proposition 3.1.6].
Definition 2.7. The polyhedral subdivision Dv is unimodular if all elements of Dv are simplices
of volume 1
(n+1)! . We say V (trop(F )) is smooth in this case.
Example 2.8. Consider the polynomial F = t−1 +x1 +x2 +x−1
1 x−1
2 . In this case, the function v
is given by v((0, 0)) = 1 and v((1, 0)) = v((0, 1)) = v((−1,−1)) = 0. The set conv(Γv) and the
polyhedral subdivision Dv are shown in Fig. 3. The polyhedral subdivision Dv is unimodular
and V (trop(F )) is smooth in this case.
We set vm := val(km) for m ∈ A. For each µ ∈ P{0}, we define the subset Aµ ⊂ A as the set
of elements of A to which the dominant terms of F at µ corresponds:
Aµ :=
{
m ∈ A | vm +m ·X = trop(F )(X) for all X ∈ µ ∩O{0}(T)
}
. (2.2)
Lemma 2.9 ([8, Lemma 6.5]). Assume that the dimension of µ ∈ P{0} is k (0 ≤ k ≤ n). If the
tropical hypersurface V (trop(F )) is smooth, then the number of elements of Aµ is n+ 2− k.
Assume that V (trop(F )) is smooth. We fix a sufficiently large R ∈ R>0 such that 1/R is
smaller than the radius of convergence of km for all m ∈ A, and set S1
R := {z ∈ C | |z| = R}.
For q ∈ S1
R, let fq ∈ C
[
x±1 , . . . , x
±
n+1
]
be the Laurent polynomial obtained by substituting 1/q
to t in F . We write the closure of
{
x ∈ O{0}(C) | fq(x) = 0
}
in XF ′(C) as Vq.
Let σ ∈ F ′ be an l-dimensional cone. For µ ∈ Pσ, let µ′ ∈ P{0} be the cell such that
µ = µ′ ∩ XF ′,σ. We assume that the dimension of µ′ is k. Here, we have l ≤ k. We define
standard coordinates on Oσ(C) and Oσ(T) with respect to µ as follows. First, we number all
elements of Aµ′ from 0 to n+ 1− k and write them as (m0, . . . ,mn+1−k). We set
x̃i := qvmixmi/qvm0xm0 , X̃i := (vmi +mi ·X)− (vm0 +m0 ·X),
for i = 1, . . . , n + 1 − k. Since V (trop(F )) is smooth, we can extend (x̃1, . . . , x̃n+1−k) and
(X̃1, . . . , X̃n+1−k) to (x̃1, . . . , x̃n+1−l) and (X̃1, . . . , X̃n+1−l) which form coordinate systems on
Oσ(C) and Oσ(T) respectively by setting
x̃i := qai
n+1∏
j=1
x
bij
j , X̃i := ai +
n+1∑
j=1
bijXj ,
for i = n + 2 − k, . . . , n + 1 − l. Here, numbers ai and bij are appropriate integral numbers.
We call (x̃1, . . . , x̃n+1−l) and (X̃1, . . . , X̃n+1−l) standard coordinates with respect to µ. There are
some ambiguities of them resulting from different numbering of (m0, . . . ,mn+1−k) and different
choices of numbers ai and bij .
Geometric Monodromy around the Tropical Limit 7
Figure 4. The tropical hypersurface defined by trop(F ) = max{1, X1, X2,−X1 −X2}.
Let Hµ : (C∗)n+1−l → (C∗)n+1−l be the map defined by
(x1, . . . , xn+1−l) 7→ (x̃1, . . . , x̃n+1−l),
and Mµ : Rn+1−l → Rn+1−l be the map defined by
(X1, . . . , Xn+1−l) 7→
(
X̃1, . . . , X̃n+1−l
)
.
Then the following diagram is commutative.
(C∗)n+1−l Hµ−−−−→ (C∗)n+1−l
LogR
y yLogR
Rn+1−l Mµ−−−−→ Rn+1−l,
where the map LogR : (C∗)n+1−l → Rn+1−l is defined by
(x1, . . . , xn+1−l)→ (logR |x1|, . . . , logR |xn+1−l|).
Example 2.10. Let us consider the polynomial F = t−1 + x1 + x2 + x−1
1 x−1
2 again. We have
fq = q+x1+x2+x−1
1 x−1
2 . The tropicalization of F is trop(F ) = max{1, X1, X2,−X1−X2}. The
tropical hypersurface defined by trop(F ) is shown in Fig. 4. Let ν and µ denote the vertex and
the edge of V (trop(F )) as shown in Fig. 4. The set Aν is given by {(0, 0), (1, 0), (−1,−1)}. We
set y1 := x1/q = q−1x1, y2 := x−1
1 x−1
2 /q = q−1x−1
1 x−1
2 and Y1 := −1 +X1, Y2 := −1−X1 −X2.
Then the sets of function (y1, y2) and (Y1, Y2) form standard coordinates with respect to ν
on O{0}(C) and O{0}(T), respectively. The set Aµ is given by {(1, 0), (−1,−1)}. We set z1 :=
x−1
1 x−1
2 /x1 = x−2
1 x−1
2 and Z1 := −2X1−X2. For instance, if we set z2 := q2x1 and Z2 := 2+X1,
then we have
det
(
−2 1
−1 0
)
= 1.
Hence, the sets of functions (z1, z2) and (Z1, Z2) form standard coordinates with respect to µ
on O{0}(C) and O{0}(T), respectively.
3 Tropical localization
Tropical localization is a way to simplify algebraic hypersurfaces around the tropical limit points
by ignoring terms which are not dominant in the tropical limit. This technique is first introduced
8 Y. Yamamoto
Figure 5. The graph of the function b.
by Mikhalkin [8]. In this section, we give a concrete defining function realizing the tropical
localization based on the idea of Mikhalkin. There is also a similar construction of the tropical
localization in [1].
Let K := C{t} be the convergent Laurent series field, equipped with the standard non-
archimedean valuation (1.1). Let further ∆ ⊂ MR be a convex lattice polytope. We set A :=
∆ ∩M . Let F =
∑
m∈A
kmx
m ∈ K
[
x±1 , . . . , x
±
n+1
]
be a polynomial over K such that km 6= 0 for
all m ∈ A. We set vm := val(km). We fix a sufficiently large R ∈ R>0 such that 1/R is smaller
than the radius of convergence of km for all m ∈ A, and set S1
R := {z ∈ C | |z| = R}. For
q ∈ S1
R, let fq ∈ C
[
x±1 , . . . , x
±
n+1
]
denote the polynomial obtained by substituting 1/q to t in F .
Let F denote the normal fan to ∆. We choose a unimodular subdivision F ′ of F . Let Vq be
the hypersurface in XF ′(C) defined by fq. Let further V (trop(F )) be the tropical hypersurface
in XF ′(C) defined by trop(F ) and P be the polyhedral complex in XF ′(T) given by V (trop(F )).
Assume that V (trop(F )) is smooth (see Definition 2.7).
Let C0, C1 ∈ R be constants such that 0 < C1 < C0 � 1. Let b : R→ R be a monotone C∞
function on R satisfying following conditions:
1) b(X) = 1 if and only if X ≤ C1,
2) b(X) = 0 if and only if X ≥ C0.
The graph of the function b is shown in Fig. 5.
We define the tropical localization of the hypersurface Vq as follows.
Definition 3.1. For each m ∈ A, let bm : O{0}(C)→ R be the function defined by
bm(x) :=
∏
i∈A
b
(
logR
∣∣qvixi∣∣− logR
∣∣qvmxm∣∣).
In addition, let f̃q : O{0}(C)→ C be the function defined by
f̃q(x) :=
∑
m∈A
bm(x)qvmxm.
