Geometric Monodromy around the Tropical Limit

Let $\{V_q\}_{q}$ 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 $\{V_q\}_q$ around $q=\infty$ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin.


Introduction
Let K := C{t} be the convergent Laurent series field, equipped with the standard non-archimedean valuation, Let n ∈ N be a natural number and M be a free Z-module of rank n+1. We write M R := M ⊗ Z R. Let further ∆ ⊂ M R be a convex lattice polytope, i.e., the convex hull of a finite subset of M . We set A := ∆∩M . Let F = m∈A k m x m ∈ K x ± 1 , . . . , x ± n+1 be a Laurent polynomial over K in n+1 variables such that k m = 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 k m for all m ∈ A, and set S 1 R := {z ∈ C | |z| = R}. For q ∈ S 1 R , let f q ∈ 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 X F (C) denote the toric manifold over C associated with F . For each q ∈ S 1 R , we define V q ⊂ X F (C) as the hypersurface defined by f q in X F (C). In this paper, we discuss the monodromy transformation of {V q } q∈S 1 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 ) : R n+1 → R be the tropicalization of F defined by trop(F )(X 1 , . . . , X n+1 ) := max m∈A val(k m ) + m 1 X 1 + · · · + m n+1 X n+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 complex of dimension n. The main theorem of this paper is Theorem 4.5, which gives a concrete description of the monodromy transformation of {V q } q∈S 1 R in terms of the tropical hypersurface V (trop(F )) in the case where V (trop(F )) is smooth (see Definition 2.7). The monodromy of {V q } q∈S 1 R is also discussed in [2, Appendix B.2] and Theorem 4.5 is covered by [2,Proposition B.17]. However, this paper aims to make the relation of the monodromy of {V q } q∈S 1 R to tropical geometry clear. We give a self-contained proof and explicit examples.

arXiv:1509.00175v2 [math.AG] 24 Jun 2016
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 {V q } q∈S 1 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 {ρ i } i∈{1,...,d} be the set of all bounded edges of V (trop(F )). For each ρ i , let ν i1 , ν i2 ∈ R n+1 be the endpoints of ρ i . Let further V ∈ Z n+1 be the primitive vector such that ν i1 − ν i2 = lV for some l ∈ R >0 . We define the length L(ρ i ) of ρ i 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 (m ij ) 1≤i,j≤2 , (r i ) i=1,2 ∈ GL 2 (Z) R 2 such that in the coordinate (Y 1 , Y 2 ) on R 2 defined by Y 1 = m 11 X 1 + m 12 X 2 + r 1 , Y 2 = m 21 X 1 + m 22 X 2 + r 2 , the tropical hypersurface V (trop(F )) coincides locally with the tropical hyperplane defined by max{0, Y 1 , Y 2 } around ν. Then we have ν i1 , ν i2 ∈ Z n+1 . The amoeba of V q converges to the tropical hypersurface V (trop(F )) as q → ∞ in the Hausdorff metric [8,9] and the hypersurface V q is obtained by 'thickening' the amoeba of V q . Let C i (i = 1, . . . , d) be the simple closed curve in V q=R turning around ρ i (see Fig. 1 for an example). Let further T i : V R → V R be the Dehn twist along C i . Corollary 1.1. If n = 1 and V (trop(F )) is smooth, then the monodromy transformation of Corollary 1.1 is conjectured by Iwao [4]. Let us illustrate this claim with a simple example. Consider the polynomial F given by Then we have The tropical hypersurface V (trop(F )) and the hypersurface V q in this case are shown in Fig. 1. Let ρ i and C i (i = 1, . . . , 7) denote edges of V (trop(F )) and simple closed curves in V q as shown in Fig. 1. Then the edges ρ 1 , . . . , ρ 7 correspond to the simple closed curves C 1 , . . . , C 7 , respectively. By simple calculations, we have It follows from Corollary 1.1 that the monodromy transformation of . 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 {V q } q∈S 1 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.  Let F be a fan in N R . We write the toric variety associated with F over C as X F (C). For each cone σ ∈ F, we set Let U σ (C) := Hom(σ ∨ ∩ M, C) denote the affine toric variety and O σ (C) := Hom(σ ⊥ ∩ M, C * ) denote the torus orbit corresponding to σ. We write the closure of O σ (C) in X F (C) as X F ,σ (C).
