Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$ and $f\colon X\rightarrow Y$ be a proper map. Assume that $Y$ is smooth.

I am interested in sufficient and necessary conditions for $f$ to be topologically locally trivial on the target, i.e., whether for each $y\in Y$ there exists an open neighborhood $V$ such that $f$ factors through a homeomorphism $f^{-1}(V){\cong} V\times f^{-1}(y)\rightarrow V$.

If $X$ is a smooth variety, the answer comes from Ehresmann's theorem: $f$ is topologically locally trivial on $Y$ if and only if it is a proper submersion. This also implies that $f$ is a smooth proper algebraic map.

But what if $X$ is singular? I could not find a reference.
Surely, $f$ must be topologically locally trivial on the $source$. Then I would like to know which tools may be used to patch together these local trivializations along a fiber, and to have a reasonable understanding of when this is not possible.


1 Answer 1


It seems that you are looking for a good notion of "equisingularity." About this I know quite little and I hope someone else will give a nice answer, but my guess is that a sufficient and necessary condition would be the existence of a horizontal Whitney stratification, at least locally on the source. Perhaps the book "Stratified Morse Theory" by Goresky and MacPherson can be of some help.

Another (necessary but perhaps insufficient) condition (at least if $Y$ is the germ of a curve) is the vanishing of vanishing cycles.

Regarding the second question (local to global topological triviality), see Corollary 6.14 in L. C. Siebenmann "Deformations of Homeomorphisms on Stratified Sets" Commentarii Mathematici Helvetici 47 (1972): a proper separated topological submersion whose fiber is stratifiable is in fact a fiber bundle.


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