Star products of Green's currents and automorphic forms

In previous work, the authors computed archimedian heights of hermitian lin e bundles on families of polarized, n-dimensional abelian varieties. In thi s paper, a detailed analysis of the results obtained in the setting of abel ian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian f ibrations and hence extend to a quite general situation. Specifically, we l et f : X --> Y be any family of smooth, projective, n-dimensional complex v arieties over some base, and consider a line bundle on X equipped with a sm ooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chem class. Under mild addi tional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n + 1)-fold star product of Green's currents associate d to the sections, integrated along the fibers of f, is the log-norm of a g lobal section of M. Furthermore, it is proven that in certain general setti ngs the global section of M can be explicitly expressed in terms of point e valuations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces w hen M is a moduli space of degree-2 polarizations. In this setting the glob al section constructed via Green's currents is equal to a power of the Phi -function first studied by R. Borcherds. Additional examples and problems a re presented.