Difference between $\mathbb{C}P^2$ and $\overline{\mathbb{C}P^2}$ Sorry for the trouble, I got confused between $\mathbb{C}P^2$ and $\overline{\mathbb{C}P^2}$, do they have the same Euler characteristic? the same signature? does $\overline{\mathbb{C}P^2}$ admit a Kahler-Einstein metric? is $\overline{\mathbb{C}P^2}$ Fano? does $\overline{\mathbb{C}P^2}$ admit Fubini-Study metric？ Thank you.
 A: Your confusion is due to not keeping track of what categories everything is living in. $\overline{\mathbb{CP}^2}$, as a smooth manifold, is still just $\mathbb{CP}^2$. So it has the same smooth-manifold invariants as $\mathbb{CP}^2$, such as Euler characteristic.
$\overline{\mathbb{CP}^2}$, as an oriented smooth manifold, is $\mathbb{CP}^2$ with the opposite orientation. The signature of an oriented manifold flips when you flip its orientation, so the signature of $\overline{\mathbb{CP}^2}$ is the negative of the signature of $\mathbb{CP}^2$ ($-1$ instead of $1$). 
As far as I know, $\overline{\mathbb{CP}^2}$ does not admit a complex structure consistent with its orientation (recall that a complex structure determines an orientation), so it is not a complex manifold in a way interestingly different from the way that $\mathbb{CP}^2$ is a complex manifold. So it's unclear how to interpret your other questions. 
Said another way, when we talk about $\mathbb{CP}^2$ we are really talking about an object living in many categories, related by a sequence of forgetful functors
$$\text{ComplexMan} \to \text{OrientedSmoothMan} \to \text{SmoothMan} \to \text{TopMan}$$
none of which reflect isomorphisms, and when we talk about $\overline{\mathbb{CP}^2}$ we are talking about an object living in all of these categories except $\text{ComplexMan}$, which is only different from $\mathbb{CP}^2$ in $\text{OrientedSmoothMan}$. It's important to keep track of these things. 
