Approximating holomorphic maps by holomorphic embeddings

Question

Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let $\mathrm{HolEmb}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the subspace of those holomorphic maps which are also embeddings, so there is an inclusion $$i : \mathrm{HolEmb}^d(\Sigma, \mathbb{C} \mathbb{P}^n) \longrightarrow \mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n).$$

If we were discussing smooth maps, instead of holomorphic, Whitney's embedding theorem would say that this map is approximately $(\frac{n}{2}-2)$-connected. Is there a connectivity range for this map of spaces of holomorphic maps?

Answer 1

I will assume that d is the degree of the pull-back of $\mathcal{O}(1)$ to $\Sigma$ and that it is sufficiently large with respect to the genus g of $\Sigma$. In this case, the dimension of the space of holomorphic maps of $\Sigma$ in $\mathbb{P}^n$ is $$ D_n := (n+1)d + n(1-g) , $$ while the dimension of the space of holomorphic maps that are not isomorphisms onto their image is $$ (n+1)d + n(1-g) -n+2 = D_n -(n-2) . $$ In particular, since the inclusion that you are interested in has complement of (complex) codimension n-2, it follows that it is "quite connected", roughly (2n-1)-connected?

Unless I made some mistakes in my computations, the estimates for the dimensions above are only valid if d is sufficiently large, otherwise they should simply be lower bounds on the actual dimensions of the spaces. In the case of n=2 you obviously must allow singularities in the image, but you can also prove that the locus where the morphism is not a local embedding (i.e. when the derivative is not injective somewhere) has codimension one.

Answer 2

At least for CP^2, HolEmb is empty unless g=(d-2)(d-1)/2 by the adjunction formula.