Theorem 9.6.6 (Resolution of singularities)4CEF0578EFE0

Theorem 9.6.6 (Resolution of singularities).4CEF0578EFE0 Every reduced complex space $X$ has a proper bimeromorphic map $\pi \colon M\to X$ from a complex manifold $M$ that is a composite of blowups with smooth centers and an isomorphism over $X_{\mathrm{reg}}$. The divisor formed by the inverse images in $M$ of the blowup centers has simple normal crossings. The sequence of blowups is finite when $X$ is compact, and finite over each relatively compact open subset of $X$ in general.