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I am seeking recommendations for a concise introduction to the theory of $D$-modules suitable for an algebraic geometer. Specifically, I am interested in understanding:

  • The role of the characteristic cycle in the theory.
  • The analogies and relations between $D$-modules, $l$-adic cohomology, and ramification theory.

All in all, I would like to properly understand the first sentences of the introduction of this excellent paper of Tomoyuki Abe. Alternatively, a response explaining these two questions, rather than a reference, would also be greatly appreciated.

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    $\begingroup$ I don't know a good reference giving an introduction to the characteristic cycle in the theory of $D$-modules (this is entirely ignorance on my part and not a reflection of a dearth of references in the literature). However, once you find such a reference, the second part, understanding the analogy, should be relatively straightforward: $\endgroup$
    – Will Sawin
    Commented May 13 at 10:35
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    $\begingroup$ Essentially every result on the characteristic cycle of $\ell$-adic sheaves in the paper of Saito (or followup papers) will have a direct analogue in the theory of holonomic $D$-modules by more-or-less one-for-one translating each term to the other setting. The converse is not quite true - in particular the key statement that characteristic cycles have Lagrangian support is false in the $\ell$-adic setting in characteristic $p$, as Saito observes. $\endgroup$
    – Will Sawin
    Commented May 13 at 10:35
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    $\begingroup$ I think sciencedirect.com/bookseries/north-holland-mathematical-library/… addresses the first of these. $\endgroup$
    – Simon Wadsley
    Commented May 13 at 11:31
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    $\begingroup$ Also for the first part, there is a book by Hotta, Tankeuchi, Tanasaki on D-modules which is quite good. $\endgroup$
    – Donu Arapura
    Commented May 13 at 11:36

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