Space of algebraic maps, homotopy type of a CW complex Considering the algebraic maps between two complex varieties denoted by $C_{alg}(X,Y)$, as a subspace of continuous maps with compact-open topology. Does $C_{alg}(X,Y)$ have homotopy type of a CW complex?
If both $X$ and $Y$ are projective then this space coincides with the complex points of the Hom scheme with the analytic topology. So it will have the homotopy type of a CW complex. I am interested in the case that $X$ is not projective?
Since the general question seems to be very difficult I am trying to focus on a special case. Let's assume $X$ is a smooth affine scheme over $\mathbb{C}$ and $Y=\mathbb{CP}^n$. Do the space of algebraic maps from $X$ to $Y$ have the homotopy type of a CW complex?
In this case $Y$ is an Oka manifold. According to "Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds" by Finnur Larusson, the space of continuous maps deformation retract to the space of holomorphic maps. So the space of holomorphic maps are going to have the homotopy type of a CW complex. I am not sure whether these results can be generalized for algebraic maps or not.
 A: I find this question interesting, and I wish I had a stronger background in algebraic geometry to fully answer it. I had to look up the definition of algebraic maps and a remark reminded me that this is like the algebraic analogue of smooth functions in differential geometry. So let me answer the topological version of the question, and maybe someone with more knowledge of algebraic maps can modify this. The argument here is the standard way to prove that spaces of maps have the homotopy type of a CW complex. I'm following Smrekar's 2015 paper CW towers and mapping spaces, just after Theorem D.
Let $M$ and $N$ be smooth finite dimensional manifolds, and let $C_s(M,N)$ be the space of smooth maps, with the weak topology. Assume there is a proper Morse exhaustion function for $M$ and denote the sublevel sets by $M_i$ (they yield a filtration of $M$ by a tower of compact submanifolds). Let $Z_i = C_s(M_i,N)$. Each $Z_i$ is paracompact and homotopy equivalent to $map(M_i,N)$, the space of continuous functions with the compact-open topology. Each $Z_i$ has the homotopy type of a CW complex, by Milnor's Theorem 3, from the paper Neil Strickland pointed out. Now, $C_s(M,N)$ is the inverse limit of the tower of $Z_i$, and each $Z_i\to Z_{i-1}$ is a fibration, so $C_s(M,N)$ also has the homotopy type of a CW complex.
In the OP, $M$ is a smooth affine scheme over $\mathbb{C}$ and so it's a Stein manifold and has the required Morse filtration. And $N$ is $\mathbb{CP}^n$. So this argument works for the space of smooth maps, and I would very much like to know if it also works for the space of algebraic maps. The proof relies on several facts from Palais's 1968 book Foundations of Global non-linear analysis (which I do not have access to), and algebraic analogues of these facts are needed.
