On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

Abuaf, Roland

Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1403-1430.

On étudie les quartiques doubles de dimension cinq du point de vue des variétés Fano de type Calabi–Yau et des représentations quaternioniques exceptionnelles. On démontre tout d’abord qu’une quartique double de dimension cinq générique peut être représentée comme un recouvrement double de ${\mathbb{P}}^5$ ramifié le long d’une section linéaire de la quartique Spin${}_{12}$-invariante dans ${\mathbb{P}}^{31}$. Le nombre de telles représentations est fini. Ensuite, en utilisant la géométrie des décompositions de Vinberg de type II de certaines représentations quaternioniques exceptionnelles et en se basant sur des calculs effectués grâce au logiciel Macaulay2, on démontre l’existence d’un fibré vectoriel sphérique de rang $6$ sur de telles quartiques doubles. On déduit finalement de l’existence de ces fibrés que l’unité homologique des catégories Calabi–Yau de dimension trois associées par Kuznetsov aux quartiques doubles de dimension cinq est $ℂ\oplus ℂ\left[3\right]$.

We study quartic double fivefolds from the perspective of Fano manifolds of Calabi–Yau type and that of exceptional quaternionic representations. We first prove that the generic quartic double fivefold can be represented, in a finite number of ways, as a double cover of ${\mathbb{P}}^5$ ramified along a linear section of the Spin${}_{12}$-invariant quartic in ${\mathbb{P}}^{31}$. Then, using the geometry of the Vinberg’s type II decomposition of some exceptional quaternionic representations, and backed by some cohomological computations performed by Macaulay2, we prove the existence of a spherical rank $6$ vector bundle on such a generic quartic double fivefold. We finally use the existence of this vector bundle to prove that the homological unit of the CY-3 category associated by Kuznetsov to the derived category of a generic quartic double fivefold is $ℂ\oplus ℂ\left[3\right]$.

Classification : 14F05, 14M17, 14M99
Keywords: Fano varieties of Calabi–Yau type, Calabi–Yau categories, spherical vector bundles, homological units.
Mot clés : Variétés Fano de type Calabi–Yau, catégories de Calabi–Yau, fibrés vectoriels sphériques, unités homologiques.

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