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Analytic Varieties with Finite Volume Amoebas Are Algebraic ANALYTIC VARIETIES WITH FINITE VOLUME AMOEBAS ARE ALGEBRAIC FARID MADANI AND MOUNIR NISSE Abstract. In this paper, we study the amoeba volume of a given k−dimensional generic analytic variety V of the complex algebraic torus ∗ n (C ) . When n ≥ 2k, we show that V is algebraic if and only if the vol- ume of its amoeba is finite. In this precise case, we establish a compari- son theorem for the volume of the amoeba and the coamoeba. Examples and applications to the k−linear spaces will be given. 1. Introduction Some fundamental questions concerning the complex logarithm lead us to study certain mathematical objects called amoebas and coamoebas, which are natural projections of complex varieties. They have a strong relations to several other areas of mathematics as real algebraic geometry, tropical geometry, complex analysis, mirror symmetry, algebraic statistics and in several other areas. Amoebas degenerate to a piecewise-linear object called tropical varieties, (see [M1-02], [M2-04], [M3-00], [FPT-00], [NS-09], [PR-04], and [PS-04]). The behavior of a coamoeba at infinity is the so called the phase limit set of the variety itself. For a k-dimensional complex algebraic variety, the phase limit set contains an arrangement of k-dimensional flat torus, which plays a crucial role on the geometry and the topology of both amoeba and coamoeba. Moreover, these objects are used as an intermediate link between the classical and the tropical geometry. In this paper, we underline that the geometry of the amoeba affects the algebraic structure of the variety. It was shown by Passare and Rullgård [PR-04] that the area of complex algebraic plane curve amoebas are finite. In [MN-11], we proved that the volume of the amoeba of a generic k-dimensional algebraic variety of the ∗ n complex algebraic torus (C ) with n ≥ 2k, is finite. There now arises the reciprocal question: Let I be an ideal in the ring of holomorphic function n on C , and V be the set of zeros of I which we assume generic of dimension arXiv:1108.1444v1 [math.AG] 6 Aug 2011 n ∗ n k ≤ 2 . If we assume that the volume of the amoeba of V = V \ (C ) is finite, then, is V algebraic? Theorem 1.1 gives an affirmative answer to this question. 1991 Mathematics Subject Classification. 14T05, 32A60. Key words and phrases. Analytic varieties, algebraic varieties, amoebas, coamoebas, logarithmic limit sets, phase limit sets, spherical polyhedrons. The first author was supported by Alexander von Humboldt Foundation and by the Mathematisches Forschungsinstitut Oberwolfach in 2011, through the program "Oberwol- fach Leibniz Fellows. Research of the second author is partially supported by NSF MCS grant DMS-0915245. 1 2 FARID MADANI AND MOUNIR NISSE The Main theorem of this paper is the following: ∗ n Theorem 1.1. Let V be a generic k-dimensional analytic variety in (C ) with n ≥ 2k. The following assertions are equivalent: (i) The variety V is algebraic; (ii) The volume of A (V ) is finite. This paper is organized as follows. In Section 4, using the geometry and the combinatorial structure of the logarithmic limit and the phase limit sets, we prove Theorem 1.1 in the special case of curves. The proof of this case is actually the crucial step to show the general one. In Section 5, we show Theorem 1.1 for varieties of higher dimensions. In Section 6, we give some examples of plane and spatial complex curves, underlying the importance of the finiteness of their amoeba areas. In Section 3, we prove a compar- ison theorem, asserting that up to rational number, the amoeba volume is bounded above and below by the coamoeba volume. Finally, in Section 7, we give an application of the comparison theorem to the k−dimensional affine ∗ 2k linear spaces in (C ) . We compute the amoeba volumes of k−dimensional ∗ 2k real affine linear spaces in (C ) . 