The geometry of a complex manifold is one of the marvels of modern day mathematics. This is not a surprise since complex geometry lays at the intersection of analysis, algebra, geometry and topology.
For a smooth complex algebraic curve, Hodge theory is a powerful tool that subsumes its period matrix and recovers the geometry of a given curve in its Jacobian variety. For a general smooth complex algebraic variety, the generalization of the above results gives a Hodge structure on the cohomology of the variety, although there is no analog of the Jacobian variety.
The relationship between Hodge theory and geometry, especially for the study of algebraic cycles on (complex) manifolds, is a rich, deep and increasingly active field. As is well known, one the main conjectures guiding this research is the (generalized) Hodge conjecture on the realization of every rational Hodge (cohomology) class as an algebraic cycle. But, as is also known, there seems not to have been any significant progress beyond the (1,1) case afforded by the Lefschetz theorem. There are, of course, many other important conjectures on the arithmetic and geometric properties of these algebraic cycles, as envisioned by Grothendieck, Bloch and Beilinson, mostly depending on deep properties of the theory of motives.
On the other hand, the geometry of homogeneous complex manifolds and the representation theory of algebraic groups are intimately intertwined, either in the finite dimensional case or in the infinite dimensional one, by the geometry of a corresponding Mumford-Tate domain.
The monograph under review has as its main subject the relationships between Hodge theory and representation theory. Thus, the connecting bridges are built on the common ground of complex geometry. The main goal of this monograph is an exposition of recent work on the subject, especially the work of Carayol and the authors. The exposition is of an advanced level, as could be expected, but the authors have tried to make the exposition as accessible as possible, by working with concrete examples throughout the lectures. For background on some of the topics, the reader is advised to look at the forthcoming collection of articles on Hodge Theory (Princeton, 2014) and Voisin’s Complex Geometry and Hodge Theory, Vol. 1 and Vol. 2, (Cambridge, 2008) or Peters and Steenbrink Mixed Hodge Structures (Springer, 2008). For Mumford-Tate groups and domains, the authors’ previous monograph Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (Princeton, 2012) is one standard reference.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.