Corollary 9.3.13F1291FB3DBB9

Proof. (1) Let $A$ first be a small representative of an irreducible germ of dimension $e$ in $\mathbb C^ n$, pure-dimensional by proposition 9.3.9, and choose generators $g_1,\ldots ,g_ N$ of $\mathcal I_ A$ on a neighborhood, by theorem 9.3.10. At a regular point, these generate the ideal of the manifold $A$, which is generated by $n-e$ normal coordinates (lemma 9.3.12); differentiating, we see that the matrix of differentials of the $g_ j$ has rank $n-e$ there. Every minor of size $n-e+1$ therefore vanishes on the dense covering locus, hence on $A$, so the rank is at most $n-e$ everywhere on $A$. Conversely, if it equals $n-e$ at $y\in A$, choose $n-e$ of the $g_ j$ with independent differentials at $y$. Together with $e$ suitable coordinate functions they form holomorphic coordinates near $y$, by the holomorphic inverse function theorem (proposition 5.2.4(2)), so their common zero set is an $e$-dimensional submanifold $N$ near $y$, and it contains $A$. Every branch of $(A,y)$ has dimension $e$ and is contained in the germ $(N,y)$, which is irreducible of dimension $e$ by lemma 9.3.12, so it equals $(N,y)$ by proposition 9.3.9; hence $A$ is a manifold near $y$. Thus $A_{\mathrm{sing}}$ is the subset of $A$ where all minors of size $n-e$ of this matrix vanish, an analytic subset.

For a general analytic set $A$ near a point $x$, let $A_1,\ldots ,A_ p$ be small representatives of the branches of $(A,x)$. A point $y$ near $x$ lying on only one $A_ i$ is regular for $A$ if and only if it is regular for $A_ i$. A point $y\in A_ i\cap A_ j$, $i\neq j$, is singular. Indeed, otherwise $(A,y)$ would be irreducible, hence equal to some $(A_ k,y)$, and we may assume $k\neq i$; then $(A_ i,y)\subset (A_ i\cap A_ k,y)$. But $\dim (A_ i\cap A_ k,x)<\dim A_ i$ by proposition 9.3.9, so $\dim (A_ i\cap A_ k,y)<\dim A_ i=\dim (A_ i,y)$ by upper semicontinuity and purity, contradicting monotonicity. This proves the formula for $A_{\mathrm{sing}}$ near $x$, which is therefore analytic by the irreducible case. Finally, let $y\in A$ and let $B$ be a branch of $(A,y)$. By corollary 9.3.6, $B$ contains points of the covering locus of a parametrization of $B$ outside the other branches of $(A,y)$, arbitrarily close to $y$, and these are regular points of $A$. Hence the germ of $A_{\mathrm{sing}}\cap B$ at $y$ differs from $B$, and it has smaller dimension than $B$, by proposition 9.3.9.

(2) By (1), each $A^{(j)}$ is analytic, and $\dim (A^{(j+1)},x)<\dim (A^{(j)},x)$ at each point $x$ of $A^{(j+1)}$, because every branch of $(A^{(j+1)},x)$ lies in a branch of $(A^{(j)},x)$ and in the singular locus. Hence $A^{(j)}$ has dimension at most $d-j$ at each of its points, and $A^{(d+1)}=\varnothing $, a set of dimension $-1$ being empty. Each $A^{(j)}_{\mathrm{reg}}=A^{(j)}\setminus A^{(j+1)}$ is a complex submanifold of $M\setminus A^{(j+1)}$ of dimension at most $d-j$, and $A$ is the union of these sets.

(3) Each of the submanifolds in (2) is locally a countable union of coordinate charts, so $A$ is locally a countable union of real submanifolds of dimensions at most $2d$. Each such chart is a countable union of images of compact cubes in $\mathbb R^ r$, $r\leq 2d$, under Lipschitz maps, because the image of the chart in $\mathbb R^ r$ is a countable union of compact cubes, on each of which the smooth inverse of the chart is Lipschitz, by the mean value inequality. Such an image is $\mathcal H^ s$-null for $s>r$: covering the cube by $O(\delta ^{-r})$ cubes of side $\delta $, whose images have diameter $O(\delta )$, we cover the image by $O(\delta ^{-r})$ balls of radii $O(\delta )$, and the sum of the $s$th powers of these radii is $O(\delta ^{s-r})$, which tends to zero with $\delta $. A countable union of $\mathcal H^ s$-null sets is $\mathcal H^ s$-null, which completes the proof. $\square$