On the Volume of Complex Amoebas

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.


Introduction
Amoebas have proven to be a very useful tool in several areas of mathematics, and they have many applications in real algebraic geometry, complex analysis, mirror symmetry, algebraic statistics and in several other areas (see [M1-02], [M2-04], [M3-00], , , ,  and [R-01]). They degenerate to piecewise-linear objects called tropical varieties (see [M1-02], [M2-04], and ). Moreover, we can use amoebas as an intermediate link between the classical and the tropical geometry.
The amoeba A of an algebraic variety V ⊂ (C * ) n is a closed subset of R n , and its (Lebesgue) volume is well-defined.  proved that the area of complex plane curve amoebas is finite and the bound is given in terms of the Newton polygon. In this paper, we prove that the amoeba area of any algebraic curve in (C * ) n is finite (the area here is with respect to the induced Euclidean metric of R n ). Moreover, we generalize our result, for any algebraic variety V of dimension k in the complex algebraic torus (C * ) 2k+m with m ≥ 0. Our main result is the following theorem: Theorem 1.1. Let V be an algebraic variety of dimension k in the complex torus (C * ) 2k+m with m ≥ 0, such that no irreducible component of V is contained in a subtorus of dimension less than 2k. Then, the volume of its amoeba is finite.
Note that if V contains an irreducible component in an algebraic subtorus of dimension strictly less than 2k, then the volume of its amoeba is infinite.
The remainder of this paper is organized as follows. In Section 2, we review some properties of the amoebas and also the structure theorem of the logarithmic limit set defined by Bergman [B-71], which is used as a tool in the proof of our results. In Section 3, we prove our main result for complex algebraic curves in (C * ) n for any n ≥ 2. In Section 4, we give an estimate of the bound for algebraic rational complex curves and some examples. In Section 5, we prove the main theorem of this paper.
Passare and Rullgård prove the following (see ): Theorem 2.1. Let f be a Laurent polynomial in two variables. Then the area of the amoeba of the curve with defining polynomial f is not greater than π 2 times the area of the Newton polytope of f .
In [MR-00], Mikhalkin and Rullgård show that up to multiplication by a constant in (C * ) 2 , the algebraic plane curves with Newton polygon ∆ with maximal amoeba area are defined over R. Furthermore, their real loci are isotopic to so-called Harnack curves (possibly singular with ordinary real isolated double points).
Recall that the Hausdorff distance between two closed subsets A, B of a metric space (E, d) is defined by where we take E = R n × (S 1 ) n , with the distance defined as the product of the Euclidean metric on R n and the flat metric on (S 1 ) n .
The logarithmic limit set of a complex algebraic variety V , denoted by L ∞ (V ), is the set of limit points of A in the sphere S n−1 = (R n ) * /R + . In other words, if S n−1 denotes the boundary of the unit ball B n and r the map defined by r : then L ∞ (V ) = r(A ) ∩ S n−1 . Bergman [B-71] proved that if V ⊂ (C * ) n is an algebraic variety of dimension k, then the cone over L ∞ (V ) is contained in a finite union of k-dimensional subspaces of R n defined over Q. On the other hand,  proved that this cone is a finite union of rational polyhedron convex cones of dimension at most k, and the maximal dimension in this union is achieved by at least one polyhedron P in this union. Moreover, one has dim R L ∞ (V ) = dim C V − 1. More precisely, we have the following structure theorem: Theorem 2.2 (Bergman, Bieri-Groves). The logarithmic limit set L ∞ (V ) of an algebraic variety V in (C * ) n is a finite union of rational spherical polyhedrons. The maximal dimension of a polyhedron in this union is achieved at least by one polyhedron P in this union, and we have dim

