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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?

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Perhaps I misunderstood which space you were referring to by "the space", because there are two. What I am asking about is the relative connectivity of two spaces, holomorphic embeddings and holomorphic maps. Surely the argument you gave applies to both of the spaces, and shoes that neither is simply connected: what does this mean for the relative connectivity though? – Oscar Randal-Williams May 3 '10 at 8:30
Oscar huge thanks, I see now that I completely misuderstood your question:) So I delit my answer. – Dmitri May 3 '10 at 9:08
up vote 4 down vote accepted

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.

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

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