Green's currents and height pairing on complex tori (Q1173827)

This paper establishes some properties of the archimedean height pairing \(\langle Z_1, Z_2\rangle_\infty\) between analytic cycles \(Z_1\) and \(Z_2\) on a compact Kähler manifold \(M\), which was introduced by \textit{A. A. Beilinson} [J. Sov. Math. 30, 2036--2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181--238 (1984; Zbl 0588.14013)] and \textit{H. Gillet} and \textit{C. Soulé} [C. R. Acad. Sci., Paris, Sér. I 299, 563--566 (1984; Zbl 0607.14003)]. First, a formula of ``reduction to the diagonal'' is proved: if \(\Delta_M\) denotes the diagonal in \(M\times M\), then \(\langle Z_1, Z_2\rangle_\infty = \langle Z_1,\times Z_2, \Delta_M\rangle_\infty\). Then this formula is used to prove that, when \(M\) is a complex torus endowed with a translation invariant Kähler metric, the following equality holds: \(\langle Z_1, Z_2\rangle_ \infty = \langle Z_1\boxminus Z_2,\{0\}\rangle_\infty\). Here \(Z_1\boxminus Z_2\) denotes the image of \(Z_1\times Z_2\) by the map \((M\times M\to M\), \((x,y)\mapsto x-y)\). This formula extends an identity due to Bloch and \textit{R. Hain} [see Duke Math. J. 61, No. 3, 859--898 (1990; Zbl 0737.14005)].