Let T := R ∪ {−∞} be the tropical semi-ring, equipped with the following arithmetic operations for any a, b ∈ T; We can also define the toric variety over T as follows. For each cone σ ∈ F, we define U σ (T) as the set of monoid homomorphisms σ ∨ ∩ M → (T, ), with the compact open topology. For cones σ, τ ∈ F such that σ ≺ τ , we have a natural immersion, where σ ≺ τ means that σ is a face of τ . By gluing {U σ (T)} σ∈F with each other, we have the tropical toric variety X F (T) associated with F, Tropical toric varieties are first introduced by Kajiwara [5], see [5] or [6] for details. For a projective toric variety, the associated tropical toric variety is homeomorphic to the moment polytope of it [6, Remark 1.3].
Example 2.1. The tropical projective space of n-dimension is homeomorphic to the n-dimensional simplex.
We define the torus orbit O σ (T) over T corresponding to σ by and write the closure of O σ (T) in X F (T) as X F ,σ (T). Let R ∈ R >0 be a positive real number and Log R : C → T denote the map defined by We have a canonical map Log R :
We write µ ≺ ρ when µ is a face of ρ.  Let F be a complete and unimodular fan in N R in the following.
is a face of ρ ∩ U σ (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 X F (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 .
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 X F (T). For each σ ∈ F, we define where relint(ρ) denotes the relative interior of ρ.

Hypersurfaces in toric varieties
Let K := C{t} be the convergent Laurent series field, equipped with the standard non-archimedean valuation (1.1). Let further ∆ ⊂ M R be a convex lattice polytope. We set A : . , x ± n+1 be a Laurent polynomial over K in n + 1 variables such that k m = 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 ) : The tropical hypersurface V (trop(F )) has a structure of a polyhedral complex in X F (T). Let P denote the polyhedral complex given by V (trop(F )) in the following.
We set v m := val(k m ) for m ∈ A. For each µ ∈ P {0} , we define the subset A µ ⊂ A as the set of elements of A to which the dominant terms of F at µ corresponds: (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 A µ 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 k m for all m ∈ A, and set S 1 . , x ± n+1 be the Laurent polynomial obtained by substituting 1/q to t in F . We write the closure of Let σ ∈ F be an l-dimensional cone. For µ ∈ P σ , let µ ∈ P {0} be the cell such that µ = µ ∩ X F ,σ . We assume that the dimension of µ is k. Here, we have l ≤ k. We define standard coordinates on O σ (C) and O σ (T) with respect to µ as follows. First, we number all elements of A µ from 0 to n + 1 − k and write them as (m 0 , . . . , m n+1−k ). We set . ,x n+1−l ) and ( X 1 , . . . , X n+1−l ) which form coordinate systems on O σ (C) and O σ (T) respectively by setting for i = n + 2 − k, . . . , n + 1 − l. Here, numbers a i and b ij are appropriate integral numbers. We call (x 1 , . . . ,x n+1−l ) and ( X 1 , . . . , X n+1−l ) standard coordinates with respect to µ. There are some ambiguities of them resulting from different numbering of (m 0 , . . . , m n+1−k ) and different choices of numbers a i and b ij .
Then the following diagram is commutative.
where the map Log R : Example 2.10. Let us consider the polynomial Hence, the sets of functions (z 1 , z 2 ) and (Z 1 , Z 2 ) form standard coordinates with respect to µ on O {0} (C) and O {0} (T), respectively.

