Basu, Suratno; Dan, Ananyo; Kaur, Inder

Degeneration of Intermediate Jacobians and the Torelli Theorem

Doc. Math. 24, 1739-1767 (2019)
DOI: 10.25537/dm.2019v24.1739-1767
Communicated by Thomas Peternell

Summary

Mumford and Newstead generalized the classical Torelli theorem to higher rank, i.e. a smooth, projective curve \(X\) is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank \(2\) bundles on \(X\), with fixed odd degree determinant. In this article we prove the analogous result in the case \(X\) is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.

Mathematics Subject Classification

14C30, 14C34, 14D07, 32G20, 32S35, 14D20, 14H40

Keywords/Phrases

Torelli theorem, intermediate Jacobians, Néron models, nodal curves, Gieseker moduli space, limit mixed Hodge structures

References

  • 1. T. Abe. The moduli stack of rank-two Gieseker bundles with fixed determinant on a nodal curve. II. International Journal of Mathematics, 20(7):859-882, 2009. DOI 10.1142/S0129167X09005558; zbl 1177.14069; MR2548402.
  • 2. V. Alexeev. Complete moduli in the presence of semiabelian group action. Annals of Mathematics (2), 155(3):611-708, 2002. DOI 10.2307/3062130; zbl 1052.14017; MR1923963; arxiv math/9905103.
  • 3. V. Alexeev. Compactified Jacobians and Torelli map. Publications of the Research Institute for Mathematical Sciences, 40(4):1241-1265, 2004. DOI 10.2977/prims/1145475446; zbl 1079.14019; MR2105707; arxiv alg-geom/9608012.
  • 4. A. Andreotti. On a theorem of Torelli. American Journal of Mathematics, 80(4):801-828, 1958. DOI 10.2307/2372835; zbl 0084.17304; MR0102518.
  • 5. V. Balaji, P. Barik, and D. S. Nagaraj. A degeneration of moduli of Hitchin pairs. International Mathematics Research Notices, 2016(21):6581-6625, 2016. DOI 10.1093/imrn/rnv356; zbl 1404.14039; MR3579973; arxiv 1308.4490.
  • 6. P. Barik, A. Dey, and B. N. Suhas. On the rationality of Nagaraj-Seshadri moduli space. Bulletin des Sciences Mathématiques, 140(8):990-1002, 2016. DOI 10.1016/j.bulsci.2016.06.001; zbl 1406.14027; MR3569200; arxiv 1609.08087.
  • 7. S. Basu. On a relative Mumford-Newstead theorem. Bulletin des Sciences Mathématiques, 140(8):953-989, 2016. DOI 10.1016/j.bulsci.2016.04.003; zbl 1369.14040; MR3569199; arxiv 1501.07347.
  • 8. L. Caporaso and F. Viviani. Torelli theorem for stable curves. Journal of the European Mathematical Society, 13(5):1289-1329, 2011. DOI 10.4171/JEMS/281; zbl 1230.14037; MR2825165; arxiv 0904.4039.
  • 9. H. Clemens. The Néron model for families of intermediate Jacobians acquiring algebraic singularities. Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 58(1):5-18, 1983. zbl 0529.14025; MR0720929.
  • 10. O. Debarre. Higher-dimensional algebraic geometry. Springer, 2001. zbl 0978.14001; MR1841091.
  • 11. R. Fringuelli. The Picard group of the universal moduli space of vector bundles on stable curves. Advances in Mathematics, 336:477-557, 2018. DOI 10.1016/j.aim.2018.07.032; zbl 1401.14068; MR3846159; arxiv 1601.04866.
  • 12. W. Fulton. Intersection theory. Springer, 1984. zbl 0885.14002; MR0732620.
  • 13. D. Gieseker. A degeneration of the moduli space of stable bundles. Journal of Differential Geometry, 19(1):173-206, 1984. DOI 10.4310/jdg/1214438427; zbl 0557.14008; MR0739786.
  • 14. T. Graber, J. Harris, and J. Starr. Families of rationally connected varieties. Journal of the American Mathematical Society, 16(1):57-67, 2003. DOI 10.1090/S0894-0347-02-00402-2; zbl 1092.14063; MR1937199; arxiv math/0109220.
  • 15. M. Green, P. Griffiths, and M. Kerr. Néron models and limits of Abel-Jacobi mappings. Compositio Mathematica, 146(2):288-366, 2010. DOI 10.1112/S0010437X09004400; zbl 1195.14006; MR2601630.
  • 16. R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, no. 52. Springer, 1977. zbl 0367.14001.
  • 17. R. Hartshorne. Deformation theory. Graduate Texts in Mathematics, no. 257. Springer, 2010. DOI 10.1007/978-1-4419-1596-2; zbl 1186.14004; MR2583634.
  • 18. D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. 2nd ed. Cambridge University Press, 2010. zbl 1206.14027; MR2665168.
