Manuel Blickle's Abstract


Principal Lecturers Participants Schedule Contact Utah Home		Manuel Blickle's Abstract TITLE: The geometry of arc spaces and motivic integration. I will outline in fair detail how one sets up a theory of motivic integration for a smooth complex variety. This is by no means the most generality in which motivic integration is defined but the smooth complex case serves as a very good model case to learn the heart of the theory (that is the geometric flavor of the theory). It is simple (and geometric) enough so that it is possible to present some/most key results with full proof, most notably the celebrated transformation rule of motivic integration. Secondly, I will describe some applications of motivic integration to the study of invariants in birtational geometry. These results shed light on how the geometry of the arc space contains information about the birational geometry of a variety. For example, certain types of singularities of a variety coming up in the minimal model program can be characterized in terms of the dimension/basic geometry of the jet-spaces over that variety. This connection can be used to study these invariants, most successfully it was applied to show a "inversion of adjunction" formula for the minimal log discrepancies. Concretely I have planned the following lectures: Recall/introduce definition and basic properties of the arc space, motivic measure and integral. Computation of the motivic integral in some simple examples. Mention (and motivation) of the key formula for the motivic integral of a function associated to a normal crossing divisor. The birational transformation rule. The birational transformation rule computed in an example (blowup of P² at a point) and an important case (blowup at a smooth center). Using the weak factorization theorem (hard) this essentially yields a full proof of the result already. Preparations for lecture 3: relative canonical divisor, Kaehler differentials and arcs. Elementary proof of the transformation rule (sketchy at times). This lecture is somewhat technical but provides a fairly detailed and fairly elementary proof of the transformation rule in the smooth case. Applications of motivic integration to log canonical threshold. This is a typical application of motivic integration which gives a characterization of the log canonical threshold (birational invariant) of a pair in terms of dimensions of the jet spaces. Some useful corollaries can be drawn from this characterization. Contact loci and valuations. Drawing from the ideas of motivic integration (but not really using motivic integration itself) Ein, Lazarsfeld and Mustata relate certain cylinders (multi-contact loci) in the arc space directly to the divisorial valuations over a smooth variety (this lecture will be more expository than the previous ones). References which I used for these talks and which might be useful for the audience to look at: -> Looijenga: Motivic Measure -> Mustata: Singularities of pairs via jet schemes. -> Ein, Lazarsfeld, Mustata: Contact loci in arc spaces. -> Blickle: Short course in geometric motivic integration.

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