# Does the argument principle generalize to projective space?

Questions:

1. Does the argument principle, which is valid for $\mathbb{C}$, generalize to higher-dimensional spaces $\mathbb{C}^n$, and to complex projective spaces $\mathbb{CP}^n$? Or at the very least to $\mathbb{CP}^2$?

2. Is it true that $\oint_C \frac{(f+g)'(z)}{(f+g)(z)}dz \ge \min \{ \oint_c \frac{f'(z)}{f(z)}dz, \oint_c \frac{g'(z)}{g(z)}dz \}$? And if so, why?

$$\oint_C\frac{f'(z)}{f(z)}dz = 2\pi i(N-P)$$ where $f(z)$ is a meromorphic function inside a closed contour $C$ with no zeros and poles on $C$, and $N$ is the number of zeros inside $C$ counted up to multiplicity, and $P$ is the number of poles inside $C$ counted up to multiplicity.

Context: I am trying to prove that for any rational functions $f,g$, that $$(N-P)_{f+g} \ge \min \{ (N-P)_f , (N-P)_g \}$$ This is necessary to show that the Riemann-Roch space of a divisor $D$ is actually a vector space. See under definition 21.2. here. Note that obviously rational functions are meromorphic.

• Yes, I believe this does answer my question -- in order to extend the generalizations discussed from $\mathbb{C}^n$ to $\mathbb{CP}^n$ one just has to use affine charts, which generally isn't difficult at all. Thank you for finding this for me -- I really appreciate it! Commented Nov 9, 2016 at 6:38