Bogomolov-Guan Manifolds are not formal
Abstract
Bogomolov--Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov--Guan manifolds of complex dimension at least four are nonformal over ℚ. The proof constructs a uniform degree-eight a-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov--Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an Sn+1-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov--Guan manifold. We also compare the Bogomolov--Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural Sn+1-equivariant finite cover of degree (n+1)2 and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov--Guan manifold of complex dimension at least four is equal to $3$.
Related
- Hodge theory and topology of Bogomolov--Guan manifolds
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BibTeX
@misc{ortiz2026bogomolovguan,
title = {{Bogomolov-Guan Manifolds are not formal}},
author = {Ángel David Ríos Ortiz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26768},
eprint = {2609.26768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26768},
}
RIS
TY - UNPB
TI - Bogomolov-Guan Manifolds are not formal
AU - Ángel David Ríos Ortiz
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26768
UR - https://arxiv.org/abs/2609.26768
AB - Bogomolov--Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov--Guan manifolds of complex dimension at least four are nonformal over ℚ. The proof constructs a uniform degree-eight a-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov--Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an S_n+1-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov--Guan manifold. We also compare the Bogomolov--Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural S_n+1-equivariant finite cover of degree (n+1)^2 and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov--Guan manifold of complex dimension at least four is equal to $3$.
ER -