PreprintarXiv licencearXiv
Complex curves in o-minimal geometry
Abstract
There has recently been considerable progress relating o-minimality to complex analytic geometry. Yet almost nothing is known about coherent cohomology or the classification of vector bundles, even for curves. In ℝan and similar structures, we show that cohomology of noncompact curves is concentrated entirely at punctures. As an application, we compute the cohomology of the structure sheaf on the affine line and describe a connection to Diophantine approximation. Finally, we use similar techniques to characterize which definable Riemann surfaces have definable compactifications. The proofs are based on a careful analysis of boundary behavior for definable holomorphic functions.
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BibTeX
@misc{dembner2026complex,
title = {{Complex curves in o-minimal geometry}},
author = {Spencer Dembner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08052},
eprint = {2607.08052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08052},
}
RIS
TY - UNPB
TI - Complex curves in o-minimal geometry
AU - Spencer Dembner
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2607.08052
UR - https://arxiv.org/abs/2607.08052
AB - There has recently been considerable progress relating o-minimality to complex analytic geometry. Yet almost nothing is known about coherent cohomology or the classification of vector bundles, even for curves. In ℝ_an and similar structures, we show that cohomology of noncompact curves is concentrated entirely at punctures. As an application, we compute the cohomology of the structure sheaf on the affine line and describe a connection to Diophantine approximation. Finally, we use similar techniques to characterize which definable Riemann surfaces have definable compactifications. The proofs are based on a careful analysis of boundary behavior for definable holomorphic functions.
ER -