We define the tropical localization Wq of Vq as the closure of {x ∈ O{0}(C) | f̃q(x) = 0} in XF ′(C).
By applying Definition 3.1 to f(x1, . . . , xn+1) = 1 + x1 + · · · + xn+1, we can construct the
tropically localized hyperplane.
Definition 3.2. We define the function f̃ : O{0}(C)→ C by
f̃(x1, . . . , xn+1) :=
n+1∏
i=1
b(logR |xi|) +
n+1∑
i=1
b(− logR |xi|)
n+1∏
j=1
b(logR |xj | − logR |xi|)
xi,
We call the submanifold defined as the zero locus of f̃ the tropically localized hyperplane.
Geometric Monodromy around the Tropical Limit 9
Figure 6. The tropical hypersurface V (trop(F ))
for F = t−1 + x1 + x2 + x−11 x−12 .
Figure 7. The regions {D̂ρ}ρ for F = t−1 + x1 +
x2 + x−11 x−12 .
Definition 3.3. For each µ ∈ P , we define Dµ ⊂ XF ′(C) and D̂µ ⊂ XF ′(T) as follows. For
µ ∈ P{0}, we define Dµ ⊂ XF ′(C) and D̂µ ⊂ XF ′(T) by
Dµ :=
{
x ∈ O{0}(C)
∣∣∣∣ bm(x) > 0 for m ∈ Aµ,
bm(x) = 0 for m ∈ A \Aµ
}
,
D̂µ :=
X ∈ O{0}(T)
∣∣∣∣∣∣
|(vm′ +m′ ·X)− (vm +m ·X)| < C0 for m,m′ ∈ Aµ,
for any m ∈ A \Aµ, there exists m′ ∈ Aµ
such that (vm′ +m′ ·X)− (vm +m ·X) ≥ C0
,
where Aµ ⊂ A is the set defined in (2.2) and the overlines mean the closure in XF ′(C)
and XF ′(T), respectively.
For µ ∈ Pσ (σ 6= {0}), let µ′ ∈ P{0} be the cell such that µ = µ′ ∩ XF ′,σ(T). We define
Dµ ⊂ XF ′,σ(C) and D̂µ ⊂ XF ′,σ(T) by
Dµ := Dµ′ ∩XF ′,σ(C), D̂µ := D̂µ′ ∩XF ′,σ(T). (3.1)
The monomial vm + m · X (m ∈ Aµ) of trop(F ) corresponds to the monomial kmx
m of F .
Hence, we have Dµ = (LogR)−1(D̂µ) for any µ ∈ P , where LogR is the map from XF ′(C)
to XF ′(T) defined in (2.1).
Example 3.4. Consider the polynomial F = t−1 + x1 + x2 + x−1
1 x−1
2 . The tropical hypersur-
face V (trop(F )) and the regions {D̂µ}µ∈P for F are shown in Figs. 6 and 7. νi and µi (i =
1, . . . , 6) denote vertices and edges of V (trop(F )) respectively as shown in Fig. 6. Each D̂νi is
the region colored in dark gray and each D̂µi is the region colored in light gray as shown in
Fig. 7.
Lemma 3.5. If C0 is sufficiently small, then Dρ∩XF ′,σ(C) 6= ∅ if and only if ρ∩XF ′,σ(T) 6= ∅
for any ρ ∈ P{0} and σ ∈ F ′.
Proof. Assume that ρ ∩ XF ′,σ(T) 6= ∅. We set µ := ρ ∩ XF ′,σ(T). We show that D̂µ =
D̂ρ∩XF ′,σ(T) 6= ∅. If C0 is sufficiently small, points in ρ which are sufficiently far from all faces
of ρ in P{0} are contained in D̂ρ. It follows that points in µ which are sufficiently far from all faces
of µ are contained in D̂ρ, and hence in D̂µ. Conversely, assume that ρ∩XF ′,σ(T) = ∅. Since the
region D̂ρ has to be near to the cell ρ if C0 is sufficiently small, we have D̂ρ∩XF ′,σ(T) = ∅. �
Lemma 3.6. If C0 is sufficiently small, then one has⋃
ρ∈P{0}
Dρ =
⋃
σ∈F ′
{ ⋃
µ∈Pσ
(Dµ ∩Oσ(C))
}
.
10 Y. Yamamoto
Proof. It is obvious that the right-hand side is contained in the left-hand side. We show that
the left-hand side is contained in the right-hand side. Let x be any point in Dρ (ρ ∈ P{0}).
There exists the unique cone σ ∈ F ′ such that x ∈ Oσ(C). Then, the point x is contained in
Dρ ∩XF ′,σ(C). From Lemma 3.5, we have ρ ∩XF ′,σ(T) 6= ∅. We set µ := ρ ∩XF ′,σ(T). Then
we have Dρ ∩XF ′,σ(C) = Dµ from (3.1). Hence one has x ∈ Dµ ∩Oσ(C). �
For each subset {m0, . . . ,mp} ⊂ A (p ∈ Z≥0), we define
Dm0,...,mp :=
{
x ∈ O{0}(C)
∣∣∣∣ bmi(x) > 0 for i = 0, . . . , p,
bm(x) = 0 for m ∈ A \ {m0, . . . ,mp}
}
,
where the overline means the closure in XF ′(C).
Lemma 3.7. Let {m0, . . . ,mp} be a subset of A such that p ≥ 1 and {m0, . . . ,mp} 6= Aρ for
any ρ ∈ P{0}. If the constant C0 is sufficiently small, then one has Dm0,...,mp = ∅.
Proof. Let H ⊂ O{0}(T) be the affine space defined by vm0 +m0 ·X = · · · = vmp +mp ·X.
First, we show that there exists a neighborhood N of H such that any of vm0 +m0 ·X, . . . , vmp +
mp ·X do not coincide with trop(F ) on N . Assume that there exists X0 ∈ H and i ∈ {1, . . . , p}
such that vmi + mi · X0 = trop(F )(X0). Then there exists ρ′ ∈ P{0} such that X0 ∈ ρ′ and
{m0, . . . ,mp} ⊂ Aρ′ . Since V (trop(F )) is smooth and locally coincides with the tropical hyper-
plane, there exists ρ ∈ P{0} such that ρ′ ≺ ρ and {m0, . . . ,mp} = Aρ. This contradicts to the
assumption. Hence, any of vm0 +m0 ·X, . . . , vmp +mp ·X do not coincide with trop(F ) on H.
Then there exists a neighborhood N of H such that any of vm0 + m0 ·X, . . . , vmp + mp ·X do
not coincide with trop(F ) on N .
Assume that Dm0,...,mp is not empty. The differences between the values of vm0 + m0 ·
X, . . . , vmp + mp · X are in the range of ±C0 on LogR(Dm0,...,mp) ∩ O{0}(T). Then the set
LogR(Dm0,...,mp)∩O{0}(T) has to be in N for a sufficiently small constant C0. The fact that any
of vm0 +m0 ·X, . . . , vmp+mp ·X do not coincide with trop(F ) on N ⊃ LogR(Dm0,...,mp)∩O{0}(T)
contradicts to the definition of Dm0,...,mp . �
Lemma 3.8. Let σ ∈ F ′ be a cone and µ1, µ2 ∈ Pσ be cells. Suppose that the constant C0
is sufficiently small. If Dµ1 ∩ Dµ2 6= ∅, then there exists µ ∈ Pσ such that µ ≺ µ1, µ2 and
Dµ1 ∩Dµ2 ⊂ Dµ.