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 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 nonarchimedean valuation (1.1). Let further ∆ ⊂ M R be a convex lattice polytope. We set A : We fix a sufficiently large R ∈ R >0 such that 1/R is smaller than the radius of convergence of k m for all m ∈ A, and set S 1 The graph of the function b is shown in Fig. 5.
We define the tropical localization of the hypersurface V q as follows.
We call the submanifold defined as the zero locus off the tropically localized hyperplane.
2) and the overlines mean the closure in X F (C) and X F (T), respectively.
The tropical hypersurface V (trop(F )) and the regions { D µ } µ∈P for F are shown in Figs. 6 and 7. ν i and µ i (i = 1, . . . , 6) denote vertices and edges of V (trop(F )) respectively as shown in Fig. 6. Each D ν i is the region colored in dark gray and each D µ i is the region colored in light gray as shown in Fig. 7.
If C 0 is sufficiently small, points in ρ which are sufficiently far from all faces of ρ in P {0} are contained in D ρ . It follows that points in µ which are sufficiently far from all faces of µ are contained in D ρ , and hence in D µ . Conversely, assume that ρ ∩ X F ,σ (T) = ∅. Since the region D ρ has to be near to the cell ρ if C 0 is sufficiently small, we have D ρ ∩ X F ,σ (T) = ∅. Lemma 3.6. If C 0 is sufficiently small, then one has 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 D ρ (ρ ∈ P {0} ). There exists the unique cone σ ∈ F such that x ∈ O σ (C). Then, the point x is contained in where the overline means the closure in X F (C).  Lemma 3.8. Let σ ∈ F be a cone and µ 1 , µ 2 ∈ P σ be cells. Suppose that the constant C 0 is sufficiently small. If D µ 1 ∩ D µ 2 = ∅, then there exists µ ∈ P σ such that µ ≺ µ 1 , µ 2 and Proof . Let µ 1 , µ 2 ∈ P {0} be the cells such that µ 1 = µ 1 ∩ X F ,σ (T) and µ 2 = µ 2 ∩ X F ,σ (T). We set {m 0 , . . . , m p } : The aim of this section is to prove the following theorem. Theorem 3.9. Fix a sufficiently small constant C 0 . For a sufficiently large R ∈ R >0 , the tropical localization W q and the family of subsets {D µ } µ∈P of X F (C) satisfy the following conditions: 1. For any q ∈ S 1 R , the submanifold W q is isotopic to V q in X F (C). 2. For any q ∈ S 1 R , one has W q ⊂ ρ∈P {0} D ρ . 3. Let σ ∈ F be a cone and µ ∈ P σ be a cell. Let further µ ∈ P {0} be the cell such that µ = µ ∩ X F ,σ (T). Assume that the dimension of σ and µ is l and k, respectively (l ≤ k).
Let (x 1 , . . . ,x n+1−l ) be a standard coordinate with respect to µ (see Section 2.3). Then, the defining equation of W q on D µ ∩ O σ (C) coincides with that of the (n − k)-dimensional where the branch of c First, we check that V q,s is contained in ρ∈P {0} D ρ for any q ∈ S 1 R and s ∈ [0, 1]. Then, we set q = R exp( √ −1θ) and consider the projection p : X F (C) × (− , 2π + ) × (0, 1) → (− , 2π + ) × (0, 1) given by where ∈ R is a small constant such that 0 < 1. Let Y be the subset of X F (C) × (− , 2π + ) × (0, 1) defined by We use the following theorem. We check that the functionf q,s has 0 as a regular value on each D µ ∩ O σ (C) for any q ∈ S 1 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 {V q,s } s∈[0,1] gives an isotopy between V q and W q . The condition 3 can be shown by a simple calculation. Hence, the functionf q,s can be written on where h p ∈ C, i p ∈ Z, j p ∈ A and each term h p q ip x jp denotes other monomial which is not dominant on D m , i.e., |q ip x jp |/|q vm x m | ≤ R −C 0 . (Each index p satisfies that either j p = m or j p = m and i p < v m .) Hence, for sufficiently large R, the functionf q,s can not be 0 on D m ∩ O {0} (C). Then we have V q,s ⊂ T for all q ∈ S 1 R and s ∈ [0, 1]. In particular, the condition 2 holds.
