The Rank of the Normal Functions of the Ceresa and Gross–Schoen Cycles

Hain, Richard 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e141

Résumé

In this paper we show that the rank of the normal function function of the genus $g$ Ceresa cycle over the moduli space of curves has the maximal rank possible, $3g-3$ , provided that $g\ge 3$. In genus 3 we show that the Green–Griffiths invariant of this normal function is a Teichmüller modular form of weight $(4,0,-1)$ and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus. We also introduce new techniques and tools for studying the behaviour of normal functions along and transverse to boundary divisors. These include the introduction of residual normal functions and the use of global monodromy arguments to compute them.

Détail

DOI : 10.1017/fms.2025.10089

Bibliographie

[1] Ceresa, G., ‘C is not algebraically equivalent to C− in its Jacobian’, Ann. of Math. 117(2) (1983), 285–291.10.2307/2007078 Google Scholar | DOI

[2] Cléry, F., Faber, C. and Van Der Geer, G., ‘Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three’, Selecta Math. (N.S.) 26 (2020), Paper No. 55, 39 pp.10.1007/s00029-020-00581-7 Google Scholar | DOI

[3] Collino, A. and Pirola, G., ‘The Griffiths infinitesimal invariant for a curve in its Jacobian’, Duke Math. J. 78 (1995), 59–88.10.1215/S0012-7094-95-07804-1 Google Scholar | DOI

[4] Deligne, P., ‘Équations différentielles à points singuliers réguliers’, lecture notes in Mathematics 163 (1970).10.1007/BFb0061194 Google Scholar | DOI

[5] Faltings, G. and Chai, C.-L., Degeneration of abelian varieties. With an appendix by David Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete 22 (Springer-Verlag, 1990).10.1007/978-3-662-02632-8 Google Scholar | DOI

[6] Gao, Z. and Zhang, S.-W., ‘Heights and periods of algebraic cycles in families’, Preprint, 2024, . Google Scholar | arXiv

[7] Van Der Geer, G., Kouvidakis, A. and Alexis, , ‘Effective divisors on projectivized Hodge bundles and modular forms’, Math. Nachr. 297 (2024), 1142–1170.10.1002/mana.202300098 Google Scholar | DOI

[8] Green, M., ‘Griffiths’ infinitesimal invariant and the Abel-Jacobi map’, J. Differential Geom. 29 (1989), 545–555.10.4310/jdg/1214443062 Google Scholar | DOI

[9] Griffiths, P., ‘Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions’, Compos. Math. 50 (1983), 267–324. Google Scholar

[10] Green, M., Griffiths, P. and Kerr, M., ‘Néron models and limits of Abel-Jacobi mappings’, Compos. Math. 146 (2010), 288–366.10.1112/S0010437X09004400 Google Scholar | DOI

[11] Gross, B. and Schoen, C., ‘The modified diagonal cycle on the triple product of a pointed curve’, Ann. Inst. Fourier (Grenoble) 45 (1995), 649–679.10.5802/aif.1469 Google Scholar | DOI

[12] Hain, R., ‘Torelli groups and geometry of moduli spaces of curves’, in Current topics in complex algebraic geometry (Cambridge University Press, Berkeley, 1992/93), 97–143, Math. Sci. Res. Inst. Publ. 28 (1995). Google Scholar

[13] Hain, R., ‘Infinitesimal presentations of the Torelli groups’, J. Amer. Math. Soc. 10 (1997), 597–651.10.1090/S0894-0347-97-00235-X Google Scholar | DOI

[14] Hain, R., ‘The rational cohomology ring of the moduli space of abelian 3-folds’, Math. Res. Lett. 9 (2002), 473–491.10.4310/MRL.2002.v9.n4.a7 Google Scholar | DOI

[15] Hain, R., ‘Relative weight filtrations on completions of mapping class groups’, in (R. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano and Y. Mitsumatsu editors) Groups of Diffeomorphisms (Mathematical Society of Japan, Tokyo), 309–368, Adv. Stud. Pure Math. 52 (2008), x+524. Google Scholar

[16] Hain, R., ‘Normal functions and the geometry of moduli spaces of curves’, in Handbook of Moduli (International Press) Vol. I, 527–578, Adv. Lect. Math. (ALM) 24 (2013). Google Scholar

[17] Hain, R., ‘Periods of real biextensions’, Algebra and Number Theory (2024). To appear. Google Scholar

[18] Harris, B., ‘Harmonic volumes’, Acta Math. 150 (1983), 91–123.10.1007/BF02392968 Google Scholar | DOI

[19] Ichikawa, T., ‘Teichmüller modular forms of degree 3’, Amer. J. Math. 117 (1995), 1057–1061.10.2307/2374959 Google Scholar | DOI

[20] Johnson, D., ‘An abelian quotient of the mapping class group Ig’, Math. Ann. 249 (1980), 225–242.10.1007/BF01363897 Google Scholar | DOI

[21] Johnson, D., ‘The structure of the Torelli group. III: The abelianization of I’, Topology 24 (1985), 127–144.10.1016/0040-9383(85)90050-3 Google Scholar | DOI

[22] Kashiwara, M., ‘A study of variation of mixed Hodge structure’, Publ. Res. Inst. Math. Sci. 22 (1986), 991–1024.10.2977/prims/1195177264 Google Scholar | DOI

[23] Nori, M., ‘Algebraic cycles and Hodge-theoretic connectivity’, Invent. Math. 111 (1993), 349–373.10.1007/BF01231292 Google Scholar | DOI

[24] Pulte, M., ‘The fundamental group of a Riemann surface: mixed Hodge structures and algebraic cycles’, Duke Math. J. 57 (1988), 721–760.10.1215/S0012-7094-88-05732-8 Google Scholar | DOI

[25] Saito, M., ‘Admissible normal functions’, J. Algebraic Geom. 5 (1996), 235–276. Google Scholar

[26] Schmid, W., ‘Variation of Hodge structure: the singularities of the period mapping’, Invent. Math. 22 (1973), 211–319.10.1007/BF01389674 Google Scholar | DOI

[27] Schnell, C., ‘Complex analytic Néron models for arbitrary families of intermediate Jacobians’, Invent. Math. 188 (2012), 1–81.10.1007/s00222-011-0341-8 Google Scholar | DOI

[28] Steenbrink, J. and Zucker, S., ‘Variation of mixed Hodge structure, I’, Invent. Math. 80 (1985), 489–542.10.1007/BF01388729 Google Scholar | DOI

[29] Taïbi, O., ‘Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula’, Ann. Sci. Éc. Norm. Supér. 50(4) (2017), 269–344.10.24033/asens.2321 Google Scholar | DOI