Proof. Let µ′1, µ
′
2 ∈ P{0} be the cells such that µ1 = µ′1∩XF ′,σ(T) and µ2 = µ′2∩XF ′,σ(T). We
set {m0, . . . ,mp} := Aµ′1 ∪ Aµ′2 . Here, we have Dµ′1
∩Dµ′2
⊂ Dm0,...,mp . If Dµ1 ∩Dµ2 6= ∅, the
set Dµ′1
∩Dµ′2
⊃ Dµ1 ∩Dµ2 is also nonempty. Hence, we have Dm0,...,mp 6= ∅. From Lemma 3.7,
there must exists a cell ρ ∈ P{0} such that Aρ = {m0, . . . ,mp} and Dm0,...,mp = Dρ. We have
Dµ′1
∩Dµ′2
⊂ Dρ and ρ ≺ µ′1, µ′2. Then the cell µ := ρ ∩XF ′,σ(T) satisfies Dµ1 ∩Dµ2 ⊂ Dµ and
µ ≺ µ1, µ2. �
The aim of this section is to prove the following theorem.
Theorem 3.9. Fix a sufficiently small constant C0. For a sufficiently large R ∈ R>0, the tropical
localization Wq and the family of subsets {Dµ}µ∈P of XF ′(C) satisfy the following conditions:
1. For any q ∈ S1
R, the submanifold Wq is isotopic to Vq in XF ′(C).
2. For any q ∈ S1
R, one has Wq ⊂
⋃
ρ∈P{0} Dρ.
3. Let σ ∈ F ′ be a cone and µ ∈ Pσ be a cell. Let further µ′ ∈ P{0} be the cell such that
µ = µ′ ∩XF ′,σ(T). Assume that the dimension of σ and µ′ is l and k, respectively (l ≤ k).
Let (x̃1, . . . , x̃n+1−l) be a standard coordinate with respect to µ (see Section 2.3). Then,
the defining equation of Wq on Dµ ∩Oσ(C) coincides with that of the (n− k)-dimensional
Geometric Monodromy around the Tropical Limit 11
Figure 8. The graph of the function d.
tropically localized hyperplane in (x̃1, . . . , x̃n+1−k) and is independent of the coordinate
(x̃n+2−k, . . . , x̃n+1−l).
The outline of the proof of Theorem 3.9 is as follows. In order to show the condition 1,
we construct an isotopy {Vq,s}s∈[0,1] which connects Vq and Wq. For F =
∑
m∈A
kmx
m, we set
km =
∑
i∈Z
cmit
i ∈ K (cmi ∈ C) and cm := cm,−vm . Let d(s) be a real valued monotone C∞
function on R which has 1 on {s ≥ 2/3} and 0 on {s ≤ 1/3}. The graph of d(s) is shown in
Fig. 8. For each s ∈ [0, 1], we define the functions bm,s : O{0}(C)→ R and f̃q,s : O{0}(C)→ C by
bm,s(x) := (1− d(s)) + d(s)bm(x),
f̃q,s(x) :=
∑
m∈A
bm,s(x)cm
(1−d(s))qvmxm + (1− d(s))
{
fq −
∑
m∈A
cmq
vmxm
}
,
where the branch of c
(1−d(s))
m is determined by 0 ≤ arg(cm) < 2π. Let Vq,s be the closure
of {x ∈ O{0}(C) | f̃q,s(x) = 0} in XF ′(C). Then we have f̃q,0 = fq, f̃q,1 = f̃q and Vq,0 = Vq,
Vq,1 = Wq.
First, we check that Vq,s is contained in
⋃
ρ∈P{0} Dρ for any q ∈ S1
R and s ∈ [0, 1]. Then,
we set q = R exp(
√
−1θ) and consider the projection p : XF ′(C) × (−ε, 2π + ε) × (0, 1) →
(−ε, 2π + ε)× (0, 1) given by
(x, θ, s) 7→ (θ, s),
where ε ∈ R is a small constant such that 0 < ε� 1. Let Y be the subset of XF ′(C)× (−ε, 2π+
ε)× (0, 1) defined by
Y :=
{
(x, θ, s) ∈ XF ′(C)× (−ε, 2π + ε)× (0, 1) | f̃q,s(x) = 0
}
.
We use the following theorem.
Theorem 3.10 (Ehresmann’s fibration theorem). Let f : E →M be a C∞ map between smooth
manifolds. If the map f is a proper submersion, then the map f is a locally trivial fibration.
We check that the function f̃q,s has 0 as a regular value on each Dµ ∩Oσ(C) for any q ∈ S1
R
and s ∈ [0, 1]. Then, it turns out that the restriction of p to Y is a submersion. In addition,
we can easily see that p|Y is proper. From Theorem 3.10, we can conclude that the family of
submanifolds {Vq,s}s∈[0,1] gives an isotopy between Vq and Wq. The condition 3 can be shown
by a simple calculation.
Proof of Theorem 3.9. We set T :=
⋃
ρ∈P{0} Dρ. First, we show that Vq,s is contained in T
for any q ∈ S1
R and s ∈ [0, 1]. Since we have
O{0}(C) =
( ⋃
{m0,...mp}⊂A
p∈Z≥0
Dm0,...,mp
)
∩O{0}(C),
12 Y. Yamamoto
it follows from Lemma 3.7 that it is enough to check that the function f̃q,s can not be 0 on
Dm∩O{0}(C) for any m ∈ A. The dominant term of f̃q,s on Dm∩O{0}(C) is only cm
(1−d(s))qvmxm
and we have
bm′,s(x) =
{
1− d(s), m′ 6= m,
1 m′ = m.
Hence, the function f̃q,s can be written on Dm ∩O{0}(C) as
cm
(1−d(s))qvmxm + (1− d(s))
{∑
p
hpq
ipxjp
}
,
where hp ∈ C, ip ∈ Z, jp ∈ A and each term hpq
ipxjp denotes other monomial which is not
dominant on Dm, i.e., |qipxjp |/|qvmxm| ≤ R−C0 . (Each index p satisfies that either jp 6= m
or jp = m and ip < vm.) Hence, for sufficiently large R, the function f̃q,s can not be 0 on
Dm ∩ O{0}(C). Then we have Vq,s ⊂ T for all q ∈ S1
R and s ∈ [0, 1]. In particular, the
condition 2 holds.
Next, we show that the projection p|Y is a proper submersion. For µ ∈ Pσ, let µ′ ∈ P{0}
be the cell such that µ = µ′ ∩ XF ′,σ. We define m0, . . . ,mn+1−k by {m0, . . . ,mn+1−k} = Aµ′
(the set Aµ′ is defined in (2.2)). For any m ∈ A \ Aµ′ , there exists mi ∈ Aµ′ such that
|qvmxm|/|qvmixmi | ≤ R−C0 on Dµ′∩O{0}(C). Then we have b
(
logR |qvmxm|−logR |qvmixmi |
)
≡ 1
and b
(
logR |qvmixmi | − logR |qvmxm|
)
≡ 0. Therefore, we have
bm,s|Dµ′∩O{0}(C)(x)
=
(1− d(s)) + d(s)
n+1−k∏
i=0
b
(
logR |qvmixmi | − logR |qvmxm|
)
, m ∈ Aµ′ ,
1− d(s) otherwise,
and
f̃q,s
∣∣
Dµ′∩O{0}(C)
(x) =
n+1−k∑
i=0
bmi,s(x)c(1−d(s))
mi qvmixmi
+ (1− d(s))
{
fq −
∑
m∈A
cmq
vmxm +
∑
m∈A\Aµ′
cm
(1−d(s))qvmxm
}
. (3.2)
Let τ ∈ F ′ be an (n+ 1)-dimensional cone having σ as its face. Let further e1, . . . , en+1 ∈ Zn+1
be the primitive generators of τ∨. We rearrange e1, . . . , en+1 if necessary, and set yi := xei
(i = 1, . . . , n + 1) so that the set of functions (y1, . . . , yn+1) forms a coordinate system on
Uτ (C) ∼= Cn+1 such that yn+2−l = · · · = yn+1 = 0 on Oσ(C). Let (x̃1, . . . , x̃n+1−l) be a standard
coordinate with respect to µ such that x̃i := qvmixmi/qvm0xm0 for i = 1, . . . , n+1−k. Then, the
set of functions (x̃1, . . . , x̃n+1−l, yn+2−l, . . . , yn+1) forms a coordinate system on
⋃
σ′≺σ Oσ′(C).
