Bimeromorphic invariance of ∂∂̅-property and blow-up formulae I: counterexamples and derived category approach

@misc{liu2026bimeromorphic,
  title = {{Bimeromorphic invariance of $\partial\bar\partial$-property and blow-up formulae I: counterexamples and derived category approach}},
  author = {Wei Liu and Sheng Rao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23469},
  eprint = {2609.23469},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23469},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Bimeromorphic invariance of ∂∂̅-property and blow-up formulae I: counterexamples and derived category approach
AU  - Wei Liu
AU  - Sheng Rao
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23469
UR  - https://arxiv.org/abs/2609.23469
AB  - Using N. Honda's construction of twistor spaces, bimeromorphic invarince of ∂∂̅-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the ∂∂̅-lemma that contains a smooth compact surface for which the lemma fails. This disproves heredity in the smallest possible ambient dimension and yields counterexamples to L. Alessandrini's modification question in every complex dimension at least four. We establish a derived blow-up formula for generalized Bott--Chern complexes with coefficients, compatible with the comparison maps to coefficient cohomology. We also obtain a Dolbeault blow-up formula with bounded derived coefficients. For compact complex manifolds, we introduce a relative ∂∂̅-property, characterize it using the Frölicher spectral sequence and Hodge filtrations, and prove a blow-up criterion. Finally, counterexamples show the limitations of the classical formulae for coherent coefficients and singular ambient spaces, and the failure of the naive Bott--Chern Künneth formula.
ER  -