Hodge numbers and Hodge structures for 3-Calabi–Yau categories

Abuaf, Roland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 2, pp. 337-369

Let $𝒜$ be a smooth proper $ℂ$-linear triangulated category which is $3$-Calabi–Yau endowed with a (non-trivial) rank function. Using the homological unit of $𝒜$ with respect to the given rank function, we define Hodge numbers for $𝒜$.

If the classes of unitary objects generate the rational numerical $\mathrm{K}$-theory of $𝒜$, it is proved that these numbers are independent of the chosen rank function : they are intrinsic invariants of the triangulated category $𝒜$.

In the special case where $𝒜$ is a semi-orthogonal component of the derived category of a smooth complex projective variety and the homological unit of $𝒜$ is $ℂ\oplus ℂ\left[3\right]$, we define a Hodge structure on the Hochschild homology of $𝒜$. 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.

Soit $𝒜$ une catégorie triangulée $ℂ$-linéaire, non singulière et propre, que l’on suppose être $3$-Calabi–Yau et munie d’une fonction rang non-triviale. En nous basant sur la notion d’unité homologique pour $𝒜$ associée à la fonction rang, nous définissons des nombres de Hodge pour $𝒜$.

Si les classes d’objets unitaires engendrent la $\mathrm{K}$-théorie numérique de $𝒜$, nous prouvons que ces nombres ne dépendent pas de la fonction rang choisie : ce sont alors des invariants intrinsèques de la catégorie $𝒜$.

Dans le cas particulier où $𝒜$ est une composante semi-orthogonale de la catégorie dérivée d’une variété projective non singulière définie sur $ℂ$ et que l’unité homologique de $𝒜$ est $ℂ\oplus ℂ\left[3\right]$, nous définissons une structure de Hodge sur l’homologie d’Hochschild de $𝒜$. Les dimensions des espaces de Hodge associés à cette structure sont les nombres de Hodge déjà mentionnés.

En conclusion, nous donnons quelques applications numériques de notre théorie en direction de la conjecture de Symétrie Miroir Homologique pour les hypersurfaces cubiques de dimension $5$ et les recouvrements doubles quartiques de ${\mathbb{P}}^5$.

Classification : 14A22
Keywords: Calabi–Yau categories, Hodge theory of non-commutative spaces, numerical invariants of triangulated categories.
Mots-clés : Catégories de Calabi–Yau, Théorie de Hodge pour les variétés non-commutatives, invariants numériques des catégories triangulées.

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