Bott-Chern cohomology ($H_{BC}^{p,q}(M)$)

For a complex manifold, Bott-Chern cohomology isIts classes are represented by $d$-closed forms of pure type. A natural map sends a Bott-Chern class to its Dolbeault cohomology class.

For a compact Kähler manifold, the natural map $H_{BC}^{p,q}\to H_{\bar{\partial }}^{p,q}$ is an isomorphism. The Kähler Laplacian identity makes the harmonic representative of every Dolbeault cohomology class $d$-closed, giving surjectivity, and the ddbar lemma gives injectivity.

On a polydisc, $H_{BC}^{p,q}=0$ for $p,q>0$. A Poincare lemma primitive for a closed pure-type form can be modified by exact terms until only bidegrees $(p-1,q)$ and $(p,q-1)$ remain. The Dolbeault-Poincaré lemma and conjugate Dolbeault-Poincaré lemma then make the original form $\partial \bar{\partial }$-exact.