Advertisement

On some differential-geometric aspects of the Torelli map

  • Alessandro GhigiEmail author
Article
  • 33 Downloads

Abstract

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside \(\mathsf {A}_g\). We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.

Mathematics Subject Classification

14C30 14D07 14H10 14H15 14H40 32G20 

Notes

Acknowledgements

The author wishes to thank Professors L. Biliotti and G. P. Pirola for very interesting discussions and Professor J. S. Milne for very interesting emails. Funding was provided by MIUR PRIN 2015, GNSAGA of INDAM.

References

  1. 1.
    Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of Algebraic Curves, vol. II, Volume 268 of Grundlehren der Mathematischen Wissenschaften. Springer, New York (2011)zbMATHGoogle Scholar
  2. 2.
    Oort, F., Steenbrink, J.: The local Torelli problem for algebraic curves. In: Journées de Géometrie Algébrique d’Angers. Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 157–204. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)Google Scholar
  3. 3.
    Colombo, E., Pirola, G.P., Tortora, A.: Hodge-Gaussian maps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30(1), 125–146 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Colombo, E., Frediani, P.: Siegel metric and curvature of the moduli space of curves. Trans. Am. Math. Soc. 362(3), 1231–1246 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colombo, E., Frediani, P., Ghigi, A.: On totally geodesic submanifolds in the Jacobian locus. Int. J. Math. 26(1), 1550005 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eschenburg, J.-H.: Lecture notes on symmetric spaces. http://myweb.rz.uni-augsburg.de/eschenbu/symspace.pdf. Accessed 7 Aug 2018
  7. 7.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, vol. 80 of Pure and Applied Mathematics. Academic Press Inc., New York (1978)Google Scholar
  8. 8.
    Freitag, E.: Funktionentheorie 2. Riemannsche Flächen, mehrere komplexe Variable, Abelsche Funktionen, höhere Modulformen. Springer, Berlin (2009)zbMATHGoogle Scholar
  9. 9.
    Moonen, B., Oort, F.: The Torelli locus and special subvarieties. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, vol. II, pp. 549–94. International Press, Boston (2013)Google Scholar
  10. 10.
    Schnell, C.: Two lectures about Mumford-Tate groups. Rend. Semin. Mat. Univ. Politec. Torino 69(2), 199–216 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Voisin, C.: Hodge loci. In: Handbook of moduli, vol. III, vol. 26 of Adv. Lect. Math. (ALM), pp. 507–546. Int. Press, Somerville (2013)Google Scholar
  12. 12.
    Voisin, C.: Théorie de Hodge et géométrie algébrique complexe, vol. 10 of Cours Spécialisés. Société Mathématique de France, Paris (2002)Google Scholar
  13. 13.
    Cattani, E., Deligne, P., Kaplan, A.: On the locus of Hodge classes. J. Am. Math. Soc. 8(2), 483–506 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mumford, D.: A note of Shimura’s paper “Discontinuous groups and abelian varieties”. Math. Ann. 181, 345–351 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moonen, B.: Linearity properties of Shimura varieties. I. J. Algebraic Geom. 7(3), 539–567 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    de Jong, J., Noot, R.: Jacobians with complex multiplication. In: van der Geer G., Oort F., Steenbrink J. (eds.) Arithmetic algebraic geometry (Texel, 1989), volume 89 of Progr. Math., pp. 177–192. Birkhäuser, Boston (1991)Google Scholar
  17. 17.
    Moonen, B.: Special subvarieties arising from families of cyclic covers of the projective line. Doc. Math. 15, 793–819 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Rohde, J.C.: Cyclic coverings, Calabi–Yau Manifolds and Complex Multiplication, vol. 1975 of Lecture Notes in Mathematics. Springer, Berlin (2009)Google Scholar
  19. 19.
    Frediani, P., Ghigi, A., Penegini, M.: Shimura varieties in the Torelli locus via Galois coverings. Int. Math. Res. Not. IMRN 20, 10595–10623 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Frediani, P., Penegini, M., Porru, P.: Shimura varieties in the Torelli locus via Galois coverings of elliptic curves. Geom. Dedicata 181, 177–192 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pirola, G.P.: On a conjecture of Xiao. J. Reine Angew. Math. 431, 75–89 (1992)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Grushevsky, S., Möller, M.: Explicit formulas for infinitely many Shimura curves in genus 4. Asian J. Math. 22(2), 381–390 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Toledo, D.: Nonexistence of certain closed complex geodesics in the moduli space of curves. Pac. J. Math. 129(1), 187–192 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hain, R.: Locally symmetric families of curves and Jacobians. In: Faber C., Looijenga E. (eds.) Moduli of curves and abelian varieties. Aspects Math., E33, pp. 91–108. Friedr. Vieweg, Braunschweig (1999)Google Scholar
  25. 25.
    de Jong, J., Zhang, S.-W.: Generic abelian varieties with real multiplication are not Jacobians. In: Diophantine geometry, vol. 4 of CRM Series, pp. 165–172. Ed. Norm., Pisa (2007)Google Scholar
  26. 26.
    Liu, K., Sun, X., Yang, X., Yau, S.-T.: Curvatures of moduli spaces of curves and applications. Asian J. Math. 21(5), 841–854 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Chen, K., Lu, X., Zuo, K.: On the Oort conjecture for Shimura varieties of unitary and orthogonal types. Compos. Math. 152(5), 889–917 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lu, X., Zuo, K.: The Oort conjecture for on Shimura curves in the Torelli locus of curves. arXiv preprint. arXiv:1405.4751 (2014)
  29. 29.
    Lu, X., Zuo, K.: The Oort conjecture on Shimura curves in the Torelli locus of hyperelliptic curves. J. Math. Pures Appl. (9) 108(4), 532–552 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Grushevsky, S., Möller, M.: Shimura curves within the locus of hyperelliptic Jacobians in genus 3. Int. Math. Res. Not. IMRN 6, 1603–1639 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mohajer, A., Zuo, K.: On Shimura subvarieties generated by families of abelian covers of \(\mathbb{P}^1\). J. Pure Appl. Algebra 222(4), 931–949 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Frediani, P., Porru, P.: On the bielliptic and bihyperelliptic loci. arXiv preprint. arXiv:1807.02073 (2018)
  33. 33.
    Colombo, E., Frediani, P., Ghigi, A., Penegini, M.: Shimura curves in the Prym locus arXiv preprint arXiv:1706.02364, To appear on Commun. Contemp. Math (2017)
  34. 34.
    Colombo, E., Frediani, P.: A bound on the dimension of a totally geodesic submanifold in the Prym locus. arXiv preprint. arXiv:1711.03421 (2017)
  35. 35.
    Frediani, P., Ghigi, A., Pirola, G.P.: Fujita decomposition and Hodge loci. J. Inst. Math. Jussieu . arXiv preprint. arXiv:1710.03531 (2017) (To appear)
  36. 36.
    Marcucci, V., Naranjo, J.C., Pirola, G.P.: Isogenies of Jacobians. Algebraic Geom. 3(4), 424–440 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Università di PaviaPaviaItaly

Personalised recommendations