The phase limit set of a variety. (English) Zbl 1277.14048

From the introduction: A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. Coamoebae are cousins to amoebae, which are images of subvarieties under the coordinatewise logarithm map \(z \mapsto \log |z|\). Amoebae were introduced by [I. M. Gel’fand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser. (1994; Zbl 0827.14036)] and have subsequently been widely studied [R. Kenyon et al., Ann. Math. (2), 163, No. 3, 1019–1056 (2006; Zbl 1154.82007); G. Mikhalkin, Ann. Math. (2) 151, No. 1, 309–326 (2000; Zbl 1073.14555); M. Passare and H. Rullgård, Duke Math. J. 121, No. 3, 481–507 (2004; Zbl 1043.32001)]. Coamoebae were introduced by Passare in a talk in 2004, and they appear to have many beautiful and interesting properties. For example, coamoebae of \(\mathcal A\)-discriminants in dimension 2 are unions of two nonconvex polyhedra [L. Nilsson and M. Passare, J. Commut. Algebra 2, No. 4, 447–471 (2010; Zbl 1237.14062)], and a hypersurface coamoeba has an associated arrangement of codimension-1 tori contained in its closure [M. Nisse, “Geometric and combinatorial structure of hypersurface coamoebas”, arxiv:0906.2729].
The result is illustrated and motivated by detailed examples of lines in three-dimensional space.


14T05 Tropical geometry (MSC2010)
32A60 Zero sets of holomorphic functions of several complex variables
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