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If $\mathbf{D}$ is the complex unit disc with coordinate function $s$ and $X \to \mathbf{D}$ is a proper flat holomorphic family (and it is smooth outside of the fiber $s=0$), will the total family $X$ deformation retract onto the fiber above $s=0$?

I don't believe this will be true, but I cannot find a counterexample. My idea was that if $X$ is a family of elliptic curves degenerating to a singular cubic, then this would fail (but, I think it actually works in this case!).

Edit: in the comments, the consensus is that this should be true - but we do not have a proof yet.

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  • $\begingroup$ Deligne (SGA 7 II, Exp. XIII, second paragraph of the introduction) suggests that something slightly stronger might be true. I'm sure it has been worked out since, but I don't know a reference. In any case, we know that in the algebraic case the inclusion of the special fiber induces an isomorphism on cohomology and fundamental groups... $\endgroup$
    – Piotr Achinger
    Commented Mar 18, 2017 at 10:56
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    $\begingroup$ This is true (i.e. $X$ does deformation retract onto the special fiber $s^{-1}(0)$). I unfortunately don't know a reference (and would also be interested in seeing one). $\endgroup$
    – John Pardon
    Commented Mar 19, 2017 at 0:47
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    $\begingroup$ If X is smooth, choose a metric and then flow by $\nabla |s|^2$ $\endgroup$
    – Vivek Shende
    Commented Mar 21, 2017 at 9:05
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    $\begingroup$ @VivekShende Perhaps I don't understand - isn't the point that the central fiber will have singularities? I'm not sure how such a flow would work in that case. $\endgroup$
    – improv305
    Commented Mar 24, 2017 at 1:24
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    $\begingroup$ If $X$ is smooth, use the flow mentioned by @VivekShende. In general, by resolution of singularities, one can find $\pi: X' \to X$, a proper birational morphism which is an isomorphism outside the special fibre. One can then define a retraction for $X$ by simply composing the flow on $X'$ with $\pi$. This clearly gives a well defined set-theoretic map and continuity follows from the properness of $\pi$. $\endgroup$
    – naf
    Commented Mar 26, 2017 at 6:23

1 Answer 1

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Here is the reference:

Persson, Ulf, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189.

Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. 2, 215-290.

I found the reference here http://web.math.ucsb.edu/~drm/papers/clemens-schmid.pdf also worth reading

Morrison, David R. The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), 101-119, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984.

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