Theorem 9.6.7 (Principalization)24330D25741D

Theorem 9.6.7 (Principalization).24330D25741D Let $M$ be a complex manifold, let $\mathfrak a\subset \mathcal O_ M$ be a coherent ideal that is not identically zero on any connected component of $M$, and let $D$ be a simple normal crossings divisor on $M$, possibly empty. Then there is a sequence of blowups $\sigma _ i\colon M_ i\to M_{i-1}$, $i\geq 1$, with $M_0=M$ and smooth centers $C_ i\subset M_{i-1}$, finite over each relatively compact open subset of $M$, with the following properties. Put $\mathfrak a_0=\mathfrak a$ and $E_ i=\sigma _ i^{-1}(C_ i)$, and, for $i\geq 0$, call the union of the strict transforms of $D,E_1,\ldots ,E_ i$ in $M_ i$ the accumulated divisor on $M_ i$. Each center $C_ i$ is contained in $V(\mathfrak a_{i-1})$ and has normal crossings with the accumulated divisor on $M_{i-1}$; the ideal

is holomorphic; and over each relatively compact open subset of $M$ the last of these ideals is the unit ideal. The accumulated divisors have simple normal crossings, and the composite is an isomorphism off $V(\mathfrak a)$.