\(\partial \overline{\partial }\)-Lemma and Double Complex

Inspired by the recent works of Stelzig and Khovanov–Qi on the structure of bounded double complex, we improve further this theory with more emphasis on the bounded double complexes possibly without real structures. We calculate out six types of indecomposable double complexes in the complex decomposition, to characterize various (weak) \(\partial \overline{\partial }\) $\partial \overline{\partial }$ -lemmata by de Rham, Dolbeault, Bott–Chern, Appeli and Varouchas’ cohomology groups. In particular, we obtain new characterizations of \(\partial \overline{\partial }\) $\partial \overline{\partial }$ -lemma and Frölicher-type inequalities, and also several new injectivity results on complex manifolds. From the double complex perspective, one can easily see the mechanism behind the validity or invalidity of the (weak) \(\partial \overline{\partial }\) $\partial \overline{\partial }$ -lemmata and injectivity.