Current Developments in Hodge Theory

This book brings together contributions by top-level experts from a wide range of topics in modern Hodge theory, originating in the authors' participation in the special years on Hodge theory at the Institute of Mathematical Sciences of the Americas (IMSA) in Miami.

One of the main themes is the study of moduli spaces and their compactifications. Several articles speak of the singularities occuring in the boundaries of geometrical or Hodge-theoretic compactifications, semistable reduction, the implications of canonical models for model theory in the sense of logic, and fundamental groups of moduli spaces and their associated Torelli groups. Other topics include Mukai lattices, derived moduli spaces, foliations, Higgs bundles and hyperbolicity, the study of pseudoconvexity properties of neighborhoods of infinity, contributions to the theory of degenerations and limiting mixed Hodge structures.

This text will provide an indispensable reference for research mathematicians and specialist graduate students, where the modern approaches to moduli spaces are illustrated by their realizations and applications in examples of interest for the interplay between Hodge theory and moduli spaces.

A collection of 10 timely research contributions by top-level specialists in Complex and Algebraic Geometry Covers a wide array of modern research directions in Hodge theory Develops new and original viewpoints on the structure and compactifications of moduli spaces

Autorentext

Phillip Griffiths is one of the foremost figures in Complex and Algebraic Geometry, with over 900 mathematical descendants and 150 influential books and articles. He is notably at the origin of Hodge Theory, a topic that has blossomed into one of the major currents of thought across many domains of modern mathematics and physics.

Ludmil Katzarkov is a world leader in Mirror Symmetry, Symplectic Geometry, Mathematical Physics and Algebraic Geometry. He initiated higher-dimensional factorization theory, gave the first significant advance on the Shafarevich holomorphic convexity conjecture, and introduced the most modern generalized viewpoint on Homological Mirror Symmetry relating Algebraic and Symplectic Geometry.

Carlos Simpson is a pioneer and main proponent of Nonabelian Hodge Theory, generalizing Griffiths' theory to character varieties and nonabelian cohomology following Hitchin's notion of Higgs bundle. He has also contributed to the theory of Higher Categories, and to the computer formalization of mathematical proofs.

Inhalt

Preface.- Introduction.- The Coble-Mukai lattice from Q-Gorenstein deformations.- Hyperbolic geometry of moduli spaces of algebraic varieties via Hodge theory, and beyond.- Semistable reduction over thick log points.- Degeneration of Hodge structures on I-surfaces.- Foliations and stable maps.- Pseudoconvexity at infinity in Hodge theory: a codimension one example.- The model theory of canonical models of Shimura curves.- Moduli spaces on Kuznetsov components are irreducible symplectic varieties.- Mapping class groups of simply connected Kahler manifolds.- Hodge theory of degenerations, (II): vanishing cohomology and geometric applications.