Seminar on periods and motives | Kummer extensions of Hodge structures

Abstract：

Kummer extensions of Hodge structures are extensions of the form 0 -> Z(1) -> K -> Z(0) ->0. Associated to a Kummer extension is the extension class, which has a tendency to be algebraic, and a unique period associated to the Hodge structure on K. The period is the log of the extension class. Kummer extensions are too simple to be of interest in and of themselves, but they often serve as the "platter" on which the meat and potatoes are served. I will discuss two examples.

1. Higher cross-ratios and functions on Hilbert schemes. Here the Kummer extension is actually a degenerate biextension. Biextensions are mixed Hodge structures with weight graded structure Q(1), H, Q(0), where H is a pure Hodge structure of weight -1. It can happen that H=(0), in which case the biextension becomes a Kummer extension. A class of such degenerate biextensions arises from the study of algebraic cycles on varieties with vanishing odd dimensional Betti cohomology. In this case, the extension class becomes an algebraic function on the Hilbert scheme.

2. The Gross-Zagier conjecture. This is a conjecture about values of suitable Green's functions where the t_i are CM points on a modular curve. The conjecture gives G(t_1,t_2) = D log(a) where D is a product of discriminants for the t_i, and a is an algebraic number. Since periods of Kummer extensions are always of the form D log(a), it is natural to look for a Kummer extension somewhere.