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It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that:

For a proper flat map $f \colon X \rightarrow \Delta$, where

  • $X$ is a complex algebraic variety, and
  • $\Delta$ is the unit disk (say), and
  • $f$ is smooth over the complement of $0 \in \Delta$ and $f^{-1}(0)$ is singular,

the total space $X$ deformation retracts onto the singular fiber $f^{-1}(0)$.

I would like to see a proof of this fact.

At first I thought that this should be "easy", but after struggling with it a while the best I have been able to come up with is an argument using a refined form of resolution of singularities for complex varieties. However, this fact seems to be used in work that predates resolution of singularities (e.g. Lefschetz's early work on Lefschetz pencils), so there must be a simpler way to go about it. So I would prefer to see a proof that doesn't use heavy machinery, but if machinery offers a significant improvement then that is great too.

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    $\begingroup$ Are you sure that it holds for arbitrary singularities of $X_0:=f^{-1}(0)$? I always see this statement in the case where $X_0$ has just ordinary double points, and in this case the proof is essentially based on the fact that we can use $|f|$ as a Morse-function on $X \setminus X_0$ in order to provide the desired retraction. $\endgroup$
    – Francesco Polizzi
    Commented Dec 15, 2015 at 9:31
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    $\begingroup$ To be more precise, together with the Morse-function argument we must also use the fact that the pair $(X, \, X_0)$ is triangulable, so $X_0$ is a deformation retract of arbitrary small neighborhoods in $X$, see [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter III, Section 14]. $\endgroup$
    – Francesco Polizzi
    Commented Dec 15, 2015 at 9:35
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    $\begingroup$ There is an extension of Ehresmann's theorem to the case that $X$ is a Kaehler manifold that is proper over $\Delta$ and the central fiber $X_0$ is a simple normal crossings divisor in $X$. This is one step in the construction of the Clemens-Schmid exact sequence. In his lecture in "Topics in Transcendental Algebraic Geometry" (the reference I have at hand), David Morrison refers the reader to the following: Clemens, H., "Degenerations of Kaehler manifolds", Duke Math. J., 44 (1977), pp. 215-290; Persson, U., "Degenerations of algebraic surfaces", Mem. AMS 189 (1977). $\endgroup$
    – Jason Starr
    Commented Dec 15, 2015 at 14:53
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    $\begingroup$ I am not sure that it holds for arbitrary singularities, but see for instance "Proposition A" of projecteuclid.org/download/pdf_1/euclid.mmj/1029005705 for an example of a stronger assertion than in the case of double points. This leads to the same reference of Clemens that Jason Starr pointed out, but that paper seems to use Hironaka's theorem as well. $\endgroup$
    – user84144
    Commented Dec 15, 2015 at 16:35
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    $\begingroup$ By the way, in discussion with some friends we convinced ourselves that it should follow from the fact that arbitrary complex varieties are locally contractible. This leads to the argument I alluded to above using resolution of singularities, but feels a little unsatisfying. $\endgroup$
    – user84144
    Commented Dec 15, 2015 at 16:47

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