Classically, higher logarithms appear as multivalued functions on
the projective line. Today they can be interpreted as entries of
the period matrix of a certain variation of Hodge structure, itself
called the "polylogarithm". The aim of the book is to
document the sheaf-theoretical foundations of the field of
polylogarithms. Earlier, partly unpublished results and
constructions of Beilinson, Deligne, and Levin on the classical and
elliptic polylog are generalized to the context of Shimura
varieties. The reader is expected to have a sound background in
algebraic geometry. Large parts of the book are expository, and
intended as a reference for the working mathematician. Where a
self-contained exposition was not possible, the author gives
references in order to make the material accessible for advanced
graduate students.
Contents: Mixed Structures on Fundamental Groups.- The Canonical Construction of Mixed Sheaves on Mixed Shimura Varieties.- Polylogarithmic Extensions on Mixed Shimura Varieties: Construction an Basic Properties.- Polylogarithmic Extensions on Mixed Shimura Varieties: The Classical Polylogarithm.- Polylogarithmic Extensions on Mixed Shimura Varieties: The Elliptic Polylogarithm.
