Ah, Jorge, it's you! Hi!!

That's exactly what I meant. A subvariety of a complex torus is Kobayashi hyperbolic if and only if it does not contain any translate of a subtorus

Recently it has been proved by Damian Brotbek that generic projective complete intersections of high multi-degree are hyperbolic. These should give other examples. Otherwise, what about subvarieties of complex tori which do not contain any translate of a sub-tours ?

In general one cannot speak about the curvature of the metric: it's not unique.

Complement: the elementary lemma I speak about, just says that the Euler characteristic of a bounded complex of finite dimensional vector spaces equals the Euler characteristic of its cohomology module.

Just a last comment, to be precise. This is not Siu's conjecture, this is the Kobayashi conjecture. The strategy to prove it is by Siu.

Let's say that Siu indicated a quite precise strategy to prove that. Up to now, we have a full proof only for surfaces in $\mathbb P^3$ and threefolds in $\mathbb P^4$. So your exemple works for the moment just for $n=4$.

I'll take a look, grazie.

Just because it is not included in every textbook on Banach space theory. Anyway, I just said that it would have been nice to have a brief account here. It doesn't matter! I'll take a look at the paper. Thanks.

Exactly Martin.

It would be nice if you could give an idea of how it works here.