Birational maps and Nori motives

  • Noémie C. Combe

    Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
  • Yuri I. Manin

    Max-Planck-Institut für Mathematik, Bonn, Germany
  • Matilde Marcolli

    California Institute of Technology, USA
Birational maps and Nori motives cover

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The monograph by A. Huber and S. Müller-Stach [in: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, (2017), 207–232] contains a systematic exposition of Nori motives that were developed and studied as the "universal (co)homology theory" of algebraic varieties (or schemes), according to the prophetic vision of A. Grothendieck. Since then, some research was dedicated to applications of Nori motives in various domains of algebraic geometry: geometries in characteristic 1 (see the work of J. F. Lieber, Yu. I. Manin, and M. Marcolli [in: Facets of Algebraic Geometry. Vol. II, (2017), 147–227] and Yu. I. Manin and M. Marcolli [in: Dynamics: topology and numbers (202), 27–56]), absolute Galois group (see the article of Yu. I. Manin and M. Marcolli [SIGMA Symmetry Integrability Geom. Methods Appl. (2020), article no. 038]), persistence formalism (see Yu. I. Manin and M. Marcolli, [Math. Comput. Sci. (2020), 77–102]).

In this chapter, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel [Invent. Math. (2019), 415–432], where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.