2. Preliminaries n Let W be a complex variety in C defined by an ideal I of holomorphic n functions on C . We say that a subvariety V of the complex algebraic torus ∗ n (C ) is analytic if there exists a complex variety W as above such that ∗ n V := W \ (C ) . All the analytic varieties considered in this paper are defined as above. The amoeba A of V is by definition (see M. Gelfand, M.M. Kapranov and A.V. Zelevinsky [GKZ-94]) the image of V under the map : ∗ n n Log : (C ) −! R (z1; : : : ; zn) 7−! (log jz1j;:::; log jznj): It is well known that the amoeba of a variety of codimension one is closed and n its complement components in R are convex (see [FPT-00] ). A. Henriques gives an analogous definition for the convexity of amoeba complements of higher codimension varieties as follows: n n Definition 2.1. A subset A ⊂ R is called l-convex if for any l-plane L ⊂ R n the induced homomorphism Hl−1(L n A) −! Hl−1(R n A) is injective. Also, he proves that if V is a variety of codimension l and L is an l-plane of rational slope and c is a non-zero (l − 1)-cycle in Hl−1(L n A ) then its n n image in Hl−1(R n A ) is non-zero, and then R n A is l-convex (see [H-03] Theorem 4.1). Let V be an analytic variety as above. We denote by L 1(V ) its loga- rithmic limit set which is the boundary of the closure of r(A (V )) in the n−dimenstional ball Bn, where r is the map defined by (see Bergman [B-71]): n n r : R −! B x x 7−! r(x) = 1+jxj : If V is an algebraic variety of dimension k, then its logarithmic limit set is a finite rational spherical polyhedron of dimension k − 1 (i.e., a finite union ANALYTIC VARIETIES WITH FINITE VOLUME AMOEBAS ARE ALGEBRAIC 3 of finite intersections of closed hemispheres and can be described in terms of a finite number of inequalities with integral coefficients). More precisely, we have the following theorem structure (see [B-71] and [BG-84]): Theorem 2.1 (Bergman, Bieri-Groves). The logarithmic limit set L 1(V ) ∗ n of an algebraic variety V in (C ) is a finite union of rational spherical poly- hedrons. The maximal dimension of a polyhedron in this union is achieved 1 at least by one polyhedron P in this union, and we have dimR L (V ) = dimR P = (dimC V ) − 1. The argument map is the map defined as follows: ∗ n 1 n Arg : (C ) −! (S ) (z1; : : : ; zn) 7−! (arg(z1);:::; arg(zn)): zj where arg(zj) = . The coamoeba of V , denoted by coA , is its image jzj j under the argument map (defined for the first time by Passare on 2004). On 2009, Sottile and the second author [NS-09] define the phase limit set of V , P1(V ) , as the set of accumulation points of arguments of sequences in V with unbounded logarithm. If V is an algebraic variety of dimension k, P1(V ) contains an arrangement of k-dimensional real sub-torus. 3. Comparison between Amoeba and Coamoeba volumes For a given map f, we denote by Jac(f) the Jacobian matrix of f, and by J(f) the determinant of Jac(f) when it exists. ∗ n Proposition 3.1. Let V be a k-dimensional complex submanifold in (C ) . The maps Log and Arg are well defined on V and 1 ¯ −1 ¯ @Log = @Arg ; @Log = @Arg ; J(Log I ) = J(Arg I ) ; Arg Arg for any I ⊂ f1; : : : ; ng with cardinal 2k. n Proof. We denote by fzjg1≤j≤n the complex coordinates on C and by ftjg1≤j≤k the complex coordinates on V given by a local chart (Ω; f) (i.e. 8z 2 Ω; tj = fj(z)), where Ω is an open set of V and f is holomorphic from n k ∗ n an open set of C to C . Since V is a complex submanifold of (C ) , the ∗ n injection map { : V,! (C ) is holomorphic. By definition, for any z 2 V we have {(z) = eLogzArg z. Since { is holomorphic, @{¯ (z) = 0 for any z 2 V (i.e. 8j ≤ k; @t¯j {(z) = 0 ). It implies that for any j = 1; : : : ; k and z 2 Ω 1 1 @¯ Log (z) = − @¯ Arg (z);@ Log (z) = @ Arg (z): tj Arg (z) tj tj Arg (z) tj where the second equality holds by conjugating the first one. The statement of the proposition follows. ∗ n n Theorem 3.1. Let V be an analytic variety of (C ) of dimension k ≤ 2 . Let A , coA be the amoeba and coamoeba of V respectively. We suppose that Log : V ! A and Arg : V ! coA are locally finite coverings. We define two rational numbers min #Arg −1fθg max #Arg −1fθg θ2coA nArg S θ2coA nArg S p = ;P = : max #Log −1fyg min #Log −1fyg y2A nLog S y2A nLog S 4 FARID MADANI AND MOUNIR NISSE Then p vol(coA ) ≤ vol(A ) ≤ P vol(coA ): In particular the volume of A is finite. As an application of Theorem 3.1, we have the following result that the authors had already proven in [MN-11]: Corollary 3.1. The amoeba of a k-dimensional generic complex algebraic ∗ n variety in (C ) , with 2k ≤ n has a finite volume. ∗ n Proof. If V is a generic complex algebraic variety in (C ) of dimension n k ≤ 2 , then Log : V ! A and Arg : V ! coA are locally finite coverings.
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