Derived categories and Hodge theory of some Fano 4-folds

I will talk about Fano 4-folds of Picard number 1, degree 10, and index 2. They are similar to cubic 4-folds in many respects. In particular, the derived category of such a 4-fold $X$ admits a semiorthogonal decomposition into an exceptional sequence and an interesting 2-Calabi–Yau category $A_X$. I will discuss how rationality of $X$ relates to the condition that $A_X$ is equivalent to the derived category of a K3 surface, and also connections between $A_X$ and the Hodge theory of $X$.