Applications of Thom's first isotopy lemma

Question

Thom's first isotopy lemma says that given a smooth map $f:M\to P$ between smooth manifolds, and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a submersion for any stratum $X$ of $S$, then $f|_S:S\to f(S)$ is a locally trivial fibration. Furthermore, $f|_X:X\to f(X)$ is a locally trivial fibration, for any stratum $X$ of $S$?

Kindly refer me to some works where the above lemma is used to prove fibrations in some concrete examples, which are not proven using some other method. I would like to see how the hypotheses in the lemma are checked in some explicit cases.

Answer 1

One way to view Thom's first isotopy lemma in algebraic geometry: it provides a way to turn local arguments into global geometry.

More precisely, suppose that you have a proper family $X \to S$ of complex varieties over a connected base. Assume that, etale-locally on the source and target, the family splits as a product $\sqcup_i U_i = \sqcup_i F_i \times V_i$, where $\sqcup_i U_i$ is an etale cover of $X$ and $\sqcup_i V_i$ is an etale cover of $S$.

There is a canonical Whitney stratification defined by Teissier (see his paper Varietes polaires. II. Multiplicites polaires, sections planes, et conditions de Whitney) whose definition can be seen to be etale local. In particular, if we use this stratification for $X$, then it follows from the local product structure that every stratum is submersion into $S$ (in fact smooth as a morphism of varieties). Therefore, the family $X \to S$ is locally topological trivial (and so in particular all fibers are homeomorphic).

This argument allows to use deformation theory to conclude global properties of the fibers of the family. For example, if the family $X \to S$ is flat and the singularities of the fibers are rigid (something that can be checked using homological computations), then the fibration is topologically trivial. An instance of this: Namikawa showed that Q-factorial symplectic singularities are rigid https://arxiv.org/abs/math/0506534, and so this type of argument would apply to certain relative moduli spaces of sheaves on a family of K3 surfaces.

As another example, in the paper https://arxiv.org/abs/2307.00350 the authors apply this type of argument to moduli spaces of principal bundles on a family of smooth projective curves, and to moduli spaces of sheaves on a family of del Pezzo surfaces.