A Chern-Class Obstruction to Measurable Eigensections of Analytic Quasi-Periodic Bundle Cocycles
Abstract
We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle L, we prove that c1(L) ≠ 0 obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: L carries a nonzero measurable eigensection if and only if c1(L)=0 and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by L is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.
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BibTeX
@misc{azimifard2026chernclass,
title = {{A Chern-Class Obstruction to Measurable Eigensections of Analytic Quasi-Periodic Bundle Cocycles}},
author = {Ahmadreza Azimifard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29577},
eprint = {2609.29577},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29577},
}
RIS
TY - UNPB
TI - A Chern-Class Obstruction to Measurable Eigensections of Analytic Quasi-Periodic Bundle Cocycles
AU - Ahmadreza Azimifard
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29577
UR - https://arxiv.org/abs/2609.29577
AB - We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle L, we prove that c_1(L) ≠ 0 obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: L carries a nonzero measurable eigensection if and only if c_1(L)=0 and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by L is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.
ER -