Area of complex algebraic curve amoebas
The main result of this section is the following theorem: Theorem 3.1. Let C ⊂ (C * ) n be an algebraic curve with n ≥ 2. Assume that no irreducible component of C is contained in a subtorus of dimension less than 2. Then, the area of its amoeba is finite.
We start by proving Theorem 3.1 in the rational curve case (see Theorem 3.2). Recall that a complex algebraic curve is contained in a subtorus of dimension one, meaning that the curve is the subtorus of dimension one itself (sometimes called a holomorphic annulus). Moreover, its amoeba is a straight line in R n , and this case is not interesting for us because it is not generic (i.e., the Jacobian matrix of the logarithmic map is not of maximal rank).
Let n and k be two positive integers such that 2k ≤ n. Let f : C k −→ (C * ) n be a rational map, and V be the variety in (C * ) n defined by the image of f . We denote by {z j } 1≤j≤k the complex coordinates of C k , and by {f j } 1≤j≤n the coordinates of f in C n . For simplicity, we denote by Log f the composition Log • f .
be the volume forms defined over U f associated to the metrics (Log f ) * E n and E 2k respectively. These two forms are related by the following formula: (1) and (2) we deduce that and area := vol if k = 1. This definition of the volume does not depend on the choice of coordinates. It is more convenient to use the following integral vol 2k defined as In fact, we know that there exist a positive integer m and a family of open connected If we define p = min 1≤α≤m p α and P = max 1≤α≤m p α , then On C k , dv E 2k and ψ 2k are given by is an orthonormal basis of T * R n and e I := e i1 ∧ · · · ∧ e i 2k . This implies that is an orthonormal basis of ∧ 2k T * R n with respect to the Euclidean metric. We denote by ∂ j = ∂ ∂zj if j ≤ k and by ∂ j = ∂ ∂z j−k if j ≥ k + 1. Then Hence, we deduce the following inequality: We have the following result: Theorem 3.2. Let f : C −→ (C * ) n be a rational map, and C be the rational curve of (C * ) n defined by the image of f . The area of A f with respect to the Euclidean metric of R n is finite.
It is trivial that this function, which is i times the Jacobian determinant of Log f , is a real rational function. Its poles are the zeros and the poles of f and their order is equal to one (even if there is a common pole or zero of f 1 and f 2 , one can check that this pole is also simple). On the other hand, if i det(∂ z Log f, ∂zLog f ) = P Q , then degQ ≥ degP + 2. This inequality can be improved. In fact, by elementary computations on the degP and degQ, we show that deg Q ≥ deg P + 3.
Proof of Theorem 3.2. First of all, we do not have to worry about the area of S f . Indeed, by Sard's theorem, we know that this area is equal to zero. If A f \ S f is empty, then the area of A f is zero. From now on, we assume that A f \ S f is not empty. Hence, it is a surface defined in R n and A f \ S f , A f have the same area.
The area of A f \ S f is given by (3). Hence, it is sufficient to prove that ψ 2 is integrable over C. Inequality (9) implies that We claim that all the functions on the right-hand side of (10) are integrable. Indeed, using Lemma 3.1, (i), we have the integrability in a neighborhood of any pole. By (ii), we get the integrability at infinity.
Proof of Theorem 3.1. Let C be an algebraic curve in (C * ) n . Then its closure C in CP n is an algebraic curve. Hence, any end of C corresponds to a local branch of C at some point p ∈ ∂C \ C. After a monomial map of (C * ) n if necessary, we may assume that p corresponding to an end of C is the origin of C n in CP n .
A local parametrization ρ p of a branch of C at p can be written in terms of vectorial Puiseux series in t near zero as follows: where (b 1 , . . . , b n ) ∈ (C * ) n , and (u 1 , . . . , u n ) ∈ Q n ≥0 . Indeed, the Bergman logarithmic limit set of a curve is a finite number of points {v i } on the sphere S n−1 (see [B-71]). By Bieri and Groves (see ), if O denotes the origin of R n , then the slope u of the real line (Ov i ) in R n is rational (the slope here means the direction vector of the line, and rational means that its coordinates are rational). Hence, there exist real lines L u, j in R n parametrized by x −→ (a j 1 + xu 1 , . . . , a j n + xu n ) with x ∈ R and (a j 1 , . . . , a j n ) ∈ R n , such that the amoebas A h of h in the defining ideal of the curve C reaches all these lines at the infinity in the direction u. So, for each line L u, j there exists (b 1 , . . . , b n ) ∈ (C * ) n such that the Hausdorff distance between the complex line L u, j parametrized by t −→ (b 1 t u1 , . . . , b n t un ), and V h ∩ Log −1 (L u, j \ B(O, R)) + tends to zero when R goes to infinity, where (L u, j \ B(O, R)) + denotes the component which is in the direction of u. Hence, the Hausdorff distance between the intersection of the curve with Log −1 (L u, j \ B(O, R)) + and L u, j tends to zero for R sufficiently large. Now, using Theorem 3.2 and the fact that the number of ends of an algebraic curve is finite, we obtain the result.

An estimate for the area of rational curve amoebas
In this section, we assume that k = 1. Recall that f is the rational map defined in Section 3 with k = 1. Hence, for any integer j ∈ [1, n]: where a j are distinct poles or zeros of f j , m j are their multiplicities (m j are negative in the case of poles) and d j is the number of distinct zeros and poles of f j . We define the positive integers n j := dj =1 |m j |. These integers represent the number of poles and zeros of f j counted with their multiplicities.
Theorem 4.1. Let p be the positive integer defined by The following inequalities always hold: Proof. Recall that vol 2 is defined by (4). If n = 2, using (4), (8), we obtain with f a 1 ,a 2 (z) = (z − a 1 , z − a 2 ). We know that vol 2 (A fa 1 ,a 2 ) = π 2 (it can be proven, using the substitution z = (a 1 − a 2 )t + a 1 and Example 1 below for m = 1, which is a plane line). Hence, |m 1 m 2 |.
A combination of (12) and (13) yields which gives the second inequality of Theorem 4.1. The first one is a consequence of (7). Assume that p · area(A f ) = vol 2 (A f ). Using (5), (6), we obtain that p α = p, for all α ≤ m. This means that the number of connected components in (Log f ) −1 R α does not depend on α and Log f is a p−sheeted covering map. If we suppose that Log f : C k \ S −→ A f \ S f is a covering with exactly p sheets, then (7) becomes an equality.

