I found that related to the Kähler cone there are many discussions on MathOverflow. Recently I am interested in the very special manifold of the one-point blow up of $\mathbb{C}P^n$ and just want to see what the general results on Fano Kähler manifolds look like when it comes to this special manifold.

My question is very concrete. Let $x$ and $y$ be the two generators of $H^2(\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n};\mathbb{Z})$ corresponding to the two components respectively. Moreover we assume $\int x^n=-\int y^n=1$. $H^2(\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n};\mathbb{R})=H^{1,1}(\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n};\mathbb{R})$ as $b_2(\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n};\mathbb{R})=2$. Thus every element in $H^{1,1}(\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n};\mathbb{R})$ can be written in the form $ax+by$, $a,b\in\mathbb{R}$. So my question is $$ \{(a,b)\in\mathbb{R}^2~|~ax+by>0\}=?$$ Of course this set is contained in $\{(a,b)~|~a^n-b^n>0\}$.

mathbinif we are going to be approaching MO like copy-editors... – Yemon Choi Jul 25 '13 at 8:34