Tailoring a Pair of Pants: The Phase Tropical Version
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
TAILORING A PAIR OF PANTS: THE PHASE TROPICAL VERSION I. Zharkov Kansas State University 138, Cardwell Hall, Manhattan 66506, USA [email protected]
UDC 515.148
We show that the phase tropical pair-of-pants P ◦ ⊂ (C∗ )n is (ambient) isotopic to the complex pair-of-pants P ◦ ⊂ (C∗ )n . The existence of an isotopy between the complex and ober-tropical pairs-of-pants was recently established by the author jointly with H. Ruddat. Thereby all the three (complex, phase tropical, and ober-tropical) versions are isotopic. Bibliography: 10 titles. Illustrations: 8 figures.
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Introduction
The (n − 1)-dimensional (complex) pair-of-pants P ◦ ⊂ (C∗ )n is the main building block for many problems in complex and symplectic geometries. Its projection under the log |z| map is called the amoeba, and its projection under the argument map is called the coamoeba. The phase tropical pair-of-pants P ◦ ⊂ (C∗ )n is the fibration over the tropical hyperplane with fibers over tropical strata given by the corresponding coamoebas. It is natural to consider closed spaces, so we compactify (C∗ )n to Δ × Tn , where Δ is the standard n-simplex and Tn is the n-torus. The main result of the paper is Theorem 3.1 which states that the closures P and P of the two verstions of the pair-of-pants in Δ × Tn are (ambient) isotopic (in the PL category). Instead of trying to build an isotopy explicitly, we build regular cell decompositions of both pairs and show that they are homeomorphic. The cell structures respect the natural stratification of Δ × Tn , so the homeomorphisms will glue well at the boundary. Thus, with a tiny bit of effort the isotopy can be extended to any general affine hypersurface by using the pair-of-pants decomposition of [1] and [2]. The proof follows the same path as for the ober-tropical case in [3]. Namely, the main ingredient is to show that, in the CW-decomposition, the pair ((C∗ )n , P) restricted to each cell is the standard ball pair. This involves two statements: local flatness (Lemma 3.1) and the complement homotopic to a circle (Lemma 3.2). The main application of the isotopy is to address the following question in mirror symmetry. Given an integral affine manifold with singularities, we want to build a topological SYZ fibration [4] with discriminant in codimension 2 (rather than codimension one) (cf. [5] for the quintic Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 101-109. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0300
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3-fold case and an announcement [6]). To compare with the ober-tropical version, the phase tropical one has the advantage that no unwiggling is required when gluing local models. On the other hand, the singular fibers are not equi-dimensional. For example, in dimension 3, the negative vertex fiber looks like S 1 -fibration over T2 , where the circle collapses to a point over two triangles (the coamoeba), rather than over its skeleton (the Θ-graph in the ober-tropical case).
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Δ◦
CW-Structure of Complex an
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