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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.

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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 $T: \text{Aut}(\mathbb{C}P^2) \to \text{Aut}(\mathbb{C}P^2)$.

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  • $\begingroup$ What is T ? Is it just any action which preserves $Aut(CP^2)$ ? $\endgroup$ – user44690 Mar 24 at 9:39
  • $\begingroup$ $T$ 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 $\text{Aut}(\mathbb{C}P^2)$. $\endgroup$ – Gary Moon Mar 24 at 16:09

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