nLab motivic integration

Idea

Motivic integration attaches to certain subsets of arc schemes of varieties an element in a Grothendieck ring of varieties in a manner close in spirit to Euler characteristics. Technically it may be viewed as one of generalizations of p-adic integration to a geometric integration theory obtained by replacing the p-adic integers by a more broad class of rings. Some examples include formal power series ring over the complex numbers, and more generally any henselian discretely valued fields of residual characteristic zero. Recently, these cases have been generalized to the case of any complete discrete valuation ring (whose residue field is perfect in the mixed characteristic case).

In all of the described cases, the idea is to construct a suitable scheme over the residue field, called the arc scheme which serves as a suitable measure space on which functions are integrated; however, the value of integrals are not real numbers, but instead they are valued in the Grothendieck ring of varieties. In this sense, the value of a “motivic” integral is often regarded as a geometric object, rather than a numerical value.

The notion of arc scheme has a further generalization, known as a Greenberg scheme.

History

Motivic integration was introduced in the talk of Maxim Kontsevich at Orsay in 1995 in order to prove that Hodge numbers of Calabi-Yau manifolds are birational invariants. This talk also dealt with orbifold cohomology as well as 2 related papers of Lev Borisov. The orbifold cohomology has been continued by Weimin Chen, Yongbin Ruan and collaborators, and later also by algebraic geometers Abramovich, Vistoli, and others. Batyrev is another contributor to both subjects in the context of physics.

Later, a more general framework of motivic integration in model theory has been put forward by Denef and Loeser, partly based on Denef’s work on $p$-adic integration. More recent work using a model-theoretic approach is by Hrushovski and Kazhdan.

Related concepts

• p-adic integration

• motivic cohomology

• arc space

• Greenberg scheme

References

A textbook account is in

• Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, Motivic integration, Progress in Mathematics 325 Birkhaeuser 2018 (doi:10.1007/978-1-4939-7887-8)

See also

• Wikipedia Motivic integration

Lectures include

• Karen Smith, Motivic integration, (pdf)
• Manuel Blickle, A short course on geometric motivic integration, math.AG/0507404
• Raf Cluckers, Motivic integration for dummies, pdf, A course on motivic integration I, II, III
• Julia Gordon, Yoav Yaffe, An overview of arithmetic motivic integration, arxiv/0811.2160
• Lou van den Dries, Lectures on motivic integration , Ms. University of Illinois at Urbana-Champaign. (dvi)
• Thomas C. Hales, What is motivic measure?, math.LO/0312229

The concept originates from

• M. Kontsevich, lecture on motivic integration, Orsay, December 7, 1995.

Original articles include the following:

• Jan Denef, François Loeser, Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc. 14 (2001), no. 2, 429–469, doi

• Jan Denef, François Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields Proc. ICM, Vol. II (Beijing, 2002), 13–23, Higher Ed. Press, Beijing, 2002.

• R. Cluckers, F. Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), 23–121 math.AG/0410203

• Jan Denef, François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232.

• D. Abramovich, M. Mariño, M. Thaddeus, R. Vakil, Enumerative invariants in algebraic geometry and string theory, Lectures from the C.I.M.E. Summer School, Cetraro, June 6–11, 2005. Edited by Kai Behrend and Marco Manetti. LNIM 1947, Springer 2008. x+201 pp.

Hrushovski and Kazhdan studied motivic integration from model theory point of view:

• Ehud Hrushovski, David Kazhdan, Motivic Poisson summation, arxiv/0902.0845

• Ehud Hrushovski, David Kazhdan, The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL_2$, math.LO/0609115; Integration in valued fields, in Algebraic geometry and number theory, 261–405, Progress. Math. 253, Birkhäuser Boston, pdf

• David Kazhdan, Lecture notes in motivic integration, with intro to logic and model theory, pdf

• R. Cluckers, J. Nicaise, J. Sebag (Editors), Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, 2 vols. London Mathematical Society Lecture Note Series 383, 384

• Raf Cluckers, Julia Gordon, Immanuel Halupczok, Motivic functions, integrability, and uniform in p bounds for orbital integrals, arxiv:1309.0594

• Julien Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132:1 (2004) 1–54, MR2005e:14017

• Takehiko Yasuda, Motivic integration over Deligne-Mumford stacks (2004) arXiv:math/0312115

• Michael Larsen, Valery Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259, math.AG/0110255, MR2005a:14026, journal; Rationality criteria for motivic zeta functions, Compos. Math. 140 (2004), no. 6, 1537–1560, math.AG/0212158

• Alexei Bondal, Michael Larsen, Valery Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 2004, no. 29, 1461–1495, math.AG/0401009

• Emmanuel Bultot, Motivic integration and logarithmic geometry, PhD thesis arxiv/1505.05688