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Harmonic tropical morphisms and approximation.

Lionel Lang1

  • 1Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.

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Summary
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Harmonic amoebas generalize amoebas in complex tori, extending tropical geometry. This study introduces harmonic morphisms from tropical curves, systematically described as limits of harmonic amoeba maps on Riemann surfaces.

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Area of Science:

  • Algebraic Geometry
  • Tropical Geometry
  • Complex Analysis

Background:

  • Harmonic amoebas generalize amoebas of algebraic curves in complex tori.
  • This work expands the scope of tropical geometry.
  • Previous results exist on approximating tropical curves in affine spaces.

Purpose of the Study:

  • Introduce harmonic morphisms from tropical curves to affine spaces.
  • Systematically describe these morphisms as limits of harmonic amoeba maps.
  • Provide a new perspective on Mikhalkin's approximation Theorem.

Main Methods:

  • Study of imaginary normalized differentials on families of punctured Riemann surfaces.
  • Analysis of limits of harmonic amoeba maps.
  • Generalization of amoebas to complex tori.

Main Results:

  • Introduction of harmonic morphisms from tropical curves.
  • Systematic description of harmonic morphisms as limits of harmonic amoeba maps.
  • Extension of results on approximating tropical curves in affine spaces.

Conclusions:

  • Harmonic morphisms offer a new viewpoint on tropical geometry and approximation theorems.
  • The study reveals connections between harmonic amoebas, Riemann surfaces, and moduli spaces.
  • This research bridges concepts from algebraic geometry, complex analysis, and tropical geometry.