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It may be difficult to give some special and non-trivial examples of Kähler cones.The examples I know are the following:

  1. for complex tori, the Kähler cone is just the set of positive hermitian matrices;
  2. for a Kähler manifold $M$ with $h^{1,1}(M)=1$, then the Kähler cone is just $\mathbb{R}_{+}$.

Thus in order to give a non-trivial example, firstly we must require $h^{1,1}(M)>1$. I would like to see an example such that:

  1. the cone equation or the boundary of the cone is clear;
  2. $c_{1}(M)>0$, or the sign of $c_{1}(M)$ is not definite (are there some intresting manifolds of this type?).

Of course, any explicit shape will be welcome.

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  • $\begingroup$ Given that the question currently has $5$ downvotes and one close vote (mine) stemming from its inauspicious beginning, it might be best to close and delete it and start again from scratch. I wonder what the moderators think of this? $\endgroup$
    – Pete L. Clark
    Commented Dec 28, 2010 at 11:38
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    $\begingroup$ This is a genuine question, I don't quite see why it is down-voted. If you want to look for non-trivial examples where the cone can be explicitly described, one case to consider is toric varieties, you might have a look at Section 6 in the book cs.amherst.edu/~dac/toric.html . Ample cone can be explicitly described as well for Fano surfaces. If you want the case when $C_1$ is not necessarily ample, you should definitely look in the article arxiv.org/abs/1008.3825 $\endgroup$
    – Dmitri Panov
    Commented Dec 28, 2010 at 12:51
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    $\begingroup$ I guess that the OP is a non-native english speaker, and this would explain the high number of grammar mistakes. However, I also think that this is a genuine question which does not deserve so many downvotes, so I took the liberty of fixing the english and the LateX (needless to say, always respecting the author). $\endgroup$
    – Francesco Polizzi
    Commented Dec 28, 2010 at 13:13
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    $\begingroup$ The question was downvoted because in its original form it was, literally, random noise (check the edit history). I agree that it's a perfectly good question now. I would like to withdraw my vote to close. $\endgroup$
    – Pete L. Clark
    Commented Dec 28, 2010 at 13:48

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