TOWARDS THE ARITHMETIC DEGREE OF LINE BUNDLES ON ABELIAN-VARIETIES

In this paper we analyze the integral of the star-product of (n + 1) G reen currents associated to (n + 1) global sections of an ample line b undle equipped with a translation invariant metric over an n-dimension al, polarized abelian variety. The integral is shown to equal the loga rithm of the Petersson norm of a certain Siegel modular form, which is explicitly described in terms of the given data. This result can be i nterpreted as evaluating an;archimedian height on a family of polarize d abelian varieties. The key ingredient to the proof of the main formu la is a tide-variational formula for the integral under consideration. In the case of dimensions n = 1, 2, 3 explicit examples in terms of c lassical Riemann theta functions are given.