RECENT ADVANCES IN HODGE THEORY. PERIOD DOMAINS, ALGEBRAIC CYCLES, AND ARITHMETIC

In its simplest form, Hodge theory is the study of periods – integrals of algebraic differential forms which arise in the study of complex geometry and moduli, number theory and physics. Organized around the basic concepts of variations of Hodge structure and period maps, this volume draws together new developments in deformation theory, mirror symmetry, Galois representations, iterated integrals, algebraic cycles and the Hodge conjecture. Its mixture of high-quality expository and research articles make it a useful resource for graduate students and seasoned researchers alike.

• Discusses recent developments in Hodge theory with a novel focus on period maps
• Thematic organisation helps the reader see how different topics and techniques fit together
• High-quality papers from world-leading academics will draw new researchers and young mathematicians to the study of period maps.

Authors
• Matt Kerr, Washington University, St Louis. Associate Professor of Mathematics at Washington University, St Louis, and an established researcher in Hodge theory and algebraic geometry. His work is supported by an FRG grant from the National Science Foundation. He is also co-author (with M. Green and P. Griffiths) of Mumford-Tate Groups and Domains: Their Geometry and Arithmetic and Hodge Theory, Complex Geometry, and Representation Theory.
• Gregory Pearlstein, Texas A & M University. Associate Professor of Mathematics at Texas A&M University. He is an established researcher in Hodge theory and algebraic geometry and his work is supported by an FRG grant from the National Science Foundation.

Table of Contents
Preface.
Introduction.
List of conference participants
• Part I. Hodge Theory at the Boundary:
• Part I(A). Period Domains and Their Compactifications: Classical period domains.
The singularities of the invariant metric on the Jacobi line bundle.
Symmetries of graded polarized mixed Hodge structures.
• Part I(B). Period Maps and Algebraic Geometry: Deformation theory and limiting mixed Hodge structures.
Studies of closed/open mirror symmetry for quintic threefolds through log mixed Hodge theory.
The 14th case VHS via K3 fibrations.
• Part II. Algebraic Cycles and Normal Functions: A simple construction of regulator indecomposable higher Chow cycles in elliptic surfaces.
A relative version of the Beilinson–Hodge conjecture.
Normal functions and spread of zero locus.
Fields of definition of Hodge loci.
Tate twists of Hodge structures arising from abelian varieties.
Some surfaces of general type for which Bloch's conjecture holds.
• Part III. The Arithmetic of Periods:
• Part III(A). Motives, Galois Representations, and Automorphic Forms: An introduction to the Langlands correspondence.
Generalized Kuga–Satake theory and rigid local systems I – the middle convolution.
On the fundamental periods of a motive.
• Part III(B). Modular Forms and Iterated Integrals: Geometric Hodge structures with prescribed Hodge numbers.
The Hodge–de Rham theory of modular groups.