Resolutions of compact balanced spaces
Abstract
Let X be a compact reduced balanced complex space of pure dimension n≥2. We prove that every resolution of singularities of X is a balanced manifold; in particular, X belongs to the class ℬ. This removes the additional cohomological assumptions appearing in earlier results of Fu-Meng-Xia. The main new ingredient is the linear independence of the Aeppli classes of the exceptional prime divisors of a projective bimeromorphic morphism onto a normal complex space. This follows from the vanishing of curve degrees for Aeppli-trivial divisors and the relative negativity lemma. For non-normal spaces, strict positivity is used to eliminate divisors mapping onto codimension-one subsets before passing to the normalization. We also recall a classical surface example of Grauert showing that the inclusion of compact balanced spaces in ℬ is strict.
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BibTeX
@misc{fu2026resolutions,
title = {{Resolutions of compact balanced spaces}},
author = {Jixiang Fu and Wei Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21728},
eprint = {2610.21728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21728},
}
RIS
TY - UNPB
TI - Resolutions of compact balanced spaces
AU - Jixiang Fu
AU - Wei Xia
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21728
UR - https://arxiv.org/abs/2610.21728
AB - Let X be a compact reduced balanced complex space of pure dimension n≥2. We prove that every resolution of singularities of X is a balanced manifold; in particular, X belongs to the class ℬ. This removes the additional cohomological assumptions appearing in earlier results of Fu-Meng-Xia. The main new ingredient is the linear independence of the Aeppli classes of the exceptional prime divisors of a projective bimeromorphic morphism onto a normal complex space. This follows from the vanishing of curve degrees for Aeppli-trivial divisors and the relative negativity lemma. For non-normal spaces, strict positivity is used to eliminate divisors mapping onto codimension-one subsets before passing to the normalization. We also recall a classical surface example of Grauert showing that the inclusion of compact balanced spaces in ℬ is strict.
ER -