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Let $\mathcal{A}$ be a smooth proper C-linear triangulated categoryCalabi-Yau of dimension 3 endowed with a (non-trivial) rank function. Using thehomological unit of $\mathcal{A}$ with respect to the given rank function, wedefine Hodge numbers for $\mathcal{A}$. If the classes of unitary objectsgenerate the complexified numerical K-theory of $\mathcal{A}$ (hypothesissatisfied for many examples of smooth proper Calabi-Yau categories of dimension3), it is proved that these numbers are independent of the chosen rank function: they are intrinsic invariants of the triangulated category $\mathcal{A}$. Inthe special case where $\mathcal{A}$ is a semi-orthogonal component of thederived category of a smooth complex projective variety and the homologicalunit of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C}[3]$ (that is$\mathcal{A}$ is strict Calabi-Yau with respect to the rank function), wedefine a Hodge structure on the Hochschild homology of $\mathcal{A}$. Thedimensions of the Hodge spaces of this structure are the Hodge numbersaforementioned. Finally, we give some numerical applications toward theHomological Mirror Symmetry conjecture for cubic sevenfolds and double quarticfivefolds.

Hodge numbers and Hodge structures for Calabi-Yau categories of dimension three