Are spaces of holomorphic maps manifolds? 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. 
 A: Here are some remarks about the last part of the question.
I wonder if the component of the space of maps $Hom(Y,Y)$ of degree $1$ and higher is not always smooth (it is not clear straight away how to consturct a contre-example). In the case of degree $1$ maps the space is clearly smooth since it is a Lie group of dimension $H^0(TY)$. In general, manifolds that admit slef-maps of degree higher than $1$ are not so common. For example if you take in $\mathbb CP^n$ ($n>3$) a hypersurface of degree $2$ and higher it does not admit self-maps of degree higher than $1$, this result is discussed in a nice article of Beauville http://math.unice.fr/~beauvill/pubs/endo.pdf. 
On the other hand, it is not hard to construct manfiolds for which some irreducible components of the space of self-maps of zero degree will be non-smooth. It is sufficient to take $Y$ such that $Hom(\mathbb CP^1,Y)$ is non smooth, and conisder $X=\mathbb CP^1\times Y$. Then you just conisder maps $X\to \mathbb CP^1\to X$. 
A: 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
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 by Akaohori and Namba.
