Quantitative and qualitative cohomological properties for non-Kähler manifolds (Q2832827)

The \(\partial \bar{\partial}\) lemma states that every \(d\)-exact pure form is \(\partial \bar{\partial}\) exact on compact Kähler manifolds. It is not generally true on non-Kähler manifolds.NEWLINENEWLINEOn non-Kähler manifolds, two cohomology groups attempt to measure how much the \(\partial \bar{\partial}\) lemma fails, namely, the Bott-Chern and the Aeppli cohomology groups. In this paper several results are proven about the dimensions of these cohomologies.NEWLINENEWLINETheorem 2.1 implies bounds on \(p_k=\sum _{p+q=k} h^{p,q}_{BC} -h^{p,q}_A\) in terms of the Hodge numbers. Theorem 3.1 states that the manifold satisfies the \(\partial \bar{\partial}\) lemma if and only if \(p_k=0\) for all \(k\).NEWLINENEWLINEThe correct substitute to the Poincaré and the Serre dualities is the Schweitzer duality. Motivated by the same, the authors define the qualitative Kodaira-Spencer-Schweitzer property as the natural pairing on the Bott-Chern cohomology to be non-degenerate. Theorem 5.2 then states that this property is equivalent to the \(\partial \bar{\partial}\) lemma.NEWLINENEWLINEFinally, similar techniques allow the authors to prove bounds (Theorem 6.2) on the symplectic cohomologies introduced by Tseng and Yau.