Projective vs. abelian geometry

Projective space and complex tori are two of the simplest types of manifolds we encounter, and in many ways they seem very different from each other. I will try to convince you however that, at least if we consider a special (but at the same time very common) class of tori called principally polarized abelian varieties, then the geometry of their subvarieties exhibits surprising, and to date mostly unexplained, similarities to the geometry of subvarieties in projective space.

Website: https://people.math.harvard.edu/~ana/ons/