Periods of Limit Mixed Hodge Structures

The first goal of this paper is to explain some important results of Wilfred Schmid from his fundamental paper [30] in which he proves very general results which govern the behaviour of the periods of a of smooth projective variety Xt as it degenerates to a singular variety. As has been known since classical times, the periods of a smooth projective variety sometimes contain significant information about the geometry of the variety, such as in the case of curves where the periods determine the curve. Likewise, information about the asymptotic behaviour of the periods of a variety as it degenerates sometimes contain significant information about the degeneration and the singular fiber. For example, the Hodge norm estimates, which are established in [30] and [3], describe the asymptotics of the Hodge norm of a cohomology class as the variety degenerates in terms of its monodromy. They are an essential ingredient in the study of the L2 cohomology of smooth varieties with coefficients in a variation of Hodge structure [37, 4]. A second goal is to give some idea of how geometric and arithmetic information can be extracted from the limit periods, both in the geometric case and in the case of the limits of the mixed Hodge structures on fundamental groups of curves. To get oriented, recall that if X is a compact Riemann surface of genus g, then for each choice of a symplectic (w.r.t. the intersection form) basis a1, . . . , ag, b1, . . . , bg of H1(X,Z), there is a basis w1, . . . , wg of the holomorphic differentials H(X,Ω) such that ∫