## Normal functions and the height of Gross-Schoen cycles.(English)Zbl 1312.14073

Let $$X$$ be a smooth geometrically connected curve of genus $$g\geq2$$ over a field $$k$$. For a divisor $$e$$ of degree $$1$$ on $$X$$, B. H. Gross and C. Schoen [Ann. Inst. Fourier 45, No. 3, 649–679 (1995; Zbl 0822.14015)] constructed a cohomologically trivial cycle $$\Delta_e$$ of codimension $$2$$ on the triple product $$X^3$$. S.-W. Zhang [Invent. Math. 179, No. 1, 1–73 (2010; Zbl 1193.14031)] derived an explicit formula for $$\langle\Delta_e,\Delta_e\rangle$$, which led to some important new results.
The present paper obtains a variant of Zhang’s formula, based on showing that the contributions from archimedean places occur as the norms at infinity of a certain canonical isomorphism of line bundles.
In more detail, let $$S$$ be a smooth quasiprojective variety over $$\mathbb C$$. Let $$\pi:\mathcal X\to S$$ be a smooth projective family of curves of genus $$g\geq2$$ over $$S$$, let $$e$$ be a flat divisor of relative degree $$1$$ on $$\mathcal X/S$$, let $$\omega$$ be the relative dualizing sheaf of $$\mathcal X/S$$, let $$x_e$$ be the divisor class (of relative degree $$0$$) given by $$(2g-2)e-c_1(\omega)$$ (note that this differs from Zhang’s notation by a factor of $$2g-2$$), and let $$\Delta_e$$ be the relative Gross-Schoen cycle on the triple fiber self-product of $$\mathcal X/S$$. Let $$\langle\omega,\omega\rangle$$ and $$\langle x_e,x_e\rangle$$ be the Deligne self-pairings of $$\omega$$ and $$x_e$$, respectively; they are line bundles on $$S$$. Also let $$\langle\Delta_e,\Delta_e\rangle$$ be the Bloch pairing; it too is a line bundle on $$X$$.
The variant of Zhang’s result proved in this paper, then, is that there is an isomorphism $\langle\Delta_e,\Delta_e\rangle^{\otimes(2g-2)} \overset\sim{} \langle\omega,\omega\rangle^{\otimes(2g+1)} \otimes\langle x_e,x_e\rangle^{\otimes(-3)}$ of line bundles on $$S$$, canonical up to sign, and whose norms at archimedean places are related to the contributions $$\varphi(X_v)$$ at those places in Zhang’s formula.
In addition, if $$g\geq3$$ then the paper calculates the value of $$\langle\Delta_e,\Delta_e\rangle$$ in the Picard group $$\text{Pic}(\mathcal M_{g,1}^c)$$.
The basic tools used in the paper are normal functions and biextensions associated to the cohomology of the universal Jacobian.

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C25 Algebraic cycles 14D06 Fibrations, degenerations in algebraic geometry

### Keywords:

Gross-Schoen cycle; Beilinson-Bloch height

### Citations:

Zbl 1193.14031; Zbl 0822.14015
Full Text:

