On the deformed Bott-Chern cohomology

Given a compact complex manifold $X$ and integrable Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$, we introduce double structure on $A^{\bullet,\bullet}(X)$ naturally determined by $\phi$ study its Bott-Chern cohomology. In particular, establish deformation theory for cohomology use it to compute the deformed Iwasawa holomorphically parallelizable Nakamura manifold. The $\partial\bar{\partial}_{\phi}$-lemma is studied show satisfying formal.