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.


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