We define the functions bmi,µ : O{0}(C)→ R by
bm0,µ(x̃) :=
n+1−k∏
j=1
b(logR |x̃j |),
bmi,µ(x̃) := b(− logR |x̃i|)
n+1−k∏
j=1
b(logR |x̃j | − logR |x̃i|) for i = 1, . . . , n+ 1− k,
Geometric Monodromy around the Tropical Limit 13
and set bmi,µ,s(x̃) := (1− d(s)) + d(s)bmi,µ(x̃). In the coordinate system (x̃1, . . . , x̃n+1−l, yn+2−l,
. . . , yn+1), we divide (3.2) by qm0xm0 to obtain
f̃q,s
∣∣
Dµ′∩O{0}(C)
qm0xm0
= bm0,µ,s(x̃)c(1−d(s))
m0
+
n+1−k∑
i=1
bmi,µ,s(x̃)c(1−d(s))
mi x̃i + (1− d(s)){other terms}.
Notice that other terms are not dominant on Dµ′ . Hence, we may assume that the subset Vq,s
is defined on Dµ ∩Oσ(C) by
bm0,µ,s(x̃)c(1−d(s))
m0
+
n+1−k∑
i=1
bmi,µ,s(x̃)c(1−d(s))
mi x̃i + (1− d(s))
{∑
p
hpq
ip x̃jp
}
= 0, (3.3)
where hp ∈ C and terms
∑
p hpq
ip x̃jp denote other terms which are not dominant on Dµ. We have∣∣qip x̃jp∣∣ ≤ R−C0 and
∣∣qip x̃jp∣∣/|x̃i| ≤ R−C0 for all p. Let G(x̃1, . . . , x̃n+1−l) denote the left-hand
side of (3.3). We show that G(x̃1, . . . , x̃n+1−l) has 0 ∈ C as a regular value on Dµ ∩Oσ(C).
We define the subset Dµ,mi := {x ∈ Dµ | |qvmixmi |/|qvmjxmj | ≥ 1 (j = 0, . . . , n+ 1− k)} for
i = 0, . . . , n + 1 − k. For any x ∈ Dµ, there exists i ∈ {1, . . . n + 1 − k} such that |qvmixmi | ≥
|qvmjxmj | for any j ∈ {1, . . . , n + 1 − k}. Then we have Dµ =
⋃n+1−k
i=0 Dµ,mi . We have only to
show that the Jacobian matrix of G(x̃1, . . . , x̃n+1−l) has the maximal rank on Dµ,m1 . On Dµ,m1 ,
we have bm1,µ,s(x̃) ≡ 1. We set x̃i = ri exp(
√
−1θi) (ri ∈ R≥0, θi ∈ [0, 2π]) and let M be a 2× 2
matrix defined by
M :=
∂
∂r1
Re(G)
∂
∂θ1
Re(G)
∂
∂r1
Im(G)
∂
∂θ1
Im(G)
.
We can show det(M) 6= 0 for a sufficiently large R by the concrete calculation. Hence, the
subsets Vq,s and Y is smooth submanifold in XF ′(C) and {XF ′(C) × (−ε, 2π + ε) × (0, 1)}
respectively. Moreover, it turns out that the projection p|Y : Y → (−ε, 2π + ε)× (0, 1) is a sub-
mersion.
In addition, for any compact subset C ⊂ (−ε, 2π+ε)×(0, 1), the inverse image (p|Y )−1(C) ⊂ Y
coincides with {(x, θ, s) ∈ XF ′(C) × C | f̃q,s(x) = 0}. Then the set (p|Y )−1(C) is compact and
the map p|Y is proper. Hence, it turns out from Theorem 3.10 that the map p|Y has a structure
of a fiber bundle with the fiber VR,1 = Wq=R =: WR. Therefore, the family of submanifolds
{Vq,s}s∈[0,1] gives an isotopy and the condition 1 holds.
Finally, we check the condition 3. In (3.3), we set s = 1 to obtain
n+1−k∏
j=1
b(logR |x̃j |) +
n+1−k∑
i=1
{
b(− logR |x̃i|)
n+1−k∏
j=1
b(logR |x̃j | − logR |x̃i|)
}
x̃i = 0.
This coincides with the defining function of the (n − k)-dimensional tropically localized hy-
perplane in (x̃1, . . . , x̃n+1−k) and the left-hand side is independent of the values of x̃n+2−k, . . . ,
x̃n+1−l. Hence, the condition 3 holds. �
4 Monodromy transformations
We use the same notation as in Section 3 and keep the assumption that V (trop(F )) is smooth.
We set WR := Wq=R. Let {ψq=R exp(
√
−1θ) : WR → Wq}θ∈[0,2π] be a family of homeomorphisms
which depends on θ continuously. It is clear that the map ψq=R exp(2π
√
−1) : WR →WR gives the
monodromy transformation of {Vq}q∈S1
R
under the identificationWR
∼= VR. Hence, it is sufficient
to construct a monodromy transformation of {Wq}q∈S1
R
in order to get that of {Vq}q∈S1
R
.
14 Y. Yamamoto
Figure 9. The region LogR(WR) for F = 1 +
x1 + x2.
Figure 10. The map φν for F = 1 + x1 + x2.
Proposition 4.1. There exists a continuous map φ : LogR(WR) → V (trop(F )) satisfying the
following condition:
(∗) φ(LogR(WR) ∩ D̂ρ ∩Oσ(T)) ⊂ ρ ∩Oσ(T) for any σ ∈ F ′ and ρ ∈ Pσ.
Moreover, such maps are unique up to homotopy.
Proof. For each cell ρ ∈ P , we construct a continuous map φρ : LogR(WR)∩ D̂ρ → V (trop(F ))
satisfying following conditions:
(i) φρ(LogR(WR) ∩ D̂ρ ∩Oσ(T)) ⊂ ρ ∩Oσ(T), where σ ∈ F ′ is a cone such that ρ ∈ Pσ.
(ii) For any face µ ≺ ρ, the map φρ coincides with φµ on LogR(WR) ∩ D̂ρ ∩ D̂µ.
We construct φρ in an ascending order of dim ρ as follows. For each vertex ρ ∈ P , we set φρ as
a constant map from LogR(WR)∩ D̂ρ to ρ. For each 1-cell ρ, let ν0 and ν1 be the endpoints of ρ.
We set each φρ as a continuous map to ρ so that φρ coincides with the constant map to νi on
LogR(WR) ∩ D̂ρ ∩ D̂νi and satisfies the condition (i). Assume that we have constructed φρ for
all cells whose dimensions are lower than k − 1. For each k-cell ρ, we define φρ as a continuous
map to ρ so that φρ coincides with φµ on LogR(WR)∩ D̂ρ ∩ D̂µ for any face µ of ρ and satisfies
the condition (i). In this way, we can construct a family of maps {φρ}ρ∈P such that each map φρ
satisfies the condition (i) and (ii).
