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 ?



Is the Kähler cone of a toric variety always simplicial?
The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of [these notes](http://people.maths.ox.ac.uk/daviesr/resources/notes/dP6.pdf).


