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Apr 22, 2018 at 17:51 comment added R. van Dobben de Bruyn Not quite what you're asking, but you might be interested in this article by Looijenga and Lunts, where they study the $\mathfrak{sl}_2$-structure on the cohomology ring when you vary the Kähler class.
Apr 22, 2018 at 17:21 history asked Gunnar Þór Magnússon CC BY-SA 3.0