The classification of taut surface singularities via representations from logarithmic vector fields
Abstract
We study logarithmic vector fields on exceptional cycles through finite-dimensional representations and recover Laufer's classification of taut normal complex surface singularities. For graphs with two nodes, restriction maps define representations of a fixed quiver, with dimensions bounded independently of the number of components and their self-intersections. We identify restriction cohomology with an extension group and use interval decomposition to reduce its vanishing to interpolation on a rational chain. These calculations determine the classification conditions, and vanishing on every analytic realization proves sufficiency, including the nonrational cases. The obstruction cycles also give a uniform criterion: a graph is taut if and only if H1(5E,SX|5E)=0 for every analytic realization, where SX is the ambient logarithmic tangent bundle. We prove that five is the smallest positive integer for which this criterion holds for all graphs.
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BibTeX
@misc{park2026classification,
title = {{The classification of taut surface singularities via representations from logarithmic vector fields}},
author = {Heesang Park and Dongsoo Shin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25654},
eprint = {2610.25654},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25654},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The classification of taut surface singularities via representations from logarithmic vector fields
AU - Heesang Park
AU - Dongsoo Shin
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25654
UR - https://arxiv.org/abs/2610.25654
AB - We study logarithmic vector fields on exceptional cycles through finite-dimensional representations and recover Laufer's classification of taut normal complex surface singularities. For graphs with two nodes, restriction maps define representations of a fixed quiver, with dimensions bounded independently of the number of components and their self-intersections. We identify restriction cohomology with an extension group and use interval decomposition to reduce its vanishing to interpolation on a rational chain. These calculations determine the classification conditions, and vanishing on every analytic realization proves sufficiency, including the nonrational cases. The obstruction cycles also give a uniform criterion: a graph is taut if and only if H^1(5E,S_X|_5E)=0 for every analytic realization, where S_X is the ambient logarithmic tangent bundle. We prove that five is the smallest positive integer for which this criterion holds for all graphs.
ER -