I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the setup, we have a proper family of varieties $\pi : X \rightarrow D$, where $D$ is thought of as a small complex disk (or really its algebro-geometric analogue). All the fibers $\pi^{-1}(t)$ are smooth except the central fiber, which has a double point.
There is a monodromy action of the fundamental group of the punctured disk on the cohomology of the "generic fiber", which is actually only non-trivial in middle dimension, i.e. if $\dim X = n$ then the monodromy action is only non-trivial on $H^n$. See the linked page for the precise formula.
Anyway, my question is if there is some nice intuition for why the monodromy action takes this shape. Most sources that I can find online jump straight into very abstract formulations in the language of derived functors and perverse sheaves. For concreteness, here are some specific questions:
- Is there some "easy" way to see that the monodromy should act as a transvection?
- Is there some "easy" way to see that the action should be trivial except in middle dimension?
- Is there some "easy" reason that one would expect the monodromy formula to depend on the parity of the dimension?
I thought that I could see 2 by applying the Lefschetz Hyperplane Theorem to the family. The point is that I get a family $Y \subset X$ over $D$, such that the cohomology of $Y$ maps isomorphically to that of $X$ except in middle degree. If I choose $Y$ generally, then it will miss the singular point of $X$ and thus be a smooth family, so the monodromy action should be trivial. But this argument had nothing to do with the critical point of $\pi$ being nondegenerate, so maybe I'm missing something?
I would also appreciate reference recommendations that treat this subject and how it naturally evolves into the modern derived category formalism.
Best Answer
The fundamental observation is that the monodromy is given by twisting by vanishing cycles. A vanishing cycle is a homology class of the generic fibre that shrinks to zero in the special fibre. There is a unique (up to sign) such vanishing cycle attached to each node in the special fibre, which lies in the middle dimension, and the Picard–Lefschetz formula is precisely the statement that the effect of monodromy around the singular fibre is of twisting along vanishing cycles. If your cycle is disjoint from any vanishing cycles it isn't affected, otherwise it picks up a contribution corresponding to its intersection numbers with each vanishing cycle $\nu_i$:
$$ \gamma \mapsto \gamma + \sum_i \langle \nu_i, \gamma \rangle \nu_i. $$ (Note that one has to be careful with signs and orientations here to get the correct formula; see the references provided at the end.)
The easiest situation to visualise is of course in relative dimension 1. The following picture shows the twisting around a single ordinary quadratic singularity (where spurious self-intersections are shown in white). The vanishing cycle is depicted in red. (Click for larger version.)
It's not totally obvious to see that the blue cycle gets twisted by (minus) the red cycle, so the following topological view of the monodromy makes it more obvious (you can see it by flipping over one of the two sheets in the above picture).
The general situation is similar: the vanishing cycles appear in the middle dimension, and the same formula for the monodromy applies. There is just one subtlety is, as you noted: the behaviour is different in even relative dimension. The reason for this is that in even relative dimension, vanishing cycles have nonzero self-intersection (their self-intersection is $\pm 2$, depending on the dimension mod 4), yet are stable under the monodromy transformation. Because of this, the monodromy transformation has order 2, instead of infinite order.
A reference I recommend for a quick overview is Simon Donaldson's article Lefschetz Pencils and Mapping Class Groups, available here. The main reference for the Picard–Lefschetz formula in $\ell$-adic étale cohomology, which seems to interest you, is of course in SGA 7. See Deligne's exposés XIII, XIV and XV, in SGA 7 tome II, which use the formalism of nearby cycles to transfer computations to the special fibre.