I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone $\mathcal{P}$. $\mathcal{P}$ is a subset of $H^{1,1}(\Sigma_1) \cap H^2(\Sigma_1, \mathbb{R}) \simeq \mathbb{R}^2$.
The answer to this overflow question describes $\mathcal{P}$ as follows:
$$ \mathcal P \simeq \lbrace (a,b) \in \mathbb R^2 \mid a > 0, \quad b > 0, \quad a > b \rbrace $$ where $(a,b) \mapsto aH - bE$. Here $H$ is the divisor of a general hyperplane in $\mathbb P^2$, pulled back to the blowup, and $E$ is the exceptional divisor of the blowup.
This describes a cone in $\mathbb{R}^2$ that is the region between the ray generated by $(1,0)$ and the ray generated by $(1,1)$.
My first question: Am I correct in assuming that I should really replace $H$ and $E$ above with their Poincare dual cohomology classes in $H^{1,1}(\Sigma_1) \cap H^2(\Sigma_1, \mathbb{R})$? If so, can one be explicit about those cohomology classes (e.g., write down a differential form in them)?
My second question: I would like to write down explicit Kähler metrics on $\Sigma_1$. I would also like to understand the Kähler cone better. Can one write down explicit differential forms (preferably ones that arise naturally) in the cohomology classes corresponding to the abstract vectors $(1,0)$ and $(1,1)$ that generate the boundary of the cone? (For example, if we view $\Sigma_1$ as a $\mathbb{P}^1$ bundle over $\mathbb{P}^1$ with the map to the base $\pi: \Sigma_1 \to \mathbb{P}^1$, is the $\pi$-pullback of a Kähler metric on $\mathbb{P}^1$ relevant here?)
Note that I am fairly new to complex and algebraic geometry; I am more comfortable with Riemannian geometry. I would also be interested in references related to this question and/or giving an accessible introduction to Kähler cones.
It is quite standard in complex geometry to confuse cycles and their Poincaré duals. (Of course, intersection of cycles corresponds to cup products of their Poincaré duals.) The key one to work out is that the Kähler form $\omega$ on $\Bbb P^n$ (which we normalize to generate $H^2(\Bbb P^n,\Bbb Z)$) corresponds to a hyperplane $\Bbb P^{n-1}$, linearly embedded. This follows because the integral of $\omega$ over any line ($\Bbb P^1\subset\Bbb P^n$) is $1$.
The other crucial ingredient is that any divisor $D$ (integral linear combination of hypersurfaces) in a compact complex manifold $M$ corresponds to a line bundle $L$, and $c_1(L)\in H^2(M,\Bbb Z)$ is Poincaré dual to $D$. This can be worked out explicitly with differential forms (or currents); see pp. 141-143 of Griffiths-Harris.
When we blow up a point in a compact $n$-dimensional complex manifold $M$, obtaining $\tilde M$, we add a generator to $H^2(M,\Bbb Z)$ (thinking of the exceptional divisor $E\in H_{2n-2}(\tilde M,\Bbb Z)$). In general, the normal bundle $N(E,\tilde M)$ is the tautological line bundle on $E\cong \Bbb P^{n-1}$.
For the case of a surface $M$, then $E\cdot E = -1$, because the self-intersection of $E$ is given by $\displaystyle\int_E c_1(N(E))=-\int_{\Bbb P^1}\omega=-1$ (where $\omega$ is the Kähler form of $\Bbb P^1$).
Now, back to your explicit problem. Thinking of $\Sigma_1$ as $\tilde{\Bbb P^2}$, we can think of $H^2$ as being generated by (the Poincaré duals of) a generic line in $\Bbb P^2$ and the exceptional divisor $E$. There is a natural projection $\pi\colon \tilde{\Bbb P^2}\to\Bbb P^2$ and we can consider $\omega_1 = \pi^*\omega$, the pullback of the Kähler form on $\Bbb P^2$. Thinking of $E$ as a divisor in $\Sigma_1$, we get a line bundle and a corresponding Chern form $\phi\in H^2_{dR}(\Sigma_1)$, and it is natural to define $\omega_1-\phi$ as a Kähler form on $\Sigma_1$.
On the other hand, we can also think of $\Sigma_1$, as you suggested, as a $\Bbb P^1$-bundle over a generic line $L\subset\Bbb P^2$ (e.g., if we are blowing up $[1,0,0]$, the origin in $\Bbb C^2\subset\Bbb P^2$, we can take $L=\{z_0=0\}$ to be the "line at infinity"). Then we naturally get (see, e.g., Bott-Tu) two generators for the cohomology: the pullback of the Kähler form on $L$ and the Kähler form on a fiber $F$. The exceptional divisor $E\subset\tilde{\Bbb P^2}$ is often then interpreted as the "infinity section" of this $\Bbb P^1$-bundle and we can see that (thinking in homology or cohomology) $E = L - F$: Since $E\cdot F = 1$, $L\cdot L = L\cdot F = 1$, and $F\cdot F=0$, writing $E = aL+bF$ and solving, we get $a=1=-b$. (See pp. 514-520 of Griffiths-Harris for way more on this.)