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Oka complements in algebraic tori and their logarithmic covers

  • Yun-Heng Du
  • Bin Guo
  • Peng-Chao Wang
  • Song-Yan Xie
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Abstract

Suppose n≥2 and K⊂Cn is compact and polynomially convex. Write q(z)=(e−iz1,…,e−izn), XK=(C*)n∖(K∩(C*)n), and YK=Cn∖ q−1(K). We prove that XK and its logarithmic covering YK are Oka. More strongly, YK admits normalized holomorphic families of entire embeddings through prescribed holomorphic sections. The proof combines polynomial pair Hamiltonian transport in multiplicative coordinates, compatible with the coordinate hyperplanes, with a fixed Fatou--Bieberbach model. Applications include Oka slab crosses over arbitrary nonempty connected planar domains, complements of periodic equal-radius Euclidean balls centered at products of planar lattices, and complements of sufficiently small closed metric balls in products of elliptic curves.

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Cite

BibTeXRIS
BibTeX
@misc{du2026complements,
  title = {{Oka complements in algebraic tori and their logarithmic covers}},
  author = {Yun-Heng Du and Bin Guo and Peng-Chao Wang and Song-Yan Xie},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15433},
  eprint = {2610.15433},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15433},
}
RIS
TY  - UNPB
TI  - Oka complements in algebraic tori and their logarithmic covers
AU  - Yun-Heng Du
AU  - Bin Guo
AU  - Peng-Chao Wang
AU  - Song-Yan Xie
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.15433
UR  - https://arxiv.org/abs/2610.15433
AB  - Suppose n≥2 and K⊂C^n is compact and polynomially convex. Write q(z)=(e^−iz_1,…,e^−iz_n), X_K=(C^*)^n∖(K∩(C^*)^n), and Y_K=C^n∖ q^−1(K). We prove that X_K and its logarithmic covering Y_K are Oka. More strongly, Y_K admits normalized holomorphic families of entire embeddings through prescribed holomorphic sections. The proof combines polynomial pair Hamiltonian transport in multiplicative coordinates, compatible with the coordinate hyperplanes, with a fixed Fatou--Bieberbach model. Applications include Oka slab crosses over arbitrary nonempty connected planar domains, complements of periodic equal-radius Euclidean balls centered at products of planar lattices, and complements of sufficiently small closed metric balls in products of elliptic curves.
ER  -