Bimeromorphic invariance of $\partial\bar\partial$-property and blow-up formulae I: counterexamples and derived category approach

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. Th