Let $M$ be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on $\Omega\_{\mathbb C}^\bullet(M)$ have isomorphic cohomology groups.


Are there any stronger relationships between these two dgas? For example, are there simple conditions on $M$ that imply that these dgas are homotopy equivalent?



When are the Dolbeault and de Rham dgas homotopy equivalent?
A relevant reference might be [J. Neisendorfer, L. Taylor: Dolbeault homotopy theory. Trans AMS 245 (1978), 183-210.](http://www.ams.org/journals/tran/1978-245-00/S0002-9947-1978-0511405-5/S0002-9947-1978-0511405-5.pdf)
One of the results (Theorem 8) states that compact connected Kähler manifolds are both Dolbeault formal and de Rham formal (strengthening the result of Deligne-Griffiths-Morgan-Sullivan). This implies that the isomorphism of cohomology groups can be promoted to a zig-zag of homotopy equivalences of dgas.


