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Two-Valued Groups, Kummer Varieties, and Integrable Billiards

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Abstract

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of \(\sigma \)-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable.

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References

  • Arnol’d, V.: Mathematical methods of classical mechanics. Springer, New York (1978)

    Book  Google Scholar 

  • Audin, M.: Spinning tops. An introduction to integrable systems, Cambridge studies in advanced mathematics 51 (1999)

  • Audin, M.: Courbes algebriques et syst‘emes integrables: geodesiques des quadriques. Expo. Math. 12, 193–226 (1994)

    MATH  Google Scholar 

  • Baker, H.F.: On the hyperelliptic sigma functions, Am. J. Math. 301–384 (1898)

  • Buchstaber, V.M., Novikov, S.P.: Formal groups, power systems and Adams operators. Mat. Sb. (N. S) 84(126), 81–118 (1971). (in Russian)

    MathSciNet  MATH  Google Scholar 

  • Buchstaber, V.M., Rees, E.G.: Multivalued groups, their representations and Hopf algebras. Transform. Groups 2, 325–249 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Buchstaber, V.M., Veselov, A.P.: Integrable correspondences and algebraic representations of multivalued groups, Internat. Math. Res. Notices, 381–400 (1996)

  • Buchstaber, V.M.: Functional equations that are associated with addition theorems for elliptic functions, and two valued groups, Uspekhi Math. Nauk 45 (1990) No 3 (273) 185–186 (Russian) English translation. Russian Math. Surveys 45(3), 213–215 (1990)

  • Buchstaber, V.M.: n-Valued groups: theory and applications. Moscow Math. J. 6, 57–84 (2006)

    MathSciNet  MATH  Google Scholar 

  • Buchstaber, V.M.: Polynomial dynamical systems and the Korteweg-de Vries equation. Proc. Steklov Inst. Math. 294, 176–200 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Buchstaber, V.M., Mikhailov, A.V.: Infinite-dimensional lie algebras determined by the space of symmetric squares of hyperelliptic curves. Funct. Anal. Appl. 51(1), 4–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Buchstaber, V.M., Enolskii, V.Z., Leikin, D.V.: Kleinian functions, hyperelliptic Jacobians and applications., Reviews in Mathematics and Math. Physics, Krichever, I.M., Novikov, S.P. Editors, v. 10, part 2, Gordon and Breach, London, 3–120 (1997)

  • Buchstaber, V.M., Enolskii, V.Z., Leikin, D.V.: Multi-dimensional sigma-functions. arXiv:1208.0990 [math-ph] (2012)

  • Buchstaber, V.M., Leykin, D.V.: Addition laws on Jacobian varieties of plane algebraic curves. Proc. Steklov Math. Inst. 251(4), 49–120 (2005)

    Google Scholar 

  • Buchstaber, V.M., Rees, E.G.: The Gelfand map and symmetric products. Selecta Math. (N.S.) 8(4), 523–535 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Darboux, G.: Principes de géométrie analytique, p. 519. Gauthier-Villars, Paris (1917)

    MATH  Google Scholar 

  • Donagi, R.: Group law on the intersection of two quadrics Ann. Sc. Norm. Sup. Pisa, 217–239 (1980)

  • Dragović, V.: Poncelet-Darboux curves, their complete decomposition and Marden theorem. Internat. Math. Res. Notes 2011, 3502–3523 (2011)

    MathSciNet  MATH  Google Scholar 

  • Dragović, V.: Multi-valued hyperelliptic continous fractions of generalized Halphen type. Internat. Math. Res. Notices 2009, 1891–1932 (2009)

    MATH  Google Scholar 

  • Dragović, V.: Generalization and Geometrization of the Kowalevski top. Comm. Math. Phys. 298(1), 37–64 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Kukić, K.: Discriminantly separable polynomials and quad-graphs. J. Geom. Mech. 6(3), 319–333 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Kukić, K.: Systems of the Kowalevski type and discriminantly separable polynomials. Regul. Chaotic Dyn. 19(2), 162–184 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Kukić, K.: The Sokolov case, integrable Kirchhoff elasticae, and genus 2 theta-functions via discriminantly separable polynomials. Proc. Steklov Math. Inst. 286, 224–239 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Geometry of integrable billiards and pencils of quadrics. J. Math. Pures Appl. 85, 758–790 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms. Adv. Math. 219, 1577–1607 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Poncelet porisms and beyond: integrable billiards, hyperelliptic Jacobians and pencils of quadrics, Frontiers in Mathematics, Birkhauser/Springer, 294, ISBN 978-3-0348-0014-3 (2011)

  • Dragović, V., Radnović, M.: Bicentennial of the Great Poncelet Theorem. Bull. Am. Math. Soc. 51, 373–445 (2014)

    Article  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Pseudo-integrable billiards and arithemetic dynamics. J. Modern Dyn. 8(1), 109–132 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Periods of pseudo-integrable billiards. Arnold Math. J. 1(1), 69–73 (2015). https://doi.org/10.1007/s40598-014-0004-0

    Article  MathSciNet  MATH  Google Scholar 

  • Dragović, V., Radnović, M.: Pseudo-integrable billiards and double reflection nets. Russian Math. Surv. 70(1), 3–34 (2015). https://doi.org/10.4213/rm9648

    MathSciNet  MATH  Google Scholar 

  • Dragović, V., Gajić, B.: Some recent generalizations of the classical rigid body systems. Arnold Math. J. 2(4), 511–578 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Dubrovin, B.A., Novikov, S.P.: A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry (Russian) Dokl. Akad. Nauk SSSR 219 (1974), 531-534, English translation: Soviet Math. Dokl. 15 (1974), no. 6, 1597–1601 (1975)

  • Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants and multidimensional deteminants. Birhauser, Boston (1994)

    Book  MATH  Google Scholar 

  • Golubev, V.V.: Lectures on the integration of motion of a heavy rigid body around a fixed point, Gostechizdat, Moscow, 1953 [in Russian] Israel program for scintific literature, English translation (1960)

  • Griffiths, Ph, Harris, J.: Principles of Algebraic Geometry. Wiley Classics library edition (1994)

  • Knörrer, H.: Geodesics on the ellipsoid. Invent. Math. 59, 119–143 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Kowalevski, S.: Sur la probleme de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889)

    Article  MathSciNet  MATH  Google Scholar 

  • Moser, J., Veselov, A.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Narasimhan, M., Ramanan, S.: Moduli of vector bundles on compact Riemann surfaces. Ann. Math. 2(89), 14–51 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  • Tyurin, A.N.: On intersection of quadrics. Russian Math. Surv. 30, 51–105 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Veselov, A.: Growth and integrability in the dynamics of mappings. Commun. Math. Phys. 145, 181–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewers for their helpful remarks. The research of one of the authors (V. D.) was partially supported by the Serbian Ministry of Education, Science, and Technological Development, Project 174020 Geometry and Topology of Manifolds, Classical Mechanics, and Integrable Dynamical Systems.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street 8, 119991, Moscow, Russia

    V. M. Buchstaber

  2. Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Rd, Richardson, TX, 75080, USA

    V. Dragović

  3. Mathematical Institute SANU, Belgrade, Serbia

    V. Dragović

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  1. V. M. Buchstaber
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  2. V. Dragović
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Correspondence to V. Dragović.

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Buchstaber, V.M., Dragović, V. Two-Valued Groups, Kummer Varieties, and Integrable Billiards. Arnold Math J. 4, 27–57 (2018). https://doi.org/10.1007/s40598-018-0085-2

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  • DOI: https://doi.org/10.1007/s40598-018-0085-2

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