nLab
motivic integration

Contents

Contents

Idea

What is called motivic integration is an upgrade of p-adic integration to a geometric integration theory obtained by replacing the p-adic integers by a formal power series ring over the complex numbers, and more generally by henselian discretely valued fields of residual characteristic zero.

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. From physical side a pioneer of both subjects is also Batyrev.

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

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

Original articles include the following:

  • Jan Denef, François Loeser, Definable sets, motives and pp-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.

  • Manuel Blickle, A short course on geometric motivic integration, math.AG/0507404

  • 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 2SL_2, math.LO/0609115; Integration in valued fields, in Algebraic geometry and number theory, 261–405, Progress. Math. 253, Birkhäuser Boston, pdf

  • Julia Gordon, Yoav Yaffe, An overview of arithmetic motivic integration, arxiv/0811.2160

  • Thomas C. Hales, What is motivic measure?, math.LO/0312229

  • 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, Motivic integration for dummies, pdf, A course on motivic integration I, II, III

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

  • Lou van den Dries, Lectures on Motivic Integration , Ms. University of Illinois at Urbana-Champaign. (dvi)

  • Julien Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (2004), no. 1, 1–54, MR2005e:14017

  • Takehiko Yasuda, Motivic Integration over Deligne-Mumford Stacks , arXiv.0312115 (2004). (pdf)

  • M. 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. M. 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

  • Karen Smith, Motivic integration, (pdf)

Last revised on September 22, 2018 at 08:04:20. See the history of this page for a list of all contributions to it.