Reference request: the geometry of vanishing cycle I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction and model problems (for example, Oda’s  A Note on Ramification of the Galois Representation on the Fundamental Group of an Algebraic Curve.)(However, I’m also a beginner on these problems.) I want to learn more about nearby cycles and vanishing cycles, but the classical reference SGA7 is not available as reading French is quite difficult for me (especially reading something new). I have learned the elementary definition and motivation. So are there any textbook, online notes or videos on this topic, with enough detail?
 A: Beyond the definitions in section 9.2 of Lei Fu's book, you also find a finiteness result for $R^q\psi\mathcal{F}$ for a finite type $S$-scheme and constructible $\mathcal{F}$ (9.4.1). Nearby cycles, as well as their finiteness feature in the proof of finiteness for $R^q f_{\ast}\mathcal{F}$, where $f$ is a morphism of finite type $S$-schemes and $\mathcal{F}$ is torsion constructible, in (9.5.1).
I guess you are aware of this. Then, I would say that this is enough for the algebraic formalism. It is better to see them in action.
Since you presumably want to avoid French, and mention models and reduction, you may want to have a look at this paper by Takeshi Saito : Vanishing Cycles and Geometry of Curves Over a Discrete Valuation Ring. There, he proves that a local analogue of the stable reduction theorem (of Grothendieck, Deligne-Mumford): for $X/S$ a normal relative curve with an isolated singularity at a closed point $x$ of the special fiber and $Y$ a regular minimal model of $X$ (defined in the paper), the special fiber of $Y$ is a normal crossing divisor of $Y$ if and only if the action of inertia on the stalk $R\psi^1(\mathbb{Q}_{\ell})_x$ is unipotent.
As you can imagine, nearby cycles feature prominently in the proof.
A: Luc Illusie has written two great expository texts on étale vanishing cycles and other topics in SGA7, one in English (with an associated set of slides) and one in French (unfortunately).
Illusie has also written two interesting articles about more advanced topics on étale vanishing cycles, which may still be valuable to you if you want to understand some recent developments: one on the Thom-Sebastiani theorem and one on vanishing cycles over higher-dimensional bases