  • 19. I. Kaur. The \({C_1}\) conjecture for the moduli space of stable vector bundles with fixed determinant on a smooth projective curve. PhD thesis, Freie Universitaet Berlin, 2016.
  • 20. I. Kaur. Existence of semistable vector bundles with fixed determinants. Journal of Geometry and Physics, 138:90-102, 2019. DOI 10.1016/j.geomphys.2018.12.023; zbl 07058911; MR3901801.
  • 21. A. Langer. Moduli spaces and Castelnuovo-Mumford regularity of sheaves on surfaces. American Journal of Mathematics, 128:373-417, 2006. DOI 10.1353/ajm.2006.0014; zbl 1102.14030; MR2214897.
  • 22. B. Moonen, F. Oort, et al. The Torelli locus and special subvarieties. Handbook of moduli. Vol. 2, pp. 549-594. International Press, Somerville, MA, 2013. zbl 1322.14065; MR3184184; arxiv 1112.0933.
  • 23. D. Mumford. Further comments on boundary points. (mimeographed note), AMS Summer school at Woods Hoole, 1964.
  • 24. D. Mumford and P. Newstead. Periods of a moduli space of bundles on curves. American Journal of Mathematics, 90(4):1200-1208, 1968. DOI 10.2307/2373296; zbl 0174.52902; MR0234958.
  • 25. D. S. Nagaraj and C. S. Seshadri. Degenerations of the moduli spaces of vector bundles on curves. II. Generalized Gieseker moduli spaces. In Proceedings of the Indian Academy of Sciences, Mathematical Sciences, volume 109, pp. 165-201. Springer, 1999. DOI 10.1007/BF02841533; zbl 0957.14021; MR1687729.
  • 26. Y. Namikawa. On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform. Nagoya Mathematical Journal, 52:197-259, 1973. DOI 10.1017/S002776300001597X; zbl 0271.14014; MR0337981.
  • 27. F. Oort and J. Steenbrink. The local Torelli problem for algebraic curves. Journées de Géometrie Algébrique d'Angers, 1979:157-204, 1979. zbl 0444.14007; MR0605341.
  • 28. R. Pandharipande. A compactification over \(M_g\) of the universal moduli space of slope-semistable vector bundles. Journal of the American Mathematical Society, 9(2):425-471, 1996. DOI 10.1090/S0894-0347-96-00173-7; zbl 0886.14002; MR1308406.
  • 29. C. Peters and J. H. M. Steenbrink. Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, volume 52. Springer, 2008. DOI 10.1007/978-3-540-77017-6; zbl 1138.14002; MR2393625.
  • 30. G. Pezzini. Lectures on spherical and wonderful varieties. Les cours du CIRM, 1(1):33-53, 2010.
  • 31. L. Rizzi and F. Zucconi. A note on Torelli-type theorems for Gorenstein curves. Mathematische Nachrichten, 290(17-18):3006-3019, 2017. DOI 10.1002/mana.201600106; zbl 1386.14052; MR3746379; arxiv 1603.09080.
  • 32. M. Saito. Admissible normal functions. Journal of Algebraic Geometry, 5(2):235-276, 1996. zbl 0918.14018; MR1374710.
  • 33. W. Schmid. Variation of Hodge structure: the singularities of the period mapping. Inventiones mathematicae, 22(3-4):211-319, 1973. DOI 10.1007/BF01389674; zbl 0278.14003; MR0382272.
  • 34. J. Steenbrink. Limits of Hodge structures. Inventiones mathematicae, 31:229-257, 1976. DOI 10.1007/BF01403146; zbl 0303.14002; MR0429885.
  • 35. M. Thaddeus. Algebraic geometry and the Verlinde formula. PhD thesis, University of Oxford, 1992.
  • 36. R. Torelli. Sulle varieta di Jacobi. I, II. Rendiconti R. Accad. dei Lincei, 22:98-103, 437-441, 1913. zbl 44.0655.03.
  • 37. C. Voisin. Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, vol. 76. Cambridge University Press, 2002. zbl 1129.14019; MR1967689.
  • 38. C. Voisin. Hodge theory and complex algebraic geometry. II. Cambridge Studies in Advanced Mathematics, vol. 77. Cambridge University Press, 2003. zbl 1129.14020; MR1997577.
  • 39. A. Weil. Zum Beweis des Torellischen Satzes. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., IIa 1957:32-53, 1957. zbl 0079.37002; MR0089483.
  • 40. S. Zucker. Generalized intermediate Jacobians and the theorem on normal functions. Inventiones mathematicae, 33(3):185-222, 1976. DOI 10.1007/BF01404203; zbl 0329.14008; MR0412186.

Affiliation

Basu, Suratno
Institute of Mathematical Sciences, HBNI, CIT Campus, Tharamani, Chennai 600113, India
Dan, Ananyo
BCAM - Basque Centre for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
Kaur, Inder
Instituto de Matemática Pura e Aplicada, Estr. Dona Castorina, 110 - Jardim Botanico, Rio de Janeiro - RJ, 22460-320, Brazil

Downloads