What is the Kahler cone of $\mathbb{C}P^1 \times \mathbb{C}P^n$ blown-up along a co-dimension two subvariety of the form $pt \times H$ where $H \subset \mathbb{C}P^n$ is a hyperplane?



Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$
The Kahler classes are the following: $a \mathbb CP^n+b H\times \mathbb CP^1-c E$ where $E$ is the exceptional divisor, and $a, b>c>0$. This can be proven using the fact that manifold is toric.


Indeed, the moment image of $\mathbb CP^1\times \mathbb CP^n$ is a segment times an $n$-simplex, and to get your manifold one blows up a codimension two face. The inequalities are clear if you draw the picture.


