The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau:
http://arxiv.org/abs/1011.1250
http://arxiv.org/abs/1402.0427
If we look now at mirror symmetry, is there any connection between the symplectic Hodge map on one side of the duality and the K\"ahler Hodge on the other.
(Sorry if this question is too broad. I see that "is there any connection" might not be enough precise for this site. So any help to make it more precise would be great!)