Introduction

Summary

In its simplest form, Hodge theory is the study of periods – integrals of algebraic differential forms which arise in the study of complex geometry, number theory and physics. Its difficulty and richness arise in part from the non-algebraicity of these integrals. According to the beautiful conjectures of Hodge, Bloch and Beilinson, what algebraic structure they do have should be explained by “algebraic cycles.” There has been much recent progress on these conjectures and on classifying spaces for periods, as well as their asymptotics and arithmetic.

The main goal of the Vancouver conference was to bring together a diverse community of world-leading mathematicians – in Hodge theory, arithmetic geometry, algebraic cycles, complex geometry, and representation theory – around the common theme of period maps (considered broadly). With an intensive summer school followed by 32 research talks, it attracted over 85 participants from the US, Canada, Mexico, Europe and Japan, and sparked several new collaborations. With this book, we hope to draw an even larger audience into this area, and to cement the impact of the conference. In particular, this volume includes careful write-ups of expository talks by Wushi Goldring, Radu Laza and Richard Hain, as well as papers presenting key recent developments in each of several focus areas. We hope that it is useful for graduate students and seasoned researchers alike.

Overview of this volume

The birth of modern Hodge theory began with the work of P. Griffiths, who devised an extension of Lefschetz's original proof of the Hodge conjecture using normal functions and variations of Hodge structure. Despite the fact that this program did not lead to the desired fruition in higher codimension, it has had a lasting impact on the subject. Alongside the study of algebraic cycles, the last half-century has seen the development of rich theories of Hodge theory at the boundary and symmetries of Hodge structures. In particular, by the recent work of Griffiths and others, the Hodge Conjecture itself can now be stated in terms of the asymptotic behavior of normal functions; while Mumford-Tate (symmetry) groups of Hodge structures have led to proofs of the Hodge and Beilinson-Hodge conjectures in special cases.