Bimeromorphic invariance of $\partial\bar\partial$-property and blow-up formulae I: counterexamples and derived category approach
Year: 2026
Type: preprint
Abstract: Using N. Honda's construction of twistor spaces, bimeromorphic invarince of $\partial\bar\partial$-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the $\partial\bar\partial$-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 $\partial\bar\partial$-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.
Source: arXiv (Cornell University)
Authors: Liu, Wei, Rao, Sheng
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G500947340
OpenAlex ID: W7213995647