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 ?

pseudoeffectivecone. $\endgroup$simplicial cone? Thanks $\endgroup$