On cubic hypersurfaces of dimension seven and eight

Atanas Iliev (1) , Laurent Manivel (2)

1 Department of Mathematics
2 IF - Institut Fourier

Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of the Cartan cubic, in finitely many ways. To each such ``Cartan representation'' we associate a rank nine vector bundle on $X$ with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov. Finally we show that the generic eight dimensional section of the Cartan cubic is rational.

Mots clés: cubic hypersurface; exceptional group; Calabi-Yau threefold; noncommutative variety; Cayley plane

Domaines: Géométrie algébrique [math.AG]

Atanas Iliev, Laurent Manivel. On cubic hypersurfaces of dimension seven and eight. 2011. ⟨hal-00566576v1⟩