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Serre-Tate theory
As an important step in his proof of the Tate conjecture for ordinary K3 surfaces, Nygaard has developed an appropriate version of Serre-Tate theory. His results were recently extended to Calabi-Yau threefolds by Ward in his thesis [War14]. In fact, the same methods can be used to yield an analog for higher-dimensional varieties, under some extra assumptions.
See Hodge polygon