MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
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
Thanks!!!!!!!!!! – Sándor Kovács May 10 '14 at 23:33
up vote 9 down vote accepted

The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of these notes.

share|cite|improve this answer
Indeed. Thanks for the example. – F. C. May 9 '14 at 16:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.