  Let $\mathcal{A}$ be a smooth proper C-linear triangulated category
Calabi-Yau of dimension 3 endowed with a (non-trivial) rank function. Using the
homological unit of $\mathcal{A}$ with respect to the given rank function, we
define Hodge numbers for $\mathcal{A}$. If the classes of unitary objects
generate the complexified numerical K-theory of $\mathcal{A}$ (hypothesis
satisfied for many examples of smooth proper Calabi-Yau categories of dimension
3), it is proved that these numbers are independent of the chosen rank function
: they are intrinsic invariants of the triangulated category $\mathcal{A}$. In
the special case where $\mathcal{A}$ is a semi-orthogonal component of the
derived category of a smooth complex projective variety and the homological
unit of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C}[3]$ (that is
$\mathcal{A}$ is strict Calabi-Yau with respect to the rank function), we
define a Hodge structure on the Hochschild homology of $\mathcal{A}$. The
dimensions of the Hodge spaces of this structure are the Hodge numbers
aforementioned. Finally, we give some numerical applications toward the
Homological Mirror Symmetry conjecture for cubic sevenfolds and double quartic
fivefolds.
