Theorem 9.7.5689628C7AF7F

Proof. Let $\mathfrak b=\mathfrak a\mathcal O_ W$. We first principalize $\mathfrak b$ by blowups of $W$, perform the same blowups on $M$, and then principalize the remaining ideal away from the strict transform of $W$. If $\dim W=0$, then $W$ is a point, $\mathfrak b$ is the unit ideal, the first sequence below is empty, and the argument applies with $H=0$.

Step 1: principalizing the restriction. Apply theorem 9.6.7 to the ideal $\mathfrak b$ on the compact manifold $W$, with empty divisor. We obtain a finite sequence of blowups $W_ i\to W_{i-1}$ with smooth centers $C_ i\subset W_{i-1}$, $1\leq i\leq t$, and ideals $\mathfrak b_ i$ with $\mathfrak b_0=\mathfrak b$, $\mathfrak b_ i=\mathcal O_{W_ i}(E^ W_ i)\mathfrak b_{i-1}\mathcal O_{W_ i}$, where $E^ W_ i$ is the inverse image of $C_ i$, and $\mathfrak b_ t=\mathcal O_{W_ t}$. By induction on $i$, each $W_{i-1}$ is connected, $\mathfrak b_{i-1}$ is nonzero, and $C_ i$ has positive codimension in $W_{i-1}$ at each of its points; for $i=1$, the first two assertions are hypotheses of the theorem. Assume them for some $i\leq t$. A nonzero coherent ideal on a connected manifold has nonzero stalks at all points, since the points at which its stalk vanishes form an open set, and so, by the identity theorem, do the points at which it does not vanish; hence its zero set is nowhere dense. So $C_ i\subset V(\mathfrak b_{i-1})$ is nowhere dense in $W_{i-1}$, that is, of positive codimension at each of its points. Then $W_ i\setminus E^ W_ i\cong W_{i-1}\setminus C_ i$ is connected by lemma 9.2.9 and dense in $W_ i$, so $W_ i$ is connected; and $\mathfrak b_ i$ is nonzero, because it agrees with $\mathfrak b_{i-1}$ on this set.

We perform the same blowups on the ambient manifolds: let $M_0=M$ and let $\sigma _ i\colon M_ i\to M_{i-1}$ be the blowup along $C_ i$, where $W_{i-1}\subset M_{i-1}$ is the strict transform of $W$. The center $C_ i$ is closed in $M_{i-1}$, because $W_{i-1}$ is closed, and it has codimension at least two in $M_{i-1}$. By proposition 9.6.3, the strict transform of $W_{i-1}$ in $M_ i$ is $\operatorname {Bl}_{C_ i}W_{i-1}=W_ i$, and $E_ i\cap W_ i=E^ W_ i$, where $E_ i=\sigma _ i^{-1}(C_ i)$. More precisely, in the charts of that proposition whose pivot is one of the coordinates $z_1,\ldots ,z_ k$ along $W_{i-1}$, which are the only ones meeting $W_ i$, a local equation $e_ i$ of $E_ i$ restricts to a local equation of $E^ W_ i$. This sequence is adapted to $W$.

Step 2: the transformed ideals. Put $H_0=0$ and $\mathfrak a_0=\mathfrak a$, and define inductively

We show by induction that $\mathfrak a_ i$ is a holomorphic ideal and that $\mathfrak a_ i\mathcal O_{W_ i}=\mathfrak b_ i$. Every local section $h$ of $\mathfrak a_{i-1}$ vanishes on $C_ i$, because its restriction to $W_{i-1}$ belongs to $\mathfrak b_{i-1}$ and $C_ i\subset V(\mathfrak b_{i-1})$. Hence $h\in \mathcal I_{C_ i}$, and $h\circ \sigma _ i$ belongs to $\mathcal I_{C_ i}\mathcal O_{M_ i}=\mathcal O_{M_ i}(-E_ i)$; that is, it is divisible by the local equation $e_ i$ of $E_ i$, and $\mathfrak a_ i$ is holomorphic. Restricting $(h\circ \sigma _ i)/e_ i$ to $W_ i$ divides $h|_{W_{i-1}}\circ \sigma _ i|_{W_ i}$ by the local equation $e_ i|_{W_ i}$ of $E^ W_ i$, which gives $\mathfrak a_ i\mathcal O_{W_ i}=\mathfrak b_ i$; this holds also when $C_ i$ is a hypersurface in $W_{i-1}$. Iterating (9.6.5) and the definition of $\mathfrak a_ i$, we obtain

Since $\mathfrak b_ t=\mathcal O_{W_ t}$, the ideal $\mathfrak a_ t$ is the unit ideal at each point of $W_ t$: a local section whose restriction to $W_ t$ does not vanish at a point is a unit of the local ring of $M_ t$ there. Hence $V(\mathfrak a_ t)$ is disjoint from $W_ t$.

Step 3: principalizing away from $W_ t$. The manifold $M_ t$ is compact, and it is connected by the argument given for the $W_ i$ in Step 1; the ideal $\mathfrak a_ t$ is not zero, being the unit ideal near $W_ t$. Apply theorem 9.6.7 to $\mathfrak a_ t$ on $M_ t$, with empty divisor. Each center lies in the zero set of the current transformed ideal; since the blowups are isomorphisms near the successive copies of $W_ t$, which do not meet the centers, these transformed ideals remain the unit ideal near these copies, and all centers are disjoint from them. Let $g\colon M'\to M_ t$ be the composite, and let $H'$ be the effective divisor with $\mathfrak a_ t\mathcal O_{M'}=\mathcal O_{M'}(-H')$ obtained by iterating (9.6.8); put $H=g^*H_ t$ and $f=\sigma _1\circ \cdots \circ \sigma _ t\circ g$. The composition $f$ is adapted to $W$, and the strict transform $W'=g^{-1}(W_ t)$ is mapped isomorphically onto $W_ t$. By proposition 9.6.3, $\mathcal I_{W_ t}\mathcal O_{M'}=\mathcal I_{W'}$, and pulling back the two identities of Step 2 gives (9.7.6) and (9.7.7). The divisor $H'$ is supported on the exceptional divisors of $g$, which are disjoint from $W'$.

Step 4: exceptional divisors and restriction. Every component of the support of $H$ maps into the union of the images in $M$ of the centers $C_ i$; each such image lies in $W$, is analytic by theorem 9.7.2, and has dimension at most $\dim C_ i<\dim W\leq \dim M-1$ by lemma 9.3.15(3), so it has codimension at least two in $M$. Hence $H$ is exceptional for $f$, which is proper and bimeromorphic by proposition 9.6.2, corollary 9.7.3.

No component of $H$ contains $W'$: by Step 1, the restriction to $W_ i$ of a local equation of $E_ i$ is a local equation of the hypersurface $E^ W_ i$ of $W_ i$, so by induction the restriction to $W_ t$ of a local equation of $H_ t$ is not identically zero, and $g$ is an isomorphism near $W'$. Hence $H|_{W'}$ is defined, and restricting (9.7.7) to $W'$, where $H'$ vanishes, gives (9.7.8). Finally, $f|_{W'}$ is the composition of the blowups $W_ i\to W_{i-1}$ followed by the isomorphism $W'\to W_ t$. It is an isomorphism outside the union of the images in $W$ of the centers $C_ i$ that have codimension at least two in $W_{i-1}$; this union is a finite union of analytic subsets of dimension smaller than $\dim W$, hence a proper analytic subset of $W$. $\square$