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The dimension of an irreducible analytic variety is defined to be the dimension of its regular locus as a complex manifold. Suppose we have an irreducible analytic subvariety $V$ of $\mathbb{C}^n$ of dimension $k$. My question is whether the following is true: For any $p\in V$, there is a neighborhood $U$ of $p$ in $\mathbb{C}^n$ such that $V\cap U$ can be given as the zero locus of $n-k$ holomorphic functions on $U$.

The case $k=1$ is claimed to be true in Principles of Algebraic Geometry by Griffiths & Harris (first sentence on p.130), but I don't see why. How do I prove this? Is this true for general $d$? What is a good reference that explains these matters? Thanks in advance!

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True for $k=1$, false for $k > 1$. There exist say codimension 2 subvarieties that require 3 functions to define. The standard example is the subvariety given by $$\text{rank} \begin{bmatrix} z_1 & z_2 & z_3 \\ z_4 & z_5 & z_6 \end{bmatrix} < 2.$$ You need all three subdeterminants to define this codimension 2 (dimension 4) subvariety of $\mathbb{C}^6$ (not totally trivial to prove).

For $k=1$ what you do is to follow the construction of the Weierstrass polynomial defining the subvariety.

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