Analytic subvariety in complex manifold

Question

I am trying to figure out a statement in a textbook

"If $M$ is any complex manifold of a projective space $\mathbb{P}^{n}$, $V\subseteq M$ an analytic subvariety of dimension $k$, then we can find a linear subspace $\mathbb{P}^{n-k}$ of $\mathbb{P}^{n}$ meeting $V$ in isolated points."

Why should this be true?

Thanks!

Answer 1

I don't know your background. $V$ will be algebraic (Chow's Theorem) of some degree $d$, and so the generic $\Bbb P^{n-k}$ will intersect $V$ transversely (Bertini's Theorem) in $d$ points. (You can deduce this by applying transversality theory to deduce that the generic $\Bbb P^{n-k}$ is transverse to all the smooth strata of $V$.)