In addition, for any compact subset C ⊂ (− , 2π+ )×(0, 1), the inverse image (p| Y ) −1 (C) ⊂ Y coincides with {(x, θ, s) ∈ X F (C) × C |f q,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 V R,1 = W q=R =: W R . Therefore, the family of submanifolds {V q,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 This coincides with the defining function of the (n − k)-dimensional tropically localized hyperplane in (x 1 , . . . ,x n+1−k ) and the left-hand side is independent of the values ofx n+2−k , . . . , x n+1−l . Hence, the condition 3 holds.

Monodromy transformations
We use the same notation as in Section 3 and keep the assumption that V (trop(F )) is smooth. We set W R := W q=R . Let {ψ q=R exp( Moreover, such maps are unique up to homotopy. Proof . For each cell ρ ∈ P , we construct a continuous map φ ρ : Log R (W R ) ∩ D ρ → V (trop(F )) satisfying following conditions: We construct φ ρ in an ascending order of dim ρ as follows. For each vertex ρ ∈ P , we set φ ρ as a constant map from Log R (W R ) ∩ D ρ to ρ. For each 1-cell ρ, let ν 0 and ν 1 be the endpoints of ρ. We set each φ ρ as a continuous map to ρ so that φ ρ coincides with the constant map to ν i on Log R (W R ) ∩ D ρ ∩ D ν i and satisfies the condition (i). Assume that we have constructed φ ρ for all cells whose dimensions are lower than k − 1. For each k-cell ρ, we define φ ρ as a continuous map to ρ so that φ ρ coincides with φ µ on Log R (W R ) ∩ D ρ ∩ D µ for any face µ of ρ and satisfies the condition (i). In this way, we can construct a family of maps {φ ρ } ρ∈P such that each map φ ρ satisfies the condition (i) and (ii).
Example 4.2. Consider the polynomial F = 1 + x 1 + x 2 . Fig. 9 shows V (trop(F )) and Log R (W R ). Let ν denote the center vertex of V (trop(F )). The region colored gray denotes is the constant map to ν as shown in Fig. 10.
The following is the main theorem of this paper. where (y 1 , . . . , y n+1−l ) is a coordinate system on O σ (C) defined as in (4.1) and In this section, we show that Corollary 1.1 follows from Theorem 4.5. We set n = 1. Let φ : Log R (W R ) → 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 Log R (W R ) ∩ D ρ 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 (x 1 ,x 2 ) and ( X 1 , X 2 ) be standard coordinates with respect to ν (see Section 2.3). On D ν , the tropical localization W q is defined by the defining equation of the 1-dimensional tropically localized hyperplane in (x 1 ,x 2 ). Since we have X 1 (ν) = X 2 (ν) = 0 and the restriction of φ to Log R (W R ) ∩ D ν is the constant map to ν, the monodromy transformation ψ in Theorem 4.5 coincides with the identity map on D ν . Similarly, it turns out that the map ψ also coincides with the identity map on D ν 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 ν 1 , ν 2 be the endpoints of µ. We set {m 0 , m 1 , m 2 } ⊂ A so that {m 0 , m 1 , m 2 } = A ν 1 and {m 0 , m 1 } = A µ , where A ν 1 and A µ are subsets of A defined in (2.2). We define the standard coordinate with respect to ν 1 bỹ for i = 1, 2. Then the coordinate systems (x 1 ,x 2 ) and ( X 1 , X 2 ) are also standard coordinates with respect to µ. On D µ , the defining equation of the tropical localization W q coincides with These equations have no solution when R is sufficiently large. In the case −C 1 ≤ log R |x 1 | ≤ C 1 , (5.1) coincides with 1 +x 1 = 0.
