Hodge theory, complex geometry, and representation theory
- Green, M. (Mark)
- Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 
- Physical description
- iv, 308 pages : illustrations ; 25 cm.
- Regional conference series in mathematics ; no. 118.
QA1 .R33 NO.118
- Unknown QA1 .R33 NO.118
- Includes bibliographical references (pages 299-302) and index.
- The classical theory (2 parts)
- Polarized Hodge structures and Mumford-Tate groups and domains
- Hodge representations and Hodge domains
- Discrete series and n-cohomology
- Geometry of flag domains (2 parts)
- Penrose transforms in the two main examples
- Automorphic cohomology
- Miscellaneous topics and some open questions.
- Publisher's Summary
- This monograph presents topics in Hodge theory and representation theory, two of the most active and important areas in contemporary mathematics. The underlying theme is the use of complex geometry to understand the two subjects and their relationships to one another--an approach that is complementary to what is in the literature. Finite-dimensional representation theory and complex geometry enter via the concept of Hodge representations and Hodge domains. Infinite-dimensional representation theory, specifically the discrete series and their limits, enters through the realisation of these representations through complex geometry as pioneered by Schmid, and in the subsequent description of automorphic cohomology. For the latter topic, of particular importance is the recent work of Carayol that potentially introduces a new perspective in arithmetic automorphic representation theory. The present work gives a treatment of Carayol's work, and some extensions of it, set in a general complex geometric framework. Additional subjects include a description of the relationship between limiting mixed Hodge structures and the boundary orbit structure of Hodge domains, a general treatment of the correspondence spaces that are used to construct Penrose transforms and selected other topics from the recent literature.
(source: Nielsen Book Data)
- Hodge theory.
- Geometry, Differential.
- Algebraic geometry -- Special varieties -- Grassmannians, Schubert varieties, flag manifolds.
- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Cohomology of Lie (super)algebras.
- Topological groups, Lie groups -- Locally compact groups and their algebras -- Unitary representations of locally compact groups.
- Several complex variables and analytic spaces -- Deformations of analytic structures -- Period matrices, variation of Hodge structure; degenerations.
- Several complex variables and analytic spaces -- Complex spaces with a group of automorphisms -- Homogeneous complex manifolds.
- Algebraic geometry -- Families, fibrations -- Variation of Hodge structures.
- Algebraic geometry -- Special varieties -- Homogeneous spaces and generalizations.
- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Lie algebras of linear algebraic groups.
- Group theory and generalizations -- Linear algebraic groups and related topics -- None of the above, but in this section.
- Topological groups, Lie groups -- Lie groups -- Representations of Lie and linear algebraic groups over real fields: analytic methods.
- Topological groups, Lie groups -- Lie groups -- Semisimple Lie groups and their representations.
- Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces.
- Several complex variables and analytic spaces -- Automorphic functions -- Automorphic forms.
- Several complex variables and analytic spaces -- Holomorphic fiber spaces -- Twistor theory, double fibrations.
- Several complex variables and analytic spaces -- Complex manifolds -- Stein manifolds.
- Differential geometry -- Global differential geometry -- Homogeneous manifolds.
- Publication date
- Mark Green, Phillip Griffiths, Matt Kerr.
- Regional conference series in mathematics / Conference Board of the Mathematical Sciences ; number 118
- "Support from the National Science Foundation."
- "NSF-CBMS Regional Conference in the Mathematical Sciences on Hodge Theory, Complex Geometry, and Representation Theory, held at Texas Christian University, June 18-22, 2012."