See ADHM original paper,
In ref. 1 Proc.Roy.Soc.Lond. A362 (1978) 425-461_Selfduality in Four-Dimensional Riemannian Geometry by M F Atiyah, Nigel J Hitchin, I M Singer,
it was shown that “instantons”, i.e. self- dual solutions of the Yang—Mills equations in the compactified euclidean 4-space $S^4$ corresponded in a precise way to certain real algebraic bundles on the complex projective 3-space $P_3(C)$.
Since some people uses the different convention for the complex projective 3-space $P_3(C)$ (like this post, but why to use this different convention?), say the $P_3(C)$ as the 4-dimensional real-space instead of 6-dimensional real-space manifold, I would like to make sure:
In ADHM and in this Atiyah-Hitchin-Singer paper, the notation $P_3(C)$ is a 4-manifold, or 6-manifold for the real dimensions?
If it is the case, why do one construct the real algebraic bundles on the complex projective 3-space $P_3(C)$ (4-dim or 6-dim manifold in the real space?) to realize an instanton solution in 4-space?
I don't see anywhere in that blog that uses a different convention. $\Bbb CP^n$ (my preferred notation for your $P_n(\Bbb C)$) always means the complex manifold of complex dimension $n$, and thus real dimension $2n$. Perhaps you refer to when Qiaochu writes that $X$ has real dimension $2n-2$, but $X$ is not $\Bbb{CP}^n$, it is a hypersurface inside that space (which has complex codimension 1 = real codimension 2).
An unrelated confusion often arises if someone writes something like $\Bbb P(\Bbb C^n)$; this would mean the projectivization of the vector space $\Bbb C^n$, and projectivization lowers dimension by 1; so $\Bbb P(\Bbb C^n) = \Bbb{CP}^{n-1}$.
The twistor construction identifies (theorem 5.2) self-dual Hermitian connections on a bundle $E$ over a 4-manifold with self-dual curvature (note that these are two different concepts, one involving the curvature of a connection, and one involving the Weyl tensor of a Riemannian 4-manifold) with holomorphic bundles over an associated complex 3-manifold equipped with an antiholomorphic automorphism.
This is useful because it is usually very hard to explicitly construct self-dual connections, much less identify the entire moduli space thereof. Of course, the ADHM construction does this very beautifully and explicitly for $S^4$ (or $\Bbb R^4$, depending on your taste), but this is pretty much one of the only cases where this is possible.
It is nice, in particular, that one can apply this result to Einstein manifolds. I don't know off the top of my head anywhere this is applied, but I am sure one can find it around in the literature.