Example 4.2. Consider the polynomial F = 1 + x1 + x2. Fig. 9 shows V (trop(F )) and
LogR(WR). Let ν denote the center vertex of V (trop(F )). The region colored gray denotes
LogR(WR) ∩ D̂ν . The map φν : LogR(WR) ∩ D̂ν → V (trop(F )) is the constant map to ν as
shown in Fig. 10.
For any σ ∈ F ′ and ρ1, ρ2 ∈ Pσ, if D̂ρ1 ∩ D̂ρ2 6= ∅, there exists ρ ∈ Pσ such that ρ ≺ ρ1, ρ2
and D̂ρ1 ∩ D̂ρ2 ⊂ D̂ρ (Lemma 3.8). From the condition (ii), the map φρi coincides with φρ
on LogR(WR) ∩ D̂ρ ∩ D̂ρi (i = 1, 2). Hence, the maps φρ1 and φρ2 coincide with each other
on LogR(WR) ∩ D̂ρ ∩ D̂ρ1 ∩ D̂ρ2=LogR(WR) ∩ D̂ρ1 ∩ D̂ρ2 . Then it turns out that we can get
the continuous map φ : LogR(WR) → V (trop(F )) by gluing {φρ}ρ. The map φ satisfies the
condition (∗).
Let φ0, φ1 : LogR(WR)→ V (trop(F )) be two maps satisfying the condition (∗). We construct
a family of continuous maps {φs : LogR(WR) → XF ′(T)}s∈[0,1] by φs(X) := (1 − s)φ0(X) +
sφ1(X), where the addition and the multiplications are taken on Oσ(T) ∼= Rl for the cone
σ ∈ F ′ such that X ∈ Oσ(T). This construction is independent of the choice of coordinates
on Oσ(T). Since each cell is convex, φs satisfies the condition (∗) for any s ∈ [0, 1]. Therefore,
each map φs is well-defined as a continuous map to V (trop(F )) and {φs}s∈[0,1] gives a homotopy
between φ0 and φ1. �
Geometric Monodromy around the Tropical Limit 15
We fix a map φ : LogR(WR) → V (trop(F )) satisfying the condition (∗) in Proposition 4.1.
Let σ ∈ F ′ be an l-dimensional cone. We choose ai, bij ∈ Z (i = 1, . . . , n+1− l, j = 1, . . . , n+1)
so that the sets of functions (y1, . . . , yn+1−l) and (Y1, . . . , Yn+1−l) defined by
yi := qai
n+1∏
j=1
x
bij
j , Yi := ai +
n+1∑
j=1
bijXj , (4.1)
form coordinate systems on Oσ(C) and Oσ(T), respectively. For each q = R exp(
√
−1θ) ∈ S1
R,
we define the map ψσ,q : WR ∩Oσ(C)→ Oσ(C) by
(y1, . . . , yn+1−l) 7→
(
φ̃1,θy1, . . . , φ̃n+1−l,θyn+1−l
)
,
where φ̃i,θ(y1, . . . , yn+1−l) := exp
(√
−1θYi ◦ φ ◦ LogR(y)
)
(i = 1, . . . , n+ 1− l).
Lemma 4.3. For any q ∈ S1
R and σ ∈ F ′, the map ψσ,q is independent of the choice of the
coordinate system (y1, . . . , yn+1−l) on Oσ(C) and the image ψσ,q(WR ∩ Oσ(C)) is contained in
Wq ∩Oσ(C).
Proof. First, we show that the map ψσ,q is independent of the choice of the coordinate. Let
(z1, . . . , zn+1−l) and (Z1, . . . , Zn+1−l) be other coordinate systems on Oσ(C) and Oσ(T) defined
just as (y1, . . . , yn+1−l) and (Y1, . . . , Yn+1−l). We can write
zi = qαi
n+1−l∏
j=1
y
βij
j , yi =
n+1−l∏
j=1
(
q−αjzj
)γij ,
where αi, βij are some integral numbers. Here, we have
n+1−l∑
j=1
βijγjk = δik. Let ψ′σ,q : WR ∩
Oσ(C) → Oσ(C) be the map defined in (z1, . . . , zn+1−l). For all z = (z1, . . . , zn+1−l) = y =
(y1, . . . , yn+1−l) ∈WR ∩Oσ(C), we have
zi
(
ψ′σ,q(z)
)
=
exp
√−1θ
αi +
n+1−l∑
j=1
βijYj
(φ ◦ LogR(z))
Rαi
n+1−l∏
j=1
y
βij
j ,
and
yi
(
ψ′σ,q(z)
)
=
n+1−l∏
k=1
(
q−αkzk(ψ
′
σ,q(z))
)γik
=
n+1−l∏
k=1
q−αk
exp
√−1θ
αk +
n+1−l∑
j=1
βkjYj
(φ ◦ LogR(z))
Rαk
n+1−l∏
j=1
y
βkj
j
γik
=
exp
√−1θ
∑
k,j
γikβkjYj (φ ◦ LogR(y))
n+1−l∏
j=1
y
∑
k γikβkj
j
= exp
(√
−1θYi (φ ◦ LogR(y))
)
yi = φ̃i,θyi.
Therefore, we have yi(ψ
′
σ,q(y)) = yi(ψσ,q(y)) = φ̃i,θyi. Hence, one has ψσ,q = ψ′σ,q.
Next, we show that the image ψσ,q(WR ∩ Oσ(C)) is contained in Wq ∩ Oσ(C). Let µ ∈ Pσ
be a cell such that µ = µ′ ∩XF ′,σ(C) for a k-cell µ′ ∈ P{0} and (x̃1, . . . , x̃n+1−l) be a standard
16 Y. Yamamoto
coordinate with respect to µ. Since X̃1 = · · · = X̃n+1−k = 0 on µ, the restriction of ψσ,q to
Dµ ∩Oσ(C) coincides with
(x̃1, . . . , x̃n+1−l) 7→
(
x̃1, . . . , x̃n+1−k, φ̃n+2−k,θx̃n+2−k, . . . , φ̃n+1−l,θx̃n+1−l
)
.
Since the defining equation of Wq on Dµ ∩Oσ(C) coincides with that of the (n−k)-dimensional
tropically localized hyperplane in (x̃1, . . . , x̃n+1−k), we have ψσ,q (WR ∩Oσ(C) ∩Dµ) ⊂ Wq ∩
Oσ(C)∩Dµ. Hence, the map ψσ,q is well-defined as a map from WR∩Oσ(C) to Wq∩Oσ(C). �
Lemma 4.4. For any q ∈ S1
R, the family of maps {ψσ,q}σ∈F ′ glues together to give the homeo-
morphism ψq : WR →Wq.
Proof. Let τ ∈ F ′ be an (n + 1)-dimensional cone. Let further e1, . . . , en+1 ∈ Zn+1 be the
primitive generators of τ∨. We set wi := xei and Wi := ei ·X (i = 1, . . . , n + 1). Then the set
of functions (w1, . . . , wn+1) and (W1, . . . ,Wn+1) form coordinate systems on Uτ (C) ∼= Cn+1 and
Uτ (T) ∼= Tn+1, respectively. We define the map Ψτ,q : WR ∩ Uτ (C)→ Uτ (C) by
(w1, . . . , wn+1) 7→
(
φ̃1,θw1, . . . , φ̃n+1,θwn+1
)
,
where φ̃i,θ(w1, . . . , wn+1) := exp
(√
−1θWi ◦ φ ◦ LogR(w)
)
(i = 1, . . . , n + 1). It is clear from
Lemma 4.3 that the map ψσ,q coincides with Ψτ,q on Oσ(C) ⊂ Uτ (C) for any face σ ≺ τ . There-
fore, the family of maps {ψσ,q}σ∈F ′ glues together to give the continuous map ψq : WR →Wq.
