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Let $C\in \mathbb{C}\mathbb{P}^{N}$ be a nonsingular complex curve of genus $g$. Let $p_{1},\dots,p_{k}$ be distinct points on $C$ and $n_{1},\dots,n_{k}$ positive integers.

1) Estimate an upper bound on the dimension of meromophic functions on $C$ which has poles of degree less than $n_{i}$ at $p_{i}$ and regular everywhere else.

2) Do the same for meromorphic differentials.

This question reminds me of the classical Riemann-Roch theorem. But obviously I need a statement that works over complex manifolds and not just Riemann Surfaces, for $\mathbb{C}\mathbb{P}^{N}$ is involved(do we just consider the embedding of the curve into $\mathbb{C}\mathbb{P}^{N}$? then it would make no difference).

On the other hand I also do not know how to prove the second statement (even if $C$ is a compact Riemann Surface). I am not familiar with meromorphic differentials, so I venture to ask at here.

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A nonsingular complex curve $C\in \mathbb{C}\mathbb{P}^{N}$ of genus $g$ is the same as a Riemann surface of genus $g$ (This is the difficult "Riemann existence theorem", but you can use it freely). –  Georges Elencwajg Aug 5 '12 at 12:56
    
Thanks! Now I only need to solve the second one... –  Bombyx mori Aug 5 '12 at 13:00

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I'll do better than give you an estimate: I'll give you the exact dimension (for the second question, since you say that you only have to solve that one now).

Let $D$ be the divisor $D=\sum n_ip_i$.You are interested in the dimension $h^0(C,\Omega (D))$ of the vector space of sections $H^0(C,\Omega (D))$, where $\Omega(D)=\Omega \otimes_{\mathcal O} \mathcal O(D)$ (and $\Omega $ is the sheaf of holomorphic differential forms, of course.)

Now Riemann-Roch says that $$ h^0(C,\Omega (D))- h^1(C,\Omega (D)) =1-g+ deg(\Omega (D))=1-g+2g-2+\sum n_i\\=g-1+\sum n_i $$ On the other hand Serre duality implies that $$h^1(C,\Omega (D))=h^0(C,\mathcal O (-D))=0$$ the last equality being due to $\mathcal O(-D)$ having negative degree (namely $-\sum n_i)$

The final result solving your problem is thus: $$h^0(C,\Omega (D))=g-1+\sum n_i$$

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Could you explain the step involving Serre duality? Also excuse me for my low level, I have never seen the statement using $h^{0}$ and $h^{1}$ before, so this feels totally over my head. –  Bombyx mori Aug 5 '12 at 22:03
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Dear user32240, Serre duality says that for a line bundle $L$ on the compact Riemann surface $C$, we have $h^1(C,L)= h^0(C,\Omega \otimes L^*)$, so that calculation of first cohomology reduces to calculation of global sections (of another, related line bundle). In my answer I have applied this to $L=\Omega(D)$ for which $\Omega\otimes L^*=\Omega\otimes \Omega^*(-D)=\mathcal O(-D)$ –  Georges Elencwajg Aug 6 '12 at 10:03

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