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

Abstract

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.

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