Resolution of singularities of analytic spaces

Question

It seems to me that the following resolution of singularities theorem (or a modification) is known to specialists but I have trouble finding references.

Let $X$ be a complex analytic space, then there exists a sequence of blowing ups giving $\tau: \tilde{X}\to X$ such that

• $\tilde{X}$ is smooth, i.e., a complex manifold
• $\tau$ is propre, i.e., the preimages of compact subsets are compact.
• $\tau$ is an isomorphism above the smooth locus of $X$.
• The preimage for singular locus of $X$ forms a normal crossing divisor of $\tilde{X}$.

It is quite an analogue of Hironaka's resolution of singularities, but his main theorem lies in the algebraic context and according to his article, he only proved the dimension 3 case for analytic spaces.

I look for therefore a reference for the desingularization theorem for analytic spaces.

p.s., A similar question is the factorisation for rational maps. We know by the work of Abramovich, Karu, Matuski and Włodarczyk (https://www.ams.org/journals/jams/2002-15-03/S0894-0347-02-00396-X/home.html) that for any birational map $f: X-\to Y$ between smooth complex varieties, there exists blowing ups $\tilde{X}\to X$ and blowing downs $\tilde{X}\to Y$ making the diagram commutative. Does an analogue of this theorem hold in complex analytic cases for $X, Y$ complex manifolds and $f$ a bimeromorphic map defined in the sense of this page for example (https://encyclopediaofmath.org/wiki/Meromorphic_mapping)?

Answer 1

Wlodarczyk's Resolution of singularities of analytic spaces available here is exactly what you want:

Theorem 2.0.1: Let $Y$ be an analytic space defined over $\Bbb R$ or $\Bbb C$. There exists a canonical desingularization of $Y$ that is a manifold $\widetilde{Y}$ together with a proper bimeromorphic morphism $\operatorname{res}_Y: \widetilde{Y}\to Y$ such that

1. $\operatorname{res}_Y: \widetilde{Y}\to Y$ is an isomorphism over the nonsingular part $Y_{ns}$ of $Y$.
2. The inverse image of the singular locus $\operatorname{res}_Y^{-1}(Y_{sing})$ is a simple normal crossings divisor.
3. $\operatorname{res}_Y$ is functorial with respect to local analytic isomorphisms. For any local analytic isomorphism $\phi: Y'\to Y$ there is a natural lifting $\widetilde{\phi}:\widetilde{Y'}\to\widetilde{Y}$ which is a local analytic isomorphism.

There's also an embedded version of this as theorem 2.0.2.