Vanishing cycles in a nutshell?

Question

To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language of derived categories), in particular as it applies to the study of certain integral models of curves?

Answer 1

For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex nonsingular plane cubics $X_t$ degenerating to a nodal cubic $X_0$. It is possible to understand the change in topology rather explicitly.

You can set up a basis of (real) curves $\alpha_t,\beta_t\in H_1(X_t,\mathbb{Z})$, so that $\beta_t\to \beta_0\in H_1(X_0)$ and $\alpha_t\to 0$ as $t\to 0$. So that $\alpha_t$ literally is a vanishing cycle. There is more. As you transport these cycles around a loop in the $t$-plane, you end up with a new basis $T(\alpha_t),T(\beta_t)$. This is related to the old basis by the Picard-Lefschetz formula $$T(\alpha_t)=\alpha_t$$ $$T(\beta_t) = \beta_t \pm (\alpha_t\cdot\beta_t)\alpha_t$$ You visualize this by cutting along $\alpha_t$ and giving a twist before regluing.

You can jazz this up in various ways of course, and this is the modern theory of vanishing cycles. In modern notation you'll see a nearby cycle functor $R\Psi$, which corresponds roughly to $H^*(X_t) = H_*(X_t)^*$ and a vanishing cycle functor $R\Phi$ which measures the difference between $H^*(X_t)$ and $H^*(X_0)$.

Answer 2

Another good summary of vanishing cycles and Lefschetz pencils is found in Sections 4 and 5 of P. Deligne's "La conjecture de Weil, I" (available online). He discusses briefly both the theory over the complex numbers and the version in étale cohomology; and then immediately proceeds to putting these to pretty good use.

Answer 3

I recommend that you read the first few sections of Ribet's article in Inventiones 100 (the one in which he proves that modularity of elliptic curves implies FLT). In these three or four sections he summarizes a large number of the results from SGA VII, in so far as they apply to the case of curves with semistable reduction.