2 made a little more precise

I guess I should take this opportunity to mention a personal hobby-horse: the Cone Conjecture of Morrison--Kawamata. (See e.g. http://arxiv.org/abs/0901.3361 for precise statements, history, and known cases.)

To give a little background, note that (as mentioned in Michael Thaddeus' answer) for Fano varieties the nef cone (i.e. closure of the ample cone) is polyhedral, and spanned by rational rays. That's the nicest possible answer one can hope for.

Once we leave the Fano realm, however, things get worse. Already for Calabi--Yau varieties, the cone of curves can fail to be polyhedral, and even be a completely "round" cone. (This happens e.g. for Abelian surfaces of Picard number at least 3.)

That seems pretty bad, but the Morrison--Kawamata conjecture predicts that there is still something we can say about the nef cone, in good cases such as Calabi--Yau varieties. The key observation is that the automorphism group Aut(X) of any variety acts on the nef or ample cone, and one can hope that this group action simplifies the picture somehow.

Morrison's form of the conjecture is roughly the following.

Cone conjecture: Let X be a Calabi--Yau variety. Then there is a rational polyhedral cone Π inside the nef cone Nef(X) which is a fundamental domain for the action of Aut(X) on Nef(X): in particular, Aut(X).Π = Nef(X).

(To give a more precise statement, one has to consider what happens with non-rational points rays on the boundary on the nef cone. Those can never be mapped to by the image of a rational ray, hence we have to throw out all points of the boundary which are not in the convex hull of the rational pointsrays. There is also an issue of what "fundamental domain" means. But the abbreviated statement above gives the essential flavour, I think.)

So the conjecture predicts that although the nef cone may be very far from being a polyhedral cone, it is "generated" by a polyhedral cone when we take into account the action of automorphims (and, as mentioned, throwing away some boundary points.) If true, that would be a very satisfying description.

The conjecture has been generalised to Calabi--Yau fibre spaces and then klt Calabi--Yau pairs. In its most general form, it includes all Fano varieties, all Calabi--Yau varieties, and many in between.

1

I guess I should take this opportunity to mention a personal hobby-horse: the Cone Conjecture of Morrison--Kawamata. (See e.g. http://arxiv.org/abs/0901.3361 for precise statements, history, and known cases.)

To give a little background, note that (as mentioned in Michael Thaddeus' answer) for Fano varieties the nef cone (i.e. closure of the ample cone) is polyhedral, and spanned by rational rays. That's the nicest possible answer one can hope for.

Once we leave the Fano realm, however, things get worse. Already for Calabi--Yau varieties, the cone of curves can fail to be polyhedral, and even be a completely "round" cone. (This happens e.g. for Abelian surfaces of Picard number at least 3.)

That seems pretty bad, but the Morrison--Kawamata conjecture predicts that there is still something we can say about the nef cone, in good cases such as Calabi--Yau varieties. The key observation is that the automorphism group Aut(X) of any variety acts on the nef or ample cone, and one can hope that this group action simplifies the picture somehow.

Morrison's form of the conjecture is roughly the following.

Cone conjecture: Let X be a Calabi--Yau variety. Then there is a rational polyhedral cone Π inside the nef cone Nef(X) which is a fundamental domain for the action of Aut(X) on Nef(X): in particular, Aut(X).Π = Nef(X).

(To give a more precise statement, one has to consider what happens with non-rational points on the boundary on the nef cone. Those can never be mapped to by a rational ray, hence we have to throw out all points of the boundary which are not in the convex hull of the rational points. There is also an issue of what "fundamental domain" means. But the abbreviated statement above gives the essential flavour, I think.)

So the conjecture predicts that although the nef cone may be very far from being a polyhedral cone, it is "generated" by a polyhedral cone when we take into account the action of automorphims (and, as mentioned, throwing away some boundary points.) If true, that would be a very satisfying description.

The conjecture has been generalised to Calabi--Yau fibre spaces and then klt Calabi--Yau pairs. In its most general form, it includes all Fano varieties, all Calabi--Yau varieties, and many in between.