Claim: Let $V \subset \mathbb{C}P^n$ be a non-singular projective algebraic variety of complex dimension $k$ and let $P \subset \mathbb{C}P^n$ be a hyperplane.
Then $V \setminus (V \cap P)$ is a complex $k$-dimensional non-singular affine algebraic variety in $\mathbb{C}^n$.
This fact is used (but not proved) in J. Milnors book Morse Theory in the proof of the Lefschetz Hyperplane Theorem (Corollary 7.3).
(Actually, there it would be enough to know that $V \setminus (V \cap P)$ is a complex manifold of dimension $k$, but still the above is stated.)
My thoughts so far:
I first proved that $\mathbb{C}P^n \setminus P \cong \mathbb{C}^n$.
Therefore the embedding in $\mathbb{C}^n$ doesn't cause any problem.
Now assume that $p_1, \ldots, p_j : \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ are complex homogeneous polynomials such that the variety $V$ is induced from the set
$$\bigcap_{r=1}^j p_r^{-1}(0) \subset \mathbb{C}^{n+1}.$$
Furthermore let $l: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ be a linear polynomial such that $P = l^{-1}(0)$.
Then I would like to write something like
\begin{align}
V \setminus (V \cap P) &= V \cap (V \cap P)^c = V \cap P^c \\
&= \left( \bigcap p_r^{-1}(0) \right) \cap \{ l \neq 0 \} \\
&\overset{?}{=} \bigcap \left( \frac{p_r}{l} \right)^{-1}(0)
\end{align}
where the $p_r$ now denote the induced functions $\mathbb{C}P^n \rightarrow \mathbb{C}P^n$.
Then $V \setminus (V \cap P)$ would be cut out by the "polynomials" $\frac{p_r}{l}$ and I'd be done. But this doesn't seem to be formally correct…
Thanks in advance for any help!
Best Answer
The algebraic projective variety $V\subset \mathbb{P}^n$ is given by the zero locus of homogeneous polynomials $f_i\in \mathbb{C}[x_0,\dots,x_n]$.
Take now the open subset $U$ of $\mathbb{P}^n$ where $x_0\not=0$, which is an affine space. You can assume that $x_0=1$ and obtain then coordinates $x_1,\dots,x_n$. This gives an isomorphism $$\begin{array}{rcl}\mathbb{C}^n&\to& U\\ (x_1,\dots,x_n)&\mapsto & [1:x_1:\dots:x_n]\end{array}$$
Then $U\cap V$ is the locus of points of the form $[x_0:\dots:x_n]$ such that $x_0=1$ and $f_i(x_0,\dots,x_n)=0$. This shows that you obtain an affine variety given by the polynomials $f_i(1,x_1,\dots,x_n)$.
This process is very classical and can be founded in any course of algebraic geometry.