## Geometry & Topology

### Phase tropical hypersurfaces

#### Abstract

We prove a conjecture of Viro (Tr. Mat. Inst. Steklova 273 (2011) 271–303) that a smooth complex hypersurface in $( ℂ ∗ ) n$ is homeomorphic to the corresponding phase tropical hypersurface.

#### Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3287-3320.

Dates
Revised: 9 February 2018
Accepted: 13 March 2018
First available in Project Euclid: 29 September 2018

https://projecteuclid.org/euclid.gt/1538186738

Digital Object Identifier
doi:10.2140/gt.2018.22.3287

Mathematical Reviews number (MathSciNet)
MR3858765

Zentralblatt MATH identifier
06945127

#### Citation

Kerr, Gabriel; Zharkov, Ilia. Phase tropical hypersurfaces. Geom. Topol. 22 (2018), no. 6, 3287--3320. doi:10.2140/gt.2018.22.3287. https://projecteuclid.org/euclid.gt/1538186738

#### References

• A Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984) 7–16
• R Forman, Morse theory for cell complexes, Adv. Math. 134 (1998) 90–145
• I M Gel'fand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, MA (1994)
• B Grünbaum, V P Sreedharan, An enumeration of simplicial $4$–polytopes with $8$ vertices, J. Combinatorial Theory 2 (1967) 437–465
• Y R Kim, M Nisse, Geometry and a natural symplectic structure of phase tropical hypersurfaces, preprint (2016)
• R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies 88, Princeton Univ. Press (1977)
• A T Lundell, S Weingram, The topology of CW complexes, Van Nostrand, New York (1969)
• D Maclagan, B Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics 161, Amer. Math. Soc., Providence, RI (2015)
• G Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004) 1035–1065
• M Nisse, F Sottile, Non-Archimedean coamoebae, from “Tropical and non-Archimedean geometry” (O Amini, M Baker, X Faber, editors), Contemp. Math. 605, Amer. Math. Soc., Providence, RI (2013) 73–91
• N Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011) 271–367
• O Viro, Gluing algebraic hypersurfaces, removing of singularities and constructions of curves, from “Tezisy Leningradskoj mezhdunarodnoj topologicheskoj konferencii” (1983) 149–197 In Russian; translated in “Patchworking real algebraic varieties”
• O Y Viro, On basic concepts of tropical geometry, Tr. Mat. Inst. Steklova 273 (2011) 271–303