Algebraic Dimension and Reduction of LVMB manifolds

Abstract

LVMB manifolds are a large class of compact complex non-Kähler and non-algebraic manifolds, constructed from an algebro-combinatorial datum. We provide an upper bound on their algebraic dimension and prove it to be sharp. Comparing our result with the lower bound found by Meersseman, we obtain the exact algebraic dimension of an LVMB manifold in many relevant cases. Pursuing our analysis further, we also obtain a description of the algebraic reduction of an LVMB manifold. From this, it emerges that Meersseman's lower bound coincides with the dimension of the rational component. This bimeromorphic invariant of an LVMB manifold is then computed in terms of the nonrationality degree of the associated toric quasifold. This is an invariant recently introduced by Battaglia and Prato, for which we provide an explicit formula.

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BibTeX RIS

BibTeX

@misc{thiella2026algebraic,
  title = {{Algebraic Dimension and Reduction of LVMB manifolds}},
  author = {Federico Thiella},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35605},
  eprint = {2609.35605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35605},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Algebraic Dimension and Reduction of LVMB manifolds
AU  - Federico Thiella
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35605
UR  - https://arxiv.org/abs/2609.35605
AB  - LVMB manifolds are a large class of compact complex non-Kähler and non-algebraic manifolds, constructed from an algebro-combinatorial datum. We provide an upper bound on their algebraic dimension and prove it to be sharp. Comparing our result with the lower bound found by Meersseman, we obtain the exact algebraic dimension of an LVMB manifold in many relevant cases. Pursuing our analysis further, we also obtain a description of the algebraic reduction of an LVMB manifold. From this, it emerges that Meersseman's lower bound coincides with the dimension of the rational component. This bimeromorphic invariant of an LVMB manifold is then computed in terms of the nonrationality degree of the associated toric quasifold. This is an invariant recently introduced by Battaglia and Prato, for which we provide an explicit formula.
ER  -

Abstract

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