# Meaning of stable $CP^2$

I came across the following phrase in arXiv:1903.08904

....in order to have a stable $$CP^2$$ , i.e., one in which all the automorphism group is fixed...

Can anyone explain to me what one means by a stable $$CP^2$$? It is mentioned in the phrase itself, but I do not know what one means by a fixed automorphism group. It must be some usual mathematical terminology which I am not aware of, so please help.

## 1 Answer

Stability normally refers to stability under some action. Given some homomorphism $$T: \text{End}(\mathbb{C}P^2) \to \text{End}(\mathbb{C}P^2)$$, we would say that $$\text{Aut}(\mathbb{C}P^2)$$ is stable under $$T$$ (or $$T$$-stable) if $$T: \text{Aut}(\mathbb{C}P^2) \to \text{Aut}(\mathbb{C}P^2)$$.

• What is T ? Is it just any action which preserves $Aut(CP^2)$ ? – user44690 Mar 24 at 9:39
• $T$ is just some homomorphism. If $T$ maps $\text{Aut}(\mathbb{C}P^2)$ to itself, then $\text{Aut}(\mathbb{C}P^2)$ is stable under $T$. So, yes if $\text{Aut}(\mathbb{C}P^2)$ is stable under $T$, then $T$ preserves $\text{Aut}(\mathbb{C}P^2)$. – Gary Moon Mar 24 at 16:09