Introduction to Variations of Hodge Structure Summer School on Hodge Theory, Ictp, June 2010

These notes are intended to accompany the course Introduction to Variations of Hodge Structure (VHS) at the 2010 ICTP Summer School on Hodge Theory. The modern theory of variations of Hodge structure (although some authors have referred to this period as the pre-history) begins with the work of Griffiths [23, 24, 25] and continues with that of Deligne [17, 18, 19], and Schmid [41]. The basic object of study are period domains which parametrize the possible polarized Hodge structures in a given smooth projective variety. An analytic family of such varieties gives rise to a holomorphic map with values in a period domain, satisfying an additional system of differential equations. Moreover, period domains are homogeneous quasi-projective varieties and, following Griffiths and Schmid, one can apply Lie theoretic techniques to study these maps. These notes are not intended as a comprehensive survey of the theory of VHS. We refer the reader to the surveys [25, 30, 2, 1, 33], the collections [26, 1], and the monographs [4, 40, 45, 46] for fuller accounts of various aspects of the theory. In these notes we will emphasize the theory of abstract variations of Hodge structure and, in particular, their asymptotic behavior. The geometric aspects will be the subject of the subsequent course by James Carlson. In §1, we study the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. We also define the Kodaira-Spencer map associated with a family of smooth projective varieties. The second section is devoted to the study of Griffiths’ period map and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. In §3, we define the classifying spaces for polarized Hodge structures and study some of their basic properties. The last two sections deal with the asymptotics of a period mapping with particular attention to Schmid’s Orbit Theorems. We emphasize throughout this discussion the relationship between nilpotent and SL2-orbits and mixed Hodge structures. In these notes I have often drawn from previous work in collaboration with Aroldo Kaplan, Wilfried Schmid, Pierre Deligne, and Javier Fernandez to all of whom I am very grateful. A final version of these notes will be posted after the conclusion of the Summer School.