TITLE: When are the Dolbeault and de Rham dgas homotopy equivalent?
QUESTION [10 upvotes]: 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?

REPLY [6 votes]: A relevant reference might be J. Neisendorfer, L. Taylor: Dolbeault homotopy theory. Trans AMS 245 (1978), 183-210.
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.