Is there a good way to think of vanishing cycles and nearby cycles?

Question

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vanishing cycles, nearby cycles, and specialization? I have a decent idea how some of it works for studying the cohomology of a one-parameter flat family of degenerating complex manifolds (see below), but the general sheaf picture still gives me a headache. Any recognition principles (e.g., "this looks like a place where I can use a vanishing cycle argument") would be most welcome.

Say I have a family of complex manifolds where the fibers are smooth over a punctured unit disc, and have some mild singularities over zero (a priori the singularities could be arbitrarily bad, but say we blow up until we have simple normal crossings). The cohomology of the fibers forms a vector bundle on the punctured disc, and it comes equipped with some extra structure, such as a pure Hodge filtration and a Gauss-Manin connection that identifies nearby fibers. When we attempt to extend the vector bundle over the whole disc, the extra structures degenerate - the Hodge structure becomes "mixed", and the connection acquires logarithmic singularities. These structures aren't immediately relevant to this question, but they seem to be interesting.

As far as I can tell, vanishing cycles and nearby cycles arise when we try to relate the cohomology of smooth fibers X_t with that of the special fiber X₀. Each smooth fiber X_t has an inclusion map to the total space X, and X is homotopy equivalent to X₀ by a fiberwise retraction. The composition yields a map from X_t to X₀, and the pushforward of a sheaf on X_t along this map yields the nearby cycles sheaf. When I start with the constant sheaf on X_t this yields a sheaf on X₀ that computes cohomology of X_t for some abstract nonsense reason. So far, I'm okay, but it seems that choosing t is not canonical enough, so one replaces X_t with the homotopy equivalent universal fiber X_oo over the universal cover of the punctured disc (the upper half plane), and defines nearby cycles by some crazy pullback-pushforward-pullback sequence. Specialization and vanishing cycles seem to be similar - I think there is a nice geometric picture somewhere, but the proliferation of upper and lower stars makes me sad. Is there a good way to see through that thicket?

Answer 1

In general I don't think there's anything easy about nearby and vanishing cycles. However, I tend to find it enlightening to just consider their topology. Namely, if $f:X \to \Bbb C$ is a function on a complex algebraic (or analytic) variety, then the stalk cohomology of the nearby cycles functor applied to some complex of sheaves $F$ at a point $x \in f^{-1}(0)$ is simply the the cohomology (with respect to $F$) of the Milnor fiber of $f$ at $x$. Given the utility of Milnor fibers in studying the topology of singular spaces, I think this is good motivation for the nearby cycles functor: in some sense it bundles the information of the all the Milnor fibers together into a single sheaf.

Now you can also use lots of neat results about Milnor fibers to actually say things about these stalk cohomology groups and their monodromy. For example, if f has an isolated singularity, then the singular Milnor fiber is homotopic to a bouquet of n-spheres, where n is given by the so-called Milnor number. The monodromy can often be calculated with the help of the Thom-Sebastiani theorem, which says that if $f = g + h$ (where $g,h$ are functions on distinct subvarieties of $X$ such that their sum makes sense as a function on $X$) then the monodromy of the Milnor fiber with respect to $f$ is the tensor product of the monodromies with respect to $g$ and $h$.

If we're also interested in vanishing cycles, then we can give a similarly topological description of the stalk cohomology groups. Letting $i$ denote the inclusion of $f^{-1}(0)$ into $X$, we have the exact triangle with the specialization map $i^*F$ to the nearby cycles of $F$ and the canonical map from the nearby cycles of $F$ to the vanishing cycles of $F$ (although perhaps it shouldn't be called canonical because it's the cone of the specialization map). Anyway, looking at the associated long exact sequence provides a topological description of the vanishing cycles (in particular if $f$ is smooth then the vanishing cycles is zero and we get an isomorphism between $i^*F$ and the nearby cycles of $f$).

I wish I could say more about the kind of recognition principle you're looking for, but the only place that I've encountered nearby cycles is in Springer theory, where the nearby cycles of the adjoint quotient map applied to the constant sheaf is isomorphic to the pushforward of the constant sheaf under the Springer resolution (and hence the monodromy action determines the action of the Weyl group on the cohomology of Springer fibers). Perhaps in this case the thing to notice is that the nilpotent cone is singular and also the fiber over zero of the adjoint quotient map, which makes it seem like a potentially good candidate for nearby cycles arguments?

Answer 2

I liked the mentioned article by Massey very much too, but I guess the best descriptions are still Deligne's SGA 7, part II, chapters 13, 14, where he even draws some figures (BTW, my impression from those chapters and some formulations in other articles by Deligne is that he thought about generalisations, apparently never published). I found Wall: "Periods of Integrals and Topology of Algebraic Varieties", Griffiths "summary" and his "Periods III" very helpful, parts of this motivic book too. Arthur Ogus studies vanishing/nearby cycles in the context of log-geometry. Related to the theme is the local monodromy theorem, about that, monodromy-weight conjecture etc., I found Illusie's article in Asterisque 223 very good.

Answer 3

I also have a lot of difficulty seeing what's going on, and am no expert, so take this with a grain of salt.

Here's one small picture (which you might already know) that I found helpful: Massey's description of "vanishing cycles at angle $\theta$" (see https://arxiv.org/abs/math/9908107, around page 23) which gives geometric intuition for the passage from $X_{\infty}$ to its universal cover. Namely, restrict your family to the segment where $t$ is in $e^{i \theta} [0,\epsilon]$, i.e., a ray emanating from the origin in the complex plane; then proceed as you did before, except no crazy covers needed, because your base is now contractible. This gives a functor (isomorphic to nearby cycles for any fixed \theta) together with an action of monodromy.

Answer 4

One thing I understand is that vanishing cycles are more than just about singularities — there's a derived version that is more interesting. I'd like to get an answer myself. This is also important for physicists, e.g. hep-th/0605206