In addition, the inverse map (ψσ,q)
−1 : Wq →WR is given by
(y1, . . . , yn+1−l) 7→
(
φ̃1,−θy1, . . . , φ̃n+1−l,−θyn+1−l
)
and {(ψσ,q)−1}σ∈F ′ forms the inverse map (ψq)
−1 : Wq →WR. It is obvious that the map (ψq)
−1
is continuous. Hence, the map ψq is a homeomorphism for any q ∈ S1
R. �
The following is the main theorem of this paper.
Theorem 4.5. Assume that V (trop(F )) is smooth. We fix a sufficiently large number R ∈ R>0.
Let φ : LogR(WR) → V (trop(F )) be a map satisfying the condition (∗) in Proposition 4.1. Let
further ψ : WR →WR be the map defined on each orbit Oσ(C) (σ ∈ F ′) by
(y1, . . . , yn+1−l) 7→
(
φ̃1y1, . . . , φ̃n+1−lyn+1−l
)
, (4.2)
where (y1, . . . , yn+1−l) is a coordinate system on Oσ(C) defined as in (4.1) and
φ̃i(y) := exp
(
2π
√
−1Yi ◦ φ ◦ LogR(y)
)
.
Then the map ψ : WR → WR gives a monodromy transformation of {Vq}q∈S1
R
under the identi-
fication VR ∼= WR. For each cell µ ∈ Pσ, the restriction of ψ to Dµ ∩Oσ(C) coincides with
(x̃1, . . . , x̃n+1−l) 7→
(
x̃1, . . . , x̃n+1−k, φ̃n+2−kx̃n+2−k, . . . , φ̃n+1−lx̃n+1−l
)
,
in a standard coordinate (x̃1, . . . , x̃n+1−l) with respect to µ.
Proof. It is clear from Lemmas 4.3 and 4.4. �
Geometric Monodromy around the Tropical Limit 17
5 Proof of Corollary 1.1
In this section, we show that Corollary 1.1 follows from Theorem 4.5. We set n = 1. Let
φ : LogR(WR) → V (trop(F )) be a map satisfying the condition (∗) in Proposition 4.1. We
set the map φ so that the restriction of φ to LogR(WR) ∩ D̂ρ gives a bijection to ρ for any
edge ρ ∈ P{0}. Let ν ∈ P{0} be a vertex of V (trop(F )) contained in O{0}(T). Let further
(x̃1, x̃2) and (X̃1, X̃2) be standard coordinates with respect to ν (see Section 2.3). On Dν ,
the tropical localization Wq is defined by the defining equation of the 1-dimensional tropically
localized hyperplane in (x̃1, x̃2). Since we have X̃1(ν) = X̃2(ν) = 0 and the restriction of φ to
LogR(WR) ∩ D̂ν is the constant map to ν, the monodromy transformation ψ in Theorem 4.5
coincides with the identity map on Dν . Similarly, it turns out that the map ψ also coincides
with the identity map on Dν′ for any vertex ν ′ ∈ P contained in a lower dimensional torus orbit.
Let µ ∈ P{0} be a bounded edge of V (trop(F )) and ν1, ν2 be the endpoints of µ. We set
{m0,m1,m2} ⊂ A so that {m0,m1,m2} = Aν1 and {m0,m1} = Aµ, where Aν1 and Aµ are
subsets of A defined in (2.2). We define the standard coordinate with respect to ν1 by
x̃i := qvmixmi/qvm0xm0 , X̃i := (vmi +miX)− (vm0 +m0X),
for i = 1, 2. Then the coordinate systems (x̃1, x̃2) and (X̃1, X̃2) are also standard coordinates
with respect to µ. On Dµ, the defining equation of the tropical localization Wq coincides with
b(logR |x̃1|) + b(− logR |x̃1|)x̃1 = 0. (5.1)
Lemma 5.1. The solution of (5.1) is x̃1 = −1.
Proof. The equation (5.1) coincides with b(logR |x̃1|) + x̃1 = 0 when C1 ≤ logR |x̃1| ≤ C0 and
1+b(− logR |x̃1|)x̃1 = 0 when −C0 ≤ logR |x̃1| ≤ −C1. These equations have no solution when R
is sufficiently large. In the case −C1 ≤ logR |x̃1| ≤ C1, (5.1) coincides with 1 + x̃1 = 0. �
Hence, the tropical localization Wq coincides with the cylinder defined by x̃1 = −1 and
x̃2 are free on Dµ. Let l ∈ Z>0 be the length of µ. In the coordinate system (x̃1, x̃2), we
have X̃1(ν1) = X̃2(ν1) = 0 and X̃1(ν2) = 0, X̃2(ν2) = −l. Note that the lengths of edges are
invariant under the coordinate transformations. Since the restriction of φ to LogR(WR)∩D̂µ gives
a bijection to µ, we can see from Theorem 4.5 that the map ψ coincides with the composition of
l-times of Dehn twists on Dµ. Similarly, it turns out that the restriction of ψ to Dµ′ coincides
with the compositions of infinitely many times of Dehn twists for any unbounded edge µ′ ∈ P{0}.
6 Examples
6.1 Example in dimension 1
Consider the polynomial F given by (1.3). The tropical hypersurface V (trop(F )) is shown in
Fig. 11. Let ν1, ν2, µ denote cells of V (trop(F )) as shown in Fig. 11 (ν1, ν2 denote 0-cells and µ
denotes the 1-cell). The regions D̂ν1 , D̂ν2 and D̂µ defined in Definition 3.3 are shown in Fig. 12.
The set Aν1 defined in (2.2) is given by {(0, 2), (1, 1), (0, 1)}. We set
x̃1 := q2x1x2/x
2
2 = q2x1x
−1
2 , x̃2 := qx2/x
2
2 = qx−1
2 ,
X̃1 := (2 +X1 +X2)− (2X2) = 2 +X1 −X2, X̃2 := (1 +X2)− 2X2 = 1−X2.
Then the sets of functions (x̃1, x̃2) and (X̃1, X̃2) form standard coordinates on O{0}(C) and
O{0}(T) with respect to ν1 defined in Section 2.3. Similarly, we have Aν2 = {(0, 2), (1, 1), (2, 1)}
and we set
x̃3 := q2x2
1x2/x
2
2 = q2x2
1x
−1
2 , X̃3 := (2 + 2X1 +X2)− (2X2) = 2 + 2X1 −X2,
18 Y. Yamamoto
Figure 11. The tropical hypersurface V (trop(F )). Figure 12. Regions D̂ν1 , D̂ν2 and D̂µ.
Table 1. Monodromy transformation ψ on each region.
region Dν1 Dν2 Dµ
standard coordinate (x̃1, x̃2) (x̃1, x̃3) (x̃1, x̃2) or (x̃1, x̃3)
tropical localization
WR
1-dimensional tropically
localized hyperplane
in (x̃1, x̃2)
1-dimensional tropically
localized hyperplane
in (x̃1, x̃3)
1-dimensional
cylinder
in x̃2 or x̃3
map φ constant map to ν1 constant map to ν2 bijection to µ
monodromy ψ identity map identity map Dehn twist
so that the sets of functions (x̃1, x̃3) and (X̃1, X̃3) form standard coordinates on O{0}(C) and
O{0}(T) with respect to ν2. In addition, we have Aµ = {(0, 2), (1, 1)}. Hence, the sets of
functions (x̃1, x̃2), (X̃1, X̃2) and (x̃1, x̃3), (X̃1, X̃3) also form standard coordinates with respect
to µ. Let WR be the tropical localization defined in Definition 3.1 and φ : LogR(WR) →
V (trop(F )) be a map satisfying the condition (∗) in Proposition 4.1. Here, we set the map φ so
that the restriction of φ to LogR(WR) ∩ D̂ρ gives a bijection to ρ for any edge ρ ∈ P{0}. The
manifold WR, the map φ and the monodromy transformation ψ : WR →WR on each region are
listed in Table 1.