### References:

 [1] S. Y. Arakelov, An intersection theory for divisors on an arithmetic surface (in Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 38 (1974), 1179-1192. · Zbl 0355.14002 [2] E. Arbarello and M. Cornalba, The Picard groups of the moduli spaces of curves , Topology 26 (1987), 153-171. · Zbl 0625.14014 [3] A. Beauville, “Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abelienne” in Algebraic Geometry (Tokyo/Kyoto, 1982) , Lecture Notes in Math. 1016 , Springer, Berlin, 1983, 238-260. · Zbl 0526.14001 [4] A. Beilinson, “Height pairing between algebraic cycles” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985) , Contemp. Math. 67 , Amer. Math. Soc., Providence, 1987, 1-24. · Zbl 0624.14005 [5] S. Bloch, “Height pairings for algebraic cycles” in Proceedings of the Luminy Conference on Algebraic $$K$$-theory (Luminy, 1983) , J. Pure Appl. Algebra 34 , 1984, 119-145. · Zbl 0577.14004 [6] S. Bloch, “Cycles and biextensions” in Algebraic $$K$$-theory and Algebraic Number Theory (Honolulu, 1987) , Contemp. Math. 83 , Amer. Math. Soc., Providence, 1989, 19-30. · Zbl 0686.14002 [7] J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms , J. Amer. Math. Soc. 7 (1994), 903-1027. · Zbl 0973.14013 [8] E. Colombo and B. van Geemen, Note on curves in a Jacobian , Compos. Math. 88 (1993), 333-353. · Zbl 0802.14002 [9] R. de Jong, Second variation of Zhang’s $$\lambda$$-invariant on the moduli space of curves , Amer. J. Math. 135 (2013), 275-290. · Zbl 1262.14028 [10] P. Deligne, “Le déterminant de la cohomologie” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985) , Contemp. Math. 67 , Amer. Math. Soc., Providence, 1987, 93-177. [11] C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform , J. Reine Angew. Math. 422 (1991), 201-219. · Zbl 0745.14003 [12] G. Faltings, Calculus on arithmetic surfaces , Ann. of Math. (2) 119 (1984), 387-424. · Zbl 0559.14005 [13] B. H. Gross and C. Schoen, The modified diagonal cycle on the triple product of a pointed curve , Ann. Inst. Fourier (Grenoble) 45 (1995), 649-679. · Zbl 0822.14015 [14] R. Hain, Biextensions and heights associated to curves of odd genus , Duke Math. J. 61 (1990), 859-898. · Zbl 0737.14005 [15] R. Hain, “Normal functions and the geometry of moduli spaces of curves” in Handbook of Moduli , Adv. Lectures Math. 24 , International Press, Boston, 2013, 527-556. · Zbl 1322.14049 [16] R. Hain and D. Reed, Geometric proofs of some results of Morita , J. Algebraic Geom. 10 (2001), 199-217. · Zbl 0986.14017 [17] R. Hain and D. Reed, On the Arakelov geometry of moduli spaces of curves , J. Differential Geom. 67 (2004), 195-228. · Zbl 1118.14029 [18] B. Harris, Harmonic volumes , Acta Math. 150 (1983), 91-123. · Zbl 0527.30032 [19] B. Iversen, Cohomology of Sheaves , Universitext, Springer, Berlin, 1986. [20] I. Kausz, A discriminant and an upper bound for $$\omega^{2}$$ for hyperelliptic arithmetic surfaces , Compos. Math. 115 (1999), 37-69. · Zbl 0934.14015 [21] O. Meyer, Über Biextensionen und Höhenpaarungen algebraischer Zykel , Ph.D. dissertation, University of Regensburg, Regensburg, Germany, 2003. [22] L. Moret-Bailly, “Métriques permises” in Seminar in Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/84) , Astérisque 127 , Soc. Math. France, Paris, 1985, 29-87. · Zbl 1182.11028 [23] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques , Invent. Math. 98 (1989), 491-498. · Zbl 0727.14014 [24] S. Müller-Stach, $$\mathbb{C}^{*}$$-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles , $$K$$-Theory 9 (1995), 395-406. · Zbl 0833.19002 [25] M. J. Pulte, The fundamental group of a Riemann surface: Mixed Hodge structures and algebraic cycles , Duke Math. J. 57 (1988), 721-760. · Zbl 0678.14005 [26] M. Seibold, Bierweiterungen für algebraische Zykel und Poincarébundel , Ph.D. dissertation, University of Regensburg, Regensburg, Germany, 2007. · Zbl 1308.76224 [27] T. Szamuely, Galois Groups and Fundamental Groups , Cambridge Stud. Adv. Math. 117 , Cambridge University Press, Cambridge, 2009. · Zbl 1189.14002 [28] S.-W. Zhang, Admissible pairing on a curve , Invent. Math. 112 (1993), 171-193. · Zbl 0795.14015 [29] S. Zhang, Gross-Schoen cycles and dualising sheaves , Invent. Math. 179 (2010), 1-73. · Zbl 1193.14031 [30] S. Zhang, Positivity of heights of codimension $$2$$ cycles over function field of characteristic $$0$$ , preprint, [math.AG]. 1001.4788v1
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