Holomorphic Tischler fibration theorem for Stein spaces
Abstract
We study a holomorphic analogue of Tischler approximation for closed differential forms normalized by a prescribed holomorphic vector field. The main setting consists of connected normal Stein spaces satisfying a degree-one reflexive Poincaré property, with finitely generated first integral homology and a holomorphic ℂ*-action admitting a closed normalized reflexive one-form. Under these hypotheses, the given form can be approximated on compact subsets of the regular locus by closed normalized reflexive forms with discrete period groups. When the first Betti number is at least two, the approximating forms may be chosen with rank-two lattice periods. We relate these approximations to holomorphic locally trivial fibrations over quotients of ℂ, with connected Stein analytic fibers that may be singular. We also establish a finite-equivariant version for suitable Stein quotients and discuss the associated normalization obstructions, including those arising from local singularity topology. Further results concern closed holomorphic time forms, transverse foliations, and the relation between the period groups and the dynamics of complete holomorphic flows. Examples illustrate both the applicability and the limitations of the hypotheses.
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BibTeX
@misc{scardua2026holomorphic,
title = {{Holomorphic Tischler fibration theorem for Stein spaces}},
author = {Bruno Scardua},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34383},
eprint = {2610.34383},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34383},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Holomorphic Tischler fibration theorem for Stein spaces
AU - Bruno Scardua
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34383
UR - https://arxiv.org/abs/2610.34383
AB - We study a holomorphic analogue of Tischler approximation for closed differential forms normalized by a prescribed holomorphic vector field. The main setting consists of connected normal Stein spaces satisfying a degree-one reflexive Poincaré property, with finitely generated first integral homology and a holomorphic ℂ^*-action admitting a closed normalized reflexive one-form. Under these hypotheses, the given form can be approximated on compact subsets of the regular locus by closed normalized reflexive forms with discrete period groups. When the first Betti number is at least two, the approximating forms may be chosen with rank-two lattice periods. We relate these approximations to holomorphic locally trivial fibrations over quotients of ℂ, with connected Stein analytic fibers that may be singular. We also establish a finite-equivariant version for suitable Stein quotients and discuss the associated normalization obstructions, including those arising from local singularity topology. Further results concern closed holomorphic time forms, transverse foliations, and the relation between the period groups and the dynamics of complete holomorphic flows. Examples illustrate both the applicability and the limitations of the hypotheses.
ER -