An Exponential History of Functions with Logarithmic Growth

We survey recent work on normal functions, including limits and singularities of admissible normal functions, the Gri ths-Green approach to the Hodge conjecture, algebraicity of the zero-locus of a normal function, Néron models, and MumfordTate groups. Some of the material and many of the examples, esp. in §§5− 6, are original. In a talk on the theory of motives, A. A. Beilinson remarked that according to his time-line of results, advances in the (relatively young) eld were apparently a logarithmic function of t; hence, one could expect to wait 100 years for the next signi cant milestone. Here we allow ourselves to be more optimistic: following on a drawn-out history which begins with Poincaré, Lefschetz, and Hodge, the theory of Normal Functions reached maturity in the programs of Bloch, Gri ths, Zucker, and others. But the recent blizzard of results and ideas, inspired by works of M. Saito on admissible normal functions, and Green and Gri ths on the Hodge Conjecture, has been impressive indeed. In addition to further papers of theirs, signi cant progress has been made in work of P. Brosnan, F. Charles, H. Clemens, H. Fang, J. Lewis, R. Thomas, Z. Nie, C. Schnell, C. Voisin, A. Young, and the authors much of this in the last 4 years. This seems like a good time to try to summarize the state of the art and speculate about the future, barring (say) 100 more results between the time of writing and the publication of this volume. In the classical algebraic geometry of curves, Abel's theorem and Jacobi inversion articulate the relationship (involving rational integrals) between con gurations of points with integer multiplicities, or zerocycles, and an abelian variety known as the Jacobian of the curve: the latter algebraically parametrizes the cycles of degree 0 modulo the subgroup arising as divisors of meromorphic functions. Given a family X of algebraic curves over a complete base curve S, with smooth bers over S∗ (S minus a nite point set Σ over which bers have double point singularities), Poincaré [P1, P2] de ned normal functions as