In what sense is an effective (Weil) Divisor an analytic subspace cut out by one equation

Question

In Chapter 2 of Harris, it states "if Y is a complex manifold and $X\subset Y$ an effective divisor (i.e. an analytic subspace given locally be one equation)…"

This confuses me; for an arbitrary divisor $D$ we have that $D=\sum_p np$, for $n\in\mathbb{Z}$ and $p\in Y$. In what sense can this be identified with an analytic subspace of $Y$?

Answer 1

Your definition of $D$ as a sum of points is what happens in the case of curves. The general definition of a (Weil) divisor is a formal linear combination of codimension-one subvarieties, and Harris' quotation is how one would adapt this to the analytic setting.