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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?

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There are plenty of references on the transcendental side. But since you seem to be arithmetically inclined, you might find chap III of Freitag-Kiehl helpful. –  Donu Arapura Feb 20 '11 at 17:42
    
Why, thanks for the tip! Sadly, of the two local copies of Freitag-Kiehl, one is on extended loan, and the other is "missing" ... –  jvo Feb 20 '11 at 18:44
    
Zoladek's "The monodromy group" contains a very good decription. Review: ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/… –  Thomas Riepe Mar 1 '11 at 16:08

3 Answers 3

up vote 37 down vote accepted

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 $H^*(X_0)$.

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Excellent answer, thanks! –  jvo Feb 20 '11 at 19:53
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How can I vote this up more than once? –  David Steinberg Feb 21 '11 at 8:59

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.

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Great idea, thanks! I've only ever looked at the last few sections of that paper, I suppose I should have read it all ... –  jvo Feb 21 '11 at 9:50

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.

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Awesome, thanks! –  jvo Feb 21 '11 at 9:31
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"pretty good use" is a nice euphemism :) –  YBL May 11 '11 at 23:33

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