On harmonic symmetries for locally conformally Kähler manifolds (Q2082720)

Let \(M\) be a compact complex manifold with \(\dim _{\mathbb{C}} (M)=n\) endowed with Hermitian metric. Some topological and geometric inequalities for \(M\) can be expressed in terms of the kernel of a certain Laplacian-type operator \(\Box\). The space of \(\Box\)-harmonic forms in degree \(k\) is defined by the relation \(\mathcal{H}_{\Box}^k=\text{ker}(\Box)\cap \Omega^k\), where \(\Omega^k\) denotes the space of \(k\)-forms. The author considers an interdependence among properties of a given Hermitian and complex structures and the dimensions of these subspaces \(\mathcal{H}_{\Box}^k\). For a compact locally conformally Kähler manifold and \(\vert k-n\vert \geq 2\), the equality \(\mathcal{H}_{\Box}^k=\{ 0\}\) is established. A sufficient condition is given such that a complex non-Kaählerian manifold admits a Hermitian metric \(g\) which is not a locally conformally Kähler metric. Furthermore, the author studies the Hodge decomposition of the forms in \(\mathcal{H}_{\Box}^k\) on a Vaisman manifold. He develops a version of Hodge theory on Vaisman manifolds to obtain a Hodge decomposition of the \(\nabla _d\)-harmonic \(n\)-forms. Some properties depending on Betti numbers of compact Vaisman manifolds are also derived.