Since X̃1(ν1) = X̃2(ν1) = 0 and X̃1(ν2) = X̃3(ν2) = 0, we can see from Theorem 4.5 that the
restrictions of ψ to Dν1 and Dν2 are identity maps. On Dµ, if we use (x̃1, x̃2) as a coordinate
system, we have X̃1 ≡ 0 on µ and X̃2(ν1) = 0, X̃2(ν2) = −1. Then we can also see from
Theorem 4.5 that the restriction of ψ to Dµ coincides with the Dehn twist in the component of
the cylinder in x̃2.
6.2 Example in dimension 2
Consider the polynomial G(x1, x2, x3) = t−1 + x1 + x2 + x3 + x−1
1 x−1
2 x−1
3 . Then we have
gq(x1, x2, x3) = q + x1 + x2 + x3 + x−1
1 x−1
2 x−1
3 ,
trop(G)(X1, X2, X3) = max{1, X1, X2, X3,−X1 −X2 −X3}.
The tropical hypersurface V (trop(G)) is shown in Fig. 13. Let ρ denote the 2-cell of V (trop(G))
contained in X1 = 1 and ν1, ν2, ν3, µ1, µ2, µ3 denote faces of ρ as shown in Fig. 13 (ρ denotes
the 2-cell colored in light gray). Fig. 14 shows the intersections of the hyperplane X1 = 1 and
regions D̂νi , D̂µi (i = 1, 2, 3) and D̂ρ defined in Definition 3.3.
The set Aν1 is given by {(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)}. We set x̃i := xi/q = q−1xi and
X̃i := −1 +Xi for i = 1, 2, 3. Then the sets of functions (x̃1, x̃2, x̃3) and (X̃1, X̃2, X̃3) form stan-
dard coordinates with respect to ν1 on O{0}(C) and O{0}(T). On the other hand, we have Aµ1 =
Geometric Monodromy around the Tropical Limit 19
Figure 13. The tropical hypersurface V (trop(G)). Figure 14. The intersections of the hyperplane
X1 = 1 and regions D̂νi , D̂µi
, D̂ρ.
{(0, 0, 0), (1, 0, 0), (0, 1, 0)}, Aµ3 = {(0, 0, 0), (1, 0, 0), (0, 0, 1)} and Aρ = {(0, 0, 0), (1, 0, 0)}. Hen-
ce, the sets of functions (x̃1, x̃2, x̃3) and (X̃1, X̃2, X̃3) also form standard coordinates with respect
to µ1, µ3 and ρ.
1. Tropical localization WR coincides with the following:
(a) On Dν1 : the 2-dimensional tropically localized hyperplane in (x̃1, x̃2, x̃3).
(b) On Dµ1 : the direct product of the 1-dimensional tropically localized hyperplane in
(x̃1, x̃2) and the 1-dimensional cylinder in x̃3.
(c) On Dµ3 : the direct product of the 1-dimensional tropically localized hyperplane in
(x̃1, x̃3) and the 1-dimensional cylinder in x̃2.
(d) On Dρ : the direct product of 1-dimensional cylinders in x̃2 and x̃3.
2. Let φ : LogR(WR)→ V (trop(G)) be a map satisfying the condition (∗) in Proposition 4.1.
We set the map φ so that the restriction of φ to D̂ρ′ gives a surjection to ρ′ for any cell
ρ′ ∈ P .
3. Monodromy transformation ψ is given as follows:
(a) On Dν1 : the identity map.
(b) On Dµ1 : the map which is identical in the component of the 1-dimensional tropically
localized hyperplane in (x̃1, x̃2) and coincides with the composition of four times of
the Dehn twists in the component of the cylinder in x̃3.
(c) On Dµ3 : the map which is identical in the component of the 1-dimensional tropically
localized hyperplane in (x̃1, x̃3) and coincides with the composition of four times of
the Dehn twists in the component of the cylinder in x̃2.
(d) On Dρ : the map which coincides with the composition of four times of the Dehn
twists in both components of the cylinders in x̃2 and x̃3.
Since the restriction of the map φ to D̂ν1 is the constant map to ν1 and X̃i(ν1) = 0 for
i = 1, 2, 3, it follows from Theorem 4.5 that the restriction of ψ to Dν1 coincides with
the identity map. Since the restriction of the map φ to Dµ1 is a surjection to µ1 and
X̃3(ν1) = 0, X̃3(ν2) = −4, we can see from Theorem 4.5 that the restriction of ψ to Dµ1
coincides with the composition of four times of the Dehn twists in the component of the
cylinder in x̃3. Similarly, it turns out that the restriction of ψ to Dµ3 coincides with
the composition of four times of the Dehn twists in the component of the cylinder in x̃2.
On Dρ, we can also see from Theorem 4.5 that the map ψ coincides with the composition
of four times of the Dehn twists in both components of the cylinders in x̃2 and x̃3.
20 Y. Yamamoto
7 Relation to Zharkov’s work
Definition 7.1. A convex lattice polytope ∆ ⊂ MR is smooth if for each vertex v of ∆, there
exists a Z-basis z1, . . . , zn+1 of M such that R≥0(∆− v) = R≥0z1 + · · ·+ R≥0zn+1.
Definition 7.2. Let ∆ ⊂ MR be a convex lattice polytope. We define the polar polytope
∆∗ ⊂ NR of ∆ by
∆∗ := {n ∈ NR | 〈m,n〉 ≥ −1 for all m ∈ ∆}.
The convex lattice polytope ∆ is called reflexive if it contains the origin 0 ∈ M as its interior
point and the polar polytope ∆∗ is also a lattice polytope in NR.
Let ∆ be a smooth and reflexive polytope in MR and B be a subset of ∆ ∩M containing 0
and all vertices of ∆. Let further T be a coherent triangulation of (∆, B). We assume that T is
central, i.e., every maximal-dimensional simplex in T has the origin 0 ∈ M as it’s vertex. Let
λ : B → Z be an integral vector which is in the interior of the secondary cone (see [3, Chapter 7,
Definition 1.4]) corresponding to T . We consider the function fq defined by
fq(x) := qλ(0) −
∑
i∈B\{0}
qλ(i)xi.
where q ∈ S1
R := {z ∈ C | |z| = R} for a sufficiently large R ∈ R>0. Let X∆ be the toric manifold
whose moment polytope is ∆ and Vq be the hypersurface in X∆ defined by fq. In this setting,
Zharkov constructed the monodromy transformation of {Vq}q∈S1
R
as follows:
(i) Let µR : X∆ → ∆ be the weighted moment map defined by
µR(x) :=
∑
m∈B
Rλ(m)|xm|m∑
m∈B
Rλ(m)|xm|
.
There exists a small neighborhood U ⊂ ∆ of the origin 0 ∈ ∆ such that µR(Vq) ⊂ ∆ \ U
for any q ∈ S1
R. We set ∆◦ := ∆ \ U . He constructs two families of regions {Uτ}τ∈∂T and
{Ũτ}τ∈∂T in ∆◦. For instance, in the case where
fq := q −
(
x+ xy + y + x−1 + x−1y−1 + y−1
)
(7.1)
and the triangulation T is given as shown in Fig. 15, the families of regions {Uτ}ρ∈∂T and
{Ũτ}τ∈∂T are as shown in Figs. 16 and 17, respectively. vi and wi (i = 1, . . . , 6) denote
vertices and edges of ∆ respectively as shown in Fig. 15. Uvi , Ũvi denote the regions
colored in light gray and Uwi , Ũwi denote the regions colored in dark gray as shown in
Figs. 16 and 17. We omit their construction here and refer the reader to [10, Section 3]
about how to construct them.
