Geometry over composition algebras : projective geometry

The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.Comment: 24 page

On Mukai flops for Scorza varieties

I give three descriptions of the Mukai flop of type $E\_{6,I}$, one in terms of Jordan algebras, one in terms of projective geometry over the octonions, and one in terms of O-blow-ups. Each description shows that it is very similar to certain flops of type $A$. The Mukai flop of type $E\_{6,II}$ is also described.Comment: 35

Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $\rho$ is a maximal representation of a uniform complex hyperbolic lattice $\Gamma\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $\rho$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506.07274). However we do not focus immediately on the exceptional cases and instead we provide a more unified perspective, as independent as possible of the classification of the simple Hermitian Lie groups. This relies on the study of the cominuscule representation of the complexification of the target group. As a by-product of our methods, when the target Hermitian group $G$ has tube type, we obtain an inequality on the Toledo invariant of the representation $\rho:\Gamma\rightarrow G$ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).Comment: Comments are welcome

On stratified Mukai flops

We construct a resolution of stratified Mukai flops of type A, D, E_{6, I} by successively blowing up smooth subvarieties. In the case of E_{6, I}, we construct a natural functor which induces an isomorphism between the Chow groups.Comment: use diagrams.st