Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann proved that the period map for polarized hyperkähler manifolds is an open embedding.

In a joint work with Macrì, we explicitly determine the image of this map in some cases. I will explain this result together with a nice application (found by Bayer and Mongardi) to the (almost complete) determination of the image of the period map for cubic fourfolds, hereby partially recovering a result of Laza.

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