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According to the article of Hauser:

The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html

The existence of resolution of complex analytic varieties was proven in:

  • Aroca, J.-M., Hironaka, H., Vicente, J.-L.: The theory of the maximal contact. Memorias Mat. Inst. Jorge Juan, Madrid 29 (1975).

  • Aroca, J.-M., Hironaka, H., Vicente, J.-L.: Desingularization theorems. Memorias Mat. Inst. Jorge Juan, Madrid 29 (1975).

Question 1. I would like to know if since then some alternative more simple proofs were found? And especially are there some (relatively) recent books, lecture notes or exposition articles covering this topic?

The result of Hironaka on resolution of algebraic varieties is now exposed pedagogically in the book of Kollar: "Lectures on resolution of singularities", but as far as I can judge (please correct me if I am wrong), Kollar does not treat the case of complex analytic varieties.

Question 2. Suppose I want to use this result on resolution of singularities of complex analytic varieties (and the answer to question 1 is NO). Should I cite these two articles, or cite nothing, pretending that everyone is aware of these two aricles? I have an impression that nowadays people tend not to cite these results on resolution unless they work exactly on this problem.

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Bierstone and Milman, Inventiones 1997, say that their proof works in the analytic case. I suspect Wlodarczyk's does as well, but I can ask him to check. –  Donu Arapura Jan 6 '11 at 22:46
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I mean I can check by asking... –  Donu Arapura Jan 6 '11 at 22:51
    
Donu, thanks! What about the second question? –  Dmitri Jan 6 '11 at 22:58
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Are you asking what I do? I tend to be sloppy and write "by Hironaka..." or "by resolution of singularities..". But you shouldn't follow my example, follow your conscience. –  Donu Arapura Jan 6 '11 at 23:15
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When in doubt, you can always toss a coin. Slightly more seriously: if you have a precise reference, why not give it? –  algori Jan 7 '11 at 0:04
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1 Answer

up vote 5 down vote accepted

I had a chance to check with Jarek Włodarczyk, who points out that the analytic case is not much harder (to him), but that it does require some extra care. In addition to the above references, the resolution of singularities of analytic spaces is treated in

  • Bierstone, Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. (1997)
  • Włodarczyk, Resolution of singularities of analytic spaces. Proceedings of Gökova Geometry-Topology Conference 2008
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Donu, thanks a lot for speaking to Włodarczyk! –  Dmitri Jan 13 '11 at 13:20
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