show that subset of complex projective space is a submanifold

Question

Let n, m $\in \mathbb{N}$.

I'm trying to show that

$M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$

is a submanifold and codim(M(n,m))=2.

My idea was to use the regular value theorem.

Thanks in advance.