This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.

Let $X$ be a smooth proper variety (over $\mathbb{C}$). Let $\pi:\mathcal{F} \to B$ be a smooth proper morphism with $B$ connected s.t. for some $b \in B$, the fiber $\mathcal{F}_b :=\pi^{-1}(b) \cong X$. By Ehresmann's theorem $\mathcal{F} \to B$ restricts to a fiber bundle on the smooth locus and automatically it follows that all the nearby fibers to $\mathcal{F}_b$ are diffeomorphic to $X$ (in particular of the same topological type).

This kind of thing fails when $X$ is singular and to my understanding this is one possible motivation for introducing the notion of an "equisingular family". As far as I understand the notion of an "equisingular family of varieties" should (in very hand-wavy terms) correspond to the idea of a family of singular varieties varying in a fixed topological type.

Naive (stupid) definition: A proper flat family $\mathcal{F} \to B$ with $B$ reduced variety is called equisingular iff over a zariski dense subset of $B$ the induced map on $\mathbb{C}$-points is a topological fibre bundle.

This is the most naive definition I could come up with and hence it is probably wrong. However even if these are the kind of objects equisingular families should be there's still a big problem with the definition and that's the requirement that $B$ be reduced! By restricting to reduced schemes we give up on any hope of having a deformation theory for the study of those equisingular families. And without deformation theory this seems to me like an intractable problem in general.

There are several inequivalent definitions in the literature for "equisingular family" and usually one assumes that the singularities are isolated. Some definitions I saw to the notion of equisingular family of curves for instance relied on deforming an entire tower of blow ups together with corresponding sections.

Question: Is there an accepted general definition of an "equisingular family of varieties" $\mathcal{F} \to B$ (maybe should be equipped with some extra data) which satisfies that:

  1. Defined for reasonably general $\mathcal{F}$ and $B$ (no reducedness, isolated singularities assumptions or restrictions on dimensions etc...)
  2. When $B$ is reduced there's a zariski dense locus for which $\mathcal{F}(\mathbb{C}) \to B(\mathbb{C})$ is a topological fibre bundle.
  3. For any reasonable algebraic variety $X$ the notion naturally gives rise to a defomration functor (functor on artin rings) of equisingular formal deformations of $X$.
  4. For $(X,x)$ a germ of an isolated singularity the singularity is simple iff the associated deformation functor from (3) is trivial. I.e. $(X,x)$ is a simple singularity iff it has no (equisingular-)moduli (or roughly - "the topological type determines the algebraic structure", for germs of plane curve singularities for instance one expects to find the $ADE$ classification etc...).

If this is a hopeless dream I'd like to understand what are the major obstacles to coming up with this kind of notion.


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