Hence, the tropical localization W q coincides with the cylinder defined byx 1 = −1 and x 2 are free on D µ . Let l ∈ Z >0 be the length of µ. In the coordinate system (x 1 ,x 2 ), we have X 1 (ν 1 ) = X 2 (ν 1 ) = 0 and X 1 (ν 2 ) = 0, X 2 (ν 2 ) = −l. Note that the lengths of edges are invariant under the coordinate transformations. Since the restriction of φ to Log R (W R )∩ D µ 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 D µ . Similarly, it turns out that the restriction of ψ to D µ coincides with the compositions of infinitely many times of Dehn twists for any unbounded edge µ ∈ P {0} .
Let ∆ be a smooth and reflexive polytope in M R 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 f q defined by where q ∈ S 1 R := {z ∈ C | |z| = R} for a sufficiently large R ∈ R >0 . Let X ∆ be the toric manifold whose moment polytope is ∆ and V q be the hypersurface in X ∆ defined by f q . In this setting, Zharkov constructed the monodromy transformation of {V q } q∈S 1 R as follows: (i) Let µ R : X ∆ → ∆ be the weighted moment map defined by There exists a small neighborhood U ⊂ ∆ of the origin 0 ∈ ∆ such that µ R (V q ) ⊂ ∆ \ U for any q ∈ S 1 R . We set ∆ • := ∆ \ U . He constructs two families of regions {U τ } τ ∈∂T and { U τ } τ ∈∂T in ∆ • . For instance, in the case where f q := q − x + xy + y + x −1 + x −1 y −1 + y −1 (7.1) and the triangulation T is given as shown in Fig. 15, the families of regions {U τ } ρ∈∂T and { U τ } τ ∈∂T are as shown in Figs. 16 and 17, respectively. v i and w i (i = 1, . . . , 6) denote vertices and edges of ∆ respectively as shown in Fig. 15. U v i ,Ũ v i denote the regions colored in light gray and U w i ,Ũ w i 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.   for any τ ∈ ∂T . Let W q denote the submanifold in X ∆ defined byf q (x) = 0. We can see from the definition of the weighted moment map µ R that if µ R (x) ∈ U τ , the dominant part of f q at x are q λ(0) − m∈τ ∩B q λ(m) x m . Since orders of terms cut off by bump functions {b m } m∈B\{0} are lower, the submanifold W q is diffeomorphic to V q .
For any γ > 0, the set ∆ ∨ γ is a convex polytope with a nonempty interior. The set ∆ ∨ γ in the case where f q is given by (7.1) is shown in Fig. 18.
The region surrounded by the center part of the tropical hypersurface coincides with n ∈ N R | λ(0) ≥ m, n + λ(m) for any vertex m in T .
(iv) Let e i := (0, . . . , 0, ǐ 1, 0, . . . , 0) ∈ M (i = 1, . . . , n + 1) be the unit vector and ψ i,γ : X ∆ → C (i = 1, . . . , n + 1) be the function defined by  As explained in (ii), Zharkov also localized the hypersurface V q to construct the monodromy transformation. He used the weighted moment map while we used the tropicalization. The regions { U τ } τ are similar to { D µ } 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 { D µ } µ are shown in Fig. 19 in the case where f q is given by (7.1). The region U v i corresponds to D µ i and U w i corresponds to D ν i (i = 1, . . . , 6), respectively. For instance, on both U v 2 and D µ 2 , the dominant terms are q and x. On both U w 1 and D ν 1 , the dominant terms are q, x, xy, and so on. Note that regions at which the term q λ(0) is not dominant in our construction are included in other regions in Zharkov's construction. For instance, in the case f q is given by (7.1), the region corresponding to D ρ i is included in U w i for i = 1, . . . , 6. This is the only major differences in the localization and the resulting manifolds W q are similar to each other.