Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms (Q2204091)

Within the area of non-Kähler geometry, the present work studies complex manifolds which satisfy the so-called \(\partial \bar\partial\)-Lemma (i.e., \(\text{ker}\,\partial\cap\text{ker}\,\bar\partial\cap\text{im}\, d= \text{im}\,\partial\bar\partial\)). The authors are interested in whether the \(\partial \bar\partial\)-Lemma property is a bimeromorphic invariant. As this property is equivalent to the existence of a natural Hodge structure, i.e., \[ H^{p,q}_{\bar\partial}(X)\simeq \overline{H^{q,p}_{\bar\partial}(X)}\ \hbox{and}\ H^{h}_{dR}(X)\simeq\bigoplus_{p+q=h}H^{p,q}_{\bar\partial}(X), \] the first main result of the present work gives a partial answer to the problem: Let X be a compact complex manifold and \(Z\subset X\) a submanifold of codimension \(k\). Let \(\tilde X\) denote the blow up of \(X\) along \(Z\). Then, there is an isomorphism \[ H^{h}_{dR}(X)\oplus\bigoplus_{i=0}^{k-2}H^{h-2i-2}_{dR}(Z)\overset\sim\longrightarrow H^{h}_{dR}(\tilde X). \] If \(Z\) has a holomorphically contractible neighbourhood and some other technical conditions are satisfied, there is also a similar isomorphism for the Dolbeault cohomology groups: \[ H^{p,q}_{\bar\partial}(X)\oplus\bigoplus_{i=0}^{k-2}H^{p-i-1,q-i-1}_{\bar\partial}(Z)\overset\sim\longrightarrow H^{p,q}_{\bar\partial}(\tilde X). \] In particular, if \(X\) and \(Z\) admit a Hodge structure, then \(\tilde X\) does. Explicit computations with Čech cohomology groups make this result interesting despite that the mentioned problem has been solved by \textit{J. Stelzig} with different techniques, see Corollary 28 in [``On the structure of double complexes'', J. London Math. Soc. (to appear); \url{doi: 10.1112/jlms.12453}]: the \(\partial \bar\partial\)-Lemma property is a bimeromorphic invariant if and only if it is invariant under restriction. Using this general result, the authors obtain the second main result of the present work: There exists a simply-connected compact complex manifold satisfying the \(\partial \bar\partial\)-Lemma which is not Kähler (not even in class \(\mathcal{C}\) of Fujiki).