The topology of the amoeba map and maximal volume

Abstract

For a smooth complete intersection V⊂(C^*)ⁿ of dimension k, n≥2k, we bound the volume of its amoeba by π^2k times a sum of mixed volumes, divided by the order of any finite group Γ of symmetries commuting with the logarithmic map. For n=2k≥4 almost every fibre of the amoeba map has a number of points divisible by 2|Γ|. For n=2k we characterise the varieties of maximal volume, show that for k≥2 they are not real, and prove that no plane in general position in (C^*)⁴ has maximal volume. For n>2k the bound is never attained, varieties of maximal volume have amoeba maps injective off a null set, and injectivity is generic. The amoeba map of almost every real k-plane is two-to-one, and that of the k-th symmetric power of a real line exactly 2^k-to-one; conversely, real k-planes in general position in (C^*)^2k with 2^k-to-one amoeba maps are symmetric powers. We also classify the real planes of (C^*)⁴ and study lines in (C^*)³.

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BibTeX RIS

BibTeX

@misc{nisse2026topology,
  title = {{The topology of the amoeba map and maximal volume}},
  author = {Mounir Nisse},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02496},
  eprint = {2611.02496},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02496},
}

RIS

TY  - UNPB
TI  - The topology of the amoeba map and maximal volume
AU  - Mounir Nisse
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02496
UR  - https://arxiv.org/abs/2611.02496
AB  - For a smooth complete intersection V⊂(C^*)^n of dimension k, n≥2k, we bound the volume of its amoeba by π^2k times a sum of mixed volumes, divided by the order of any finite group Γ of symmetries commuting with the logarithmic map. For n=2k≥4 almost every fibre of the amoeba map has a number of points divisible by 2|Γ|. For n=2k we characterise the varieties of maximal volume, show that for k≥2 they are not real, and prove that no plane in general position in (C^*)^4 has maximal volume. For n>2k the bound is never attained, varieties of maximal volume have amoeba maps injective off a null set, and injectivity is generic. The amoeba map of almost every real k-plane is two-to-one, and that of the k-th symmetric power of a real line exactly 2^k-to-one; conversely, real k-planes in general position in (C^*)^2k with 2^k-to-one amoeba maps are symmetric powers. We also classify the real planes of (C^*)^4 and study lines in (C^*)^3.
ER  -

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