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.



Proper family deformation retracts onto special fiber
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. 


