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.