I am working on a familly of toric varieties which seem to have the following property:
- the closure of the Kähler cone is a simplicial cone (and even a smooth cone with respect to the natural lattice).
This has the interesting aspect that it provides a natural basis of the $H^2$ cohomology group.
I was wondering whether this is or not a trivial property.
Is the closed Kähler cone of any smooth toric variety a simplicial cone ?
I would guess that the answer is no, but I do not know a counter-example.
It is known that this cone is polyhedral for all toric varieties. But maybe one can find examples where it has more generators ? Do you know one ?