Examples.
(1) Let us compute the area of the amoeba A f with f (z) = (z, z m − 1) and m ∈ Z * . The set of singular points S is the union of the half lines given by The set of critical values is S f = Log f (S), which is a curve in R 2 and bounds A f . The map Log f is a covering map with exactly 2|m| sheets and for any integer j ∈ [1, 2|m|] we have that Hence, Log f is a 2|m|−sheeted covering map. By (3) and (8) we obtain (2) Now, let us consider the real line in (C * ) 3 parametrized by g(z) = (z, z + 1 2 , z − 3 2 ). The amoeba A g is a surface in R 3 with boundary as we can see in Figure 1 (this fact is proven in ). The set of singular points is the line of real points, and Log g is a 2-sheeted covering map. It is complicated to compute the area of A g . However, using the estimate given in Theorem 4.1, we deduce that 2 area(A g ) = vol 2 (A g ) ≤ 3π 2 .
(3) Let h(z) = (z, z + 1, z − 2i) be the parametrization of a complex line in (C * ) 3 . The amoeba A h is a surface in R 3 without boundary as we can see in Figure    Passare and Rullgård give an estimate for amoeba areas of complex algebraic plane curves (see Theorem 2.1). Our estimate works only for rational curves immersed in (C * ) n . However, in this case we have a finer estimate (see Theorem 4.1). Indeed, if we consider Example 1, area(A f ) = π 2 2m , n 1 = 1, n 2 = m, p = 2m and the area of Newton's polygon is m 2 . Inequality (11) gives area(A f ) ≤ π 2 2 , and the Passare-Rullgård estimate gives area(A f ) ≤ mπ 2 2 .

Volume of generic complex algebraic variety amoebas
In this section, we assume that n = 2k + m is an integer with k ≥ 1 and m ≥ 0. Let V ⊂ (C * ) n be an algebraic variety of dimension k with defining ideal I(V ) and L ∞ (V ) be its logarithmic limit set. We denote by Vert(L ∞ (V )) the set of vertices of L ∞ (V ). Let V ⊂ (C * ) n be a generic algebraic variety of dimension k. Let v ∈ Vert(L ∞ (V )) and let us denote by H R, v the hyperplane in R n with normal the vector Ov such that d(O, H R, v ) = R, where O is the origin of R n and R ∈ R + is sufficiently large. We denote by H − R, v the half space with boundary H R, v containing the origin.
In this section, we prove Theorem 1.1, using the following proposition: Proposition 5.1. With the above notation, V \ Log −1 (H − R, v ) is a fibration over an algebraic variety V v of dimension k − 1 contained in (C * ) n−1 , and its fibers are the ends of algebraic curves in (C * ) n . Moreover, these ends have a rational parametrization with the same slope (i.e., their image under Log are lines with the same slope).
We start by proving the following lemma: Lemma 5.1. Let V ⊂ (C * ) n be an algebraic variety of dimension k. Then, for each vertex v of its logarithmic limit set L ∞ (V ), we have the following: there exists a complex algebraic variety V v ⊂ (C * ) n−1 of dimension k − 1 such that the boundary of the Zariski closure V of V in (C * ) n−1 × C is equal to V v (i.e., ∂V = V \ V = V v , where (C * ) n−1 = (C * ) n−1 × {0} ⊂ (C * ) n−1 × C).
where the coefficients b x, j depend on the holomorphic branch of C x at x, and the powers u j depend only on v. Moreover, for any x 1 = x 2 in V v the end of the curve C x1 corresponding to x 1 , and the end of the curve C x2 corresponding to x 2 have an empty intersection. It may be that the curves C x1 and C x2 are the same. In fact, in this case this means that the curve has more than one end corresponding to v. In other words, if C x1 is the Zariski closure of C x1 in CP n , then it has more than one holomorphic branch at v. Indeed, if the intersection of these ends is not empty, then from the fact that they are holomorphic and with the same slope, they should be equal. This is a contradiction with the assumption on x 1 and x 2 . Hence, for R sufficiently large, V \ H − v, R is a fibration over V v .
Proof of Theorem 1.1. Recall that the volume is always computed with respect to the induced measure of the ambient space. Using induction on the dimension k of the variety, Proposition 5.1 and Theorem 4.1, there exists a rational number q v depending only on v and on the variety V v such that the inequality vol(Log (V \ H − v, R )) ≤ π 2 q v vol(A (V v )) holds. There is a finite number of vertices of the logarithmic limit set, so there exists a positive real number K such that the following inequality holds: vol(A (V )) ≤ K + v∈Vert(L ∞ (V )) π 2 q v vol(A (V v )).