(ii) He sets bump functions bm : ∆◦ → [0, 1] (m ∈ B\{0}) so that the function f̃q : (C∗)n+1 → C
defined by
f̃q(x) := qλ(0) −
∑
m∈B\{0}
(bm ◦ µR)(x)qλ(m)xm
coincides with
qλ(0) −
∑
m∈τ∩B
qλ(m)xm on µ−1
R (Uτ ) ∩ (C∗)n+1,
qλ(0) −
∑
m∈τ∩B
(bm ◦ µR)(x)qλ(m)xm on µ−1
R (Ũτ ) ∩ (C∗)n+1,
Geometric Monodromy around the Tropical Limit 21
Figure 15. The triangulation T given by (7.1).
Figure 16. Regions {Uτ}τ∈∂T . Figure 17. Regions {Ũτ}τ∈∂T .
for any τ ∈ ∂T . Let Wq denote the submanifold in X∆ defined by f̃q(x) = 0. We can
see from the definition of the weighted moment map µR that if µR(x) ∈ Ũτ , the dominant
part of fq at x are qλ(0)−
∑
m∈τ∩B
qλ(m)xm. Since orders of terms cut off by bump functions
{bm}m∈B\{0} are lower, the submanifold Wq is diffeomorphic to Vq.
(iii) He defines the family of subsets {∆∨γ ⊂ NR}γ∈[0,1] by
∆∨γ :=
{
n ∈ NR | −〈m,n〉 ≥ γ(λ(m)− λ(0)) for any vertex m in T
}
.
For any γ > 0, the set ∆∨γ is a convex polytope with a nonempty interior. The set ∆∨γ in
the case where fq is given by (7.1) is shown in Fig. 18.
The region surrounded by the center part of the tropical hypersurface coincides with{
n ∈ NR |λ(0) ≥ 〈m,n〉+ λ(m) for any vertex m in T
}
.
Hence when we set γ = 1, the boundary of the convex polytope ∆∨γ=1 coincides with the
center part of the tropical hypersurface. For each k-dimensional simplex τ ∈ ∂T , we define
an (n− k)-dimensional face τ∨ of ∆∨γ by
τ∨ :=
{
n ∈ ∆∨γ | −〈n,m〉 = γ(λ(m)− λ(0)) for any vertex m in τ
}
.
There is a bijective correspondence between simplices in ∂T and faces of ∆∨γ given by
τ ↔ τ∨. Then he constructs a family of maps {vγ : ∆◦ → ∂∆∨γ }γ∈[0,1] which depends on γ
smoothly and satisfies vγ(Ũτ ) ⊂ τ∨ for any τ ∈ ∂T .
(iv) Let ei := (0, . . . , 0,
i
1̌, 0, . . . , 0) ∈M (i = 1, . . . , n+ 1) be the unit vector and ψi,γ : X∆ → C
(i = 1, . . . , n+ 1) be the function defined by
ψi,γ := exp
(
2π
√
−1〈(vγ ◦ µR)(x), ei〉
)
.
22 Y. Yamamoto
Figure 18. The convex polytope ∆∨γ in the case where fq is given by (7.1).
Figure 19. The tropical hypersurface and {D̂µ}µ in the case where fq is given by (7.1).
He defines a family of diffeomorphisms {Dγ : X∆ → X∆}γ∈[0,1] by
(x1, . . . , xn+1)→ (ψ1,γx1, . . . , ψn+1,γxn+1). (7.2)
For any element x ∈WR ∩ µ−1
R (Ũτ ), we have µR(Dγ(x)) = µR(x) ∈ Ũτ and
f̃q(Dγ(x)) = qλ(0) −
∑
m∈τ∩B
(bm ◦ µR)(x)qλ(m)xm exp
(
2π
√
−1〈vγ(µR(x)),m〉
)
= qλ(0) −
∑
m∈τ∩B
(bm ◦ µR)(x)qλ(m)xm exp
(
−2π
√
−1γ(λ(m)− λ(0))
)
= exp(2π
√
−1γλ(0))
{
Rλ(0) −
∑
m∈τ∩B
(bm ◦ µR)(x)Rλ(m)xm
}
= 0.
Hence, the family of maps {Dγ : X∆ → X∆}γ∈[0,1] induces the monodromy transformation{
Dγ : WR →Wq=R exp(2π
√
−1γ)
}
γ∈[0,1]
.
As explained in (ii), Zharkov also localized the hypersurface Vq to construct the monodromy
transformation. He used the weighted moment map while we used the tropicalization. The
regions {Ũτ}τ are similar to {D̂µ}µ constructed in Definition 3.3. Moreover, terms which we cut
off at each region are also the same. The tropical hypersurface and the family of regions {D̂µ}µ
are shown in Fig. 19 in the case where fq is given by (7.1). The region Ũvi corresponds to D̂µi
and Ũwi corresponds to D̂νi (i = 1, . . . , 6), respectively. For instance, on both Ũv2 and D̂µ2 , the
dominant terms are q and x. On both Ũw1 and Dν1 , the dominant terms are q, x, xy, and so on.
Note that regions at which the term qλ(0) is not dominant in our construction are included in
other regions in Zharkov’s construction. For instance, in the case fq is given by (7.1), the region
corresponding to D̂ρi is included in Ũwi for i = 1, . . . , 6. This is the only major differences in
the localization and the resulting manifolds Wq are similar to each other.
Geometric Monodromy around the Tropical Limit 23
His construction of the monodromy transformation is also similar to ours. We can con-
struct the family of maps {vγ : ∆◦ → ∂∆∨γ }γ∈[0,1] as follows. First, we construct vγ=1 satisfying
vγ(Ũρ) ⊂ ρ∨ for any ρ ∈ ∂T . We set
Sγ : MR →MR, (X1, . . . , Xn+1)→ (γX1, . . . , γXn+1).
for each γ ∈ [0, 1]. Then the map vγ := Sγ ◦ vγ=1 satisfies requested conditions. The map
φ : LogR(WR) → V (trop(F )) in Proposition 4.1 plays the same role as vγ=1. Moreover, the
monodromy transformation given by (4.2) in our construction coincides with (7.2). It can be
said that our construction is a natural generalization of Zharkov’s construction.
Acknowledgements
The author would like to express his gratitude to Kazushi Ueda for encouragement and helpful
advices. The author thanks to Tatsuki Kuwagaki for explaining the context of the paper [2].
The author also thanks the anonymous referees for reading this paper carefully and giving many
helpful comments. This research is supported by the Program for Leading Graduate Schools,
MEXT, Japan.
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http://dx.doi.org/10.2140/gt.2006.10.1097
http://arxiv.org/abs/math.SG/0511644
http://arxiv.org/abs/1204.2233
http://mathsoc.jp/videos/2010shuuki.html
http://dx.doi.org/10.1090/conm/460/09018
http://dx.doi.org/10.1016/j.top.2003.11.006
http://arxiv.org/abs/math.GT/0205011
http://www2.math.su.se/reports/2001/8/2001-8.pdf
http://www2.math.su.se/reports/2001/8/2001-8.pdf
http://dx.doi.org/10.1215/S0012-7094-00-10124-X
http://arxiv.org/abs/math.AG/9806091
1 Introduction
2 Preliminaries
2.1 Tropical toric varieties
2.2 Polyhedral complex
2.3 Hypersurfaces in toric varieties
3 Tropical localization
4 Monodromy transformations
5 Proof of Corollary 1.1
6 Examples
6.1 Example in dimension 1
6.2 Example in dimension 2
7 Relation to Zharkov's work
References
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