# Is the Kähler cone of a toric variety always simplicial?

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 ?

• Perhaps you are thinking of the pseudoeffective cone. – Jason Starr May 9 '14 at 13:20
• Could you tell me the definition of a simplicial cone? Thanks – Sándor Kovács May 10 '14 at 17:46
• A simplicial cone is a cone of dimension d with d rays. for some integer d. – F. C. May 10 '14 at 20:31