Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures.

There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of symplectic forms on $X$. Using this, one can interpolate between Hard Lefschetz symplectic forms and just plain symplectic forms.

There is also a paper by Draghici that constructs on $4$-manifolds a path of cohomology classes that starts in the Kaehler cone and then definitely leaves it.

However, I am having trouble finding results in the literature that would help me to start with a Hard Lefschetz symplectic form and then somehow interpolate from that to a Kaehler form. Do people have any experience with this question? Or know some good resources for further reading on this topic? Perhaps I am not framing my question in a good way? Maybe relinquishing control of the Kaehler structures and attempting to reach them (instead of starting with them) is a little bit too intractable?

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