I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, my questions are:

  1. Which papers and results are the most important for the theory of motivic integrations?
  2. What are the main applications of motivic integration?
  3. Would it be wise to avoid all quantifier elimination and related things in the course?
up vote 21 down vote accepted

As for the first two questions (papers, results, and applications): For motivation, I'd recommend understanding the content of Batyrev's paper "Birational Calabi-Yau n-folds have equal Betti numbers" which proves the claim in its title. Using motivic integration techniques analogous to the $p$-adic techniques in Batyrev's paper, Kontsevich proved that $K$-equivalent varieties have the same Hodge numbers, which was the original motivation for motivic integration. Strictly speaking, one may deduce this result without motivic integration, using only $p$-adic methods as in Batyrev. But the proof with motivic integration is much easier. Here are some nice notes on this approach, by Manuel Blickle.

Aside from this sort of thing, the applications of motivic integration I'm aware of have the following flavor: one takes a "numerical" generating function for some invariants of geometric data (e.g. the $p$-adic Igusa zeta function or topological zeta function) and replaces them with motivic versions which specialize to the numerical zeta function on applying some homomorphism out of the Grothendieck ring of varieties. (Sometimes this latter feature is not quite satisfied, and the motivic versions are only analogues of the numerical versions). This allows one to formulate (and sometimes prove) stronger versions of features of the original numerical zeta functions. For example, see this paper of Denef and Loeser. (They have written many interesting papers on these subjects.) This paper of Chambert-Loir and Loeser is a cool example of a rather different kind.

I'm not an expert, so I may have missed some flavors of applications. (There are many interesting "motivic invariants" which I haven't talked about, e.g. "motivic Milnor fibers" which are related to the Zeta functions studied by Denef and Loeser above; the Kapranov motivic zeta function, which is a motivic analogue of the Weil zeta functions; and "motivic characteristic classes," but as far as I know the Kapranov zeta function and "motivic characteristic classes" haven't really been studied via motivic integration techniques.)

Here are some other references I like: this survey of Looijenga gives a slightly more sophisticated (e.g. equivariant etc.) version of some of these motivic invariants. This is an excellent reference on the Grothendieck ring of varieties by Mustata, though the proof of Theorem 3.1 is not correct for curves with no rational points. (The issue is that the map $\operatorname{Sym}^n(C)\to \operatorname{Pic}^n(C)$ is not a bundle of projective spaces in this case, but rather a Severi-Brauer variety over $\operatorname{Pic}^n(C)$; as far as I know a correct proof of Theorem 3.1 does not appear in the literature.) There are also plenty of generally cool papers about the Grothendieck ring of varieties, e.g. Poonen's "The Grothendieck Ring of Varieties is not a Domain", this paper of Liu and Sebag, and this great counterexample of Larsen and Lunts.

As for your last question, my advice would be to avoid the model-theoretic language for now (but take this advice with a grain of salt--my background in model theory is very weak and yours may very well be stronger). Unfortunately lots of cool papers (some of the Denef and Loeser stuff and all of Hrushovski's stuff) are written in this language. My understanding of the advantage of the model-theoretic language is as follows: motivic integrals as described in the references I've given are valued in certain completions of the Grothendieck ring of varieties or its localizations; the model-theoretic language allows us to obtain values in the non-completed rings. (Also I hear that the model-theoretic simplifies some recent work connecting non-archimedean geometry and motivic integration.) That said, I don't believe any of the applications I've mentioned rely on these advantages. Also, I think one can find expositions of all the results above which avoid the model-theoretic language; for example, Hrushovski's "motivic Poisson summation" is described in the paper of Chambert-Loir and Loeser I link to above.

Good luck with your reading course!

  • Thank you very much for such a comprehensive answer!! I decided not to read such a course now; yet this information could be very useful in future. – Mikhail Bondarko Sep 15 '13 at 13:10

Other very significant applications have to do with 'transfer theorems' of statements between local fields of positive and zero characteristic, in particular a different proof that the "fundamental lemma", as proved by Ngo in positive characteristic, is valid in characteristic zero: the first proof is by Waldspurger, but it can be proved using the (exponential) motivic integration theory of Cluckers and Loeser (see a survey of Cluckers, Hales and Loeser, e.g. on Cluckers's homepage; the foundational papers of Cluckers and Loeser are also there).

Even more recent applications of these ideas (due in particular to Cluckers, Gordon and Halupczok) involve transfering integrability statements and have applications to harmonic analysis over local fields. (Interestingly, in some cases, the transfer goes in the other direction here: from known results in characteristic zero to results in positive characteristic).

These results use a fair amount of model theory (in particular things known as the Denef-Pas language, and Presburger arithmetic).

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