Hello,


Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$.
What is known in general about the (topological) space of holomorphic maps $\mathrm{Hol}(X(\mathbf{C}),Y(\mathbf{C}))$ from $X(\mathbf{C})$ to $Y(\mathbf{C})$?


In particular, I would be interested to know under what hypothesis on $X$ and $Y$ the space 
$\mathrm{Hol}(X(\mathbf{C}),Y(\mathbf{C}))$ is a smooth manifold (and to have a formula to compute the dimension of its connected components).


One example I have in mind is : if $X=Y=\mathbf{P}^1$ (the projective line) then $\mathrm{Hol}(\mathbf{P}^1(\mathbf{C}),\mathbf{P}^1(\mathbf{C}))$, in which case one gets the space of complex rational functions (which has connected components indexed by the positive integers (the degree), but each is a smooth algebraic variety).


I think this might be an (easy?) application of the theory of Grothendieck $\mathrm{Hom}$-schemes, but I don't feel very at ease with this. 
I would also be interested to know when $\mathrm{Hom}(X,Y)$ is a smooth algebraic variety.


Many thanks,


K. 



Are spaces of holomorphic maps manifolds?
Let $X,Y$ be analytic spaces with $X$ compact reduced and $Y$ arbitrary (i.e. maybe with nilpotents in its structure sheaf). Then Douady showed in his [thesis](http://archive.numdam.org/item/AIF_1966__16_1_1_0/)
that the set of holomorphic maps $Hol(X,Y)$ can be endowed with the structure of an analytic space whose underlying topology is the compact-open topology. If $X,Y$ are compact manifolds, the Zariski tangent space at $f:X\to Y$ is a subspace of the finite-dimensional vector space of sections of the tangent bundle to $Y$ pulled-back to $X$ viz. $T\_f(Hol(X,Y))\subset \Gamma(X,f^\star TY)$.
I don't know any good general criterion for $Hol(X,Y)$ to be smooth at $f$.


**Edit** Here is a class of examples which might interest you, where smoothness occurs. Let $X$ be a Riemann surface of genus $g$. The space of ramified covers $f:X\to \mathbb P^1$ of degree $d$ is non-empty and smooth of dimension $2d+1-g$ as soon as $d\geq g+1$. But there are explicit cases for smaller $d$ where the corresponding space is singular. You can read about these results in [this article](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-54/issue-7/Examples-of-obstructed-holomorphic-maps/10.3792/pjaa.54.189.full) by Akaohori and Namba.


