eBook

Hodge theory (Mathematical notes 49)

Griffiths, Phillip A, Tráng, Lê Dũng, El Zein, Fouad, Cattani, Eduardo

2014

Overview

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck’s algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne’s theorem on absolute Hodge cycles), and variation of mixed Hodge structures. The contributors include Patrick Brosnan, James Carlson, Eduardo Cattani, François Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark L. Green, Phillip A. Griffiths, Matt Kerr, Lê Dũng Tráng, Luca Migliorini, Jacob P. Murre, Christian Schnell, and Loring W. Tu.

• Publisher: Princeton University Press
• Subject: Abelian category / Abelian group / Alexander Grothendieck / Algebraic curve / Algebraic cycle / Algebraic geometry / Algebraic variety / Analytic manifold / Automorphism / Bilinear form / Chow group / Codimension / Coefficient / Cohomology / Complex manifold / Complex number / Congresses / Conjecture / Cup product / De Rham cohomology / Diagram (category theory) / Differentiable manifold / Differential form / Dimension (vector space) / Direct sum / Divisor / Embedding / Endomorphism / Equivalence relation / Exact sequence / Existential quantification / Fundamental group / Geometry / Group Theory / Hodge conjecture / Hodge structure / Hodge theory / Homotopy / Hypersurface / Intersection form (4-manifold) / Irreducible component / Line bundle / Linear algebra / Linear map / Local system / Manifolds (Mathematics) / MATHEMATICS / MATHEMATICS / Group Theory / MATHEMATICS / Research / MATHEMATICS / Topology / Monodromy / Morphism / Nilpotent orbit / Normal function / Open set / Perverse sheaf / Projective space / Projective variety / Quasi-projective variety / Sheaf (mathematics) / Smoothness / Special case / Spectral sequence / Subgroup / Submanifold / Subset / Summation / Surjective function / Tangent space / Tensor product / Theorem / Topological space / Topology / Transversality (mathematics) / Vector bundle / Vector space / Zariski topology
• ISBN: 9780691161341, 1400851475, 9781400851478, 0691161348