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We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta \in (0,1)$ and a complex cell $\mathcal{C}$, we define its holomorphic extension $\mathcal{C}\subset \mathcal{C}^\delta $, which is again a complex cell. The hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^\delta $ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved.
@article{10_4007_annals_2019_190_1_3,
author = {Gal Binyamini and Dmitry Novikov},
title = {Complex cellular structures},
journal = {Annals of mathematics},
pages = {145--248},
publisher = {mathdoc},
volume = {190},
number = {1},
year = {2019},
doi = {10.4007/annals.2019.190.1.3},
mrnumber = {3990603},
zbl = {07097498},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.1.3/}
}
TY - JOUR AU - Gal Binyamini AU - Dmitry Novikov TI - Complex cellular structures JO - Annals of mathematics PY - 2019 SP - 145 EP - 248 VL - 190 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.1.3/ DO - 10.4007/annals.2019.190.1.3 LA - en ID - 10_4007_annals_2019_190_1_3 ER -
Gal Binyamini; Dmitry Novikov. Complex cellular structures. Annals of mathematics, Tome 190 (2019) no. 1, pp. 145-248. doi: 10.4007/annals.2019.190.1.3
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