Polarization complete left invariant connections
Abstract
Let (M,∇) be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood N of the zero section in TM is a complex structure on N such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization P on N is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to P. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply 𝒫-complete) if the ac-structure (resp. the ac-polarization) exists on TM. Although on a small enough N an ac-structure always exists, entire connections are rear and if a maximal domain of definition Nmax of an ac-structure exists (different from TM), Nmax is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole TM. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are 𝒫-complete.
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BibTeX
@misc{forman2026polarization,
title = {{Polarization complete left invariant connections}},
author = {Balázs Forman and Róbert Szőke},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15673},
eprint = {2609.15673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15673},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Polarization complete left invariant connections
AU - Balázs Forman
AU - Róbert Szőke
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15673
UR - https://arxiv.org/abs/2609.15673
AB - Let (M,∇) be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood N of the zero section in TM is a complex structure on N such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization P on N is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to P. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply 𝒫-complete) if the ac-structure (resp. the ac-polarization) exists on TM. Although on a small enough N an ac-structure always exists, entire connections are rear and if a maximal domain of definition N_max of an ac-structure exists (different from TM), N_max is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole TM. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are 𝒫-complete.
ER -