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Mixed Hodge Structures
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Book Synopsis Mixed Hodge Structures by : Chris A.M. Peters
Download or read book Mixed Hodge Structures written by Chris A.M. Peters and published by Springer Science & Business Media. This book was released on 2008-02-27 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is comprehensive basic monograph on mixed Hodge structures. Building up from basic Hodge theory the book explains Delingne's mixed Hodge theory in a detailed fashion. Then both Hain's and Morgan's approaches to mixed Hodge theory related to homotopy theory are sketched. Next comes the relative theory, and then the all encompassing theory of mixed Hodge modules. The book is interlaced with chapters containing applications. Three large appendices complete the book.
Book Synopsis Mixed Hodge Structures and Singularities by : Valentine S. Kulikov
Download or read book Mixed Hodge Structures and Singularities written by Valentine S. Kulikov and published by Cambridge University Press. This book was released on 1998-04-27 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: This vital work is both an introduction to, and a survey of singularity theory, in particular, studying singularities by means of differential forms. Here, some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and the theory of period maps, are developed in the local situation to study the case of isolated singularities of holomorphic functions. The author introduces the Gauss-Manin connection on the vanishing cohomology of a singularity, that is on the cohomology fibration associated to the Milnor fibration, and draws on the work of Brieskorn and Steenbrink to calculate this connection, and the limit mixed Hodge structure. This is an excellent resource for all researchers in singularity theory, algebraic or differential geometry.
Book Synopsis Hodge Theory (MN-49) by : Eduardo Cattani
Download or read book Hodge Theory (MN-49) written by Eduardo Cattani and published by Princeton University Press. This book was released on 2014-07-21 with total page 608 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Period Mappings and Period Domains by : James Carlson
Download or read book Period Mappings and Period Domains written by James Carlson and published by Cambridge University Press. This book was released on 2017-08-24 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to Griffiths' theory of period maps and domains, focused on algebraic, group-theoretic and differential geometric aspects.
Book Synopsis Algebraic Geometry by : Robin Hartshorne
Download or read book Algebraic Geometry written by Robin Hartshorne and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.
Book Synopsis Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106 by : Phillip A. Griffiths
Download or read book Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106 written by Phillip A. Griffiths and published by Princeton University Press. This book was released on 2016-03-02 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: The description for this book, Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106, will be forthcoming.
Book Synopsis Recent Advances in Hodge Theory by : Matt Kerr
Download or read book Recent Advances in Hodge Theory written by Matt Kerr and published by Cambridge University Press. This book was released on 2016-02-04 with total page 533 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combines cutting-edge research and expository articles in Hodge theory. An essential reference for graduate students and researchers.
Book Synopsis Motivic Aspects of Hodge Theory by : Chris Peters
Download or read book Motivic Aspects of Hodge Theory written by Chris Peters and published by . This book was released on 2010 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are based on a series of lectures given at the Tata Institute of Fundamental Research, Mumbai, in 2007, on the theme of Hodge theoretic motives associated to various geometric objects. Starting with the topological setting, the notes go on to Hodge theory and mixed Hodge theory on the cohomology of varieties. Degenerations, limiting mixed Hodge structures and the relation to singularities are addressed next. The original proof of Bittner's theorem on the Grothendieck group of varieties, with some applications, is presented as an appendix to one of the chapters. The situation of relative varieties is addressed next using the machinery of mixed Hodge modules. Chern classes for singular varieties are explained in the motivic setting using Bittner's approach, and their full functorial meaning is made apparent using mixed Hodge modules. An appendix explains the treatment of Hodge characteristic in relation with motivic integration and string theory. Throughout these notes, emphasis is placed on explaining concepts and giving examples.
Book Synopsis Hodge Theory, Complex Geometry, and Representation Theory by : Mark Green
Download or read book Hodge Theory, Complex Geometry, and Representation Theory written by Mark Green and published by American Mathematical Soc.. This book was released on 2013-11-05 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 realization 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. A co-publication of the AMS and CBMS.
Book Synopsis Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187) by : Claire Voisin
Download or read book Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187) written by Claire Voisin and published by Princeton University Press. This book was released on 2014-02-23 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.
Book Synopsis Differential Forms on Singular Varieties by : Vincenzo Ancona
Download or read book Differential Forms on Singular Varieties written by Vincenzo Ancona and published by CRC Press. This book was released on 2005-08-24 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified uses complexes of differential forms to give a complete treatment of the Deligne theory of mixed Hodge structures on the cohomology of singular spaces. This book features an approach that employs recursive arguments on dimension and does not introduce spaces of hig
Book Synopsis Mumford-Tate Groups and Domains by : Mark Green
Download or read book Mumford-Tate Groups and Domains written by Mark Green and published by Princeton University Press. This book was released on 2012-04-22 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.
Book Synopsis Introduction to Hodge Theory by : José Bertin
Download or read book Introduction to Hodge Theory written by José Bertin and published by American Mathematical Soc.. This book was released on 2002 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hodge theory originated as an application of harmonic theory to the study of the geometry of compact complex manifolds. The ideas have proved to be quite powerful, leading to fundamentally important results throughout algebraic geometry. This book consists of expositions of various aspects of modern Hodge theory. Its purpose is to provide the nonexpert reader with a precise idea of the current status of the subject. The three chapters develop distinct but closely related subjects:$L2$ Hodge theory and vanishing theorems; Frobenius and Hodge degeneration; variations of Hodge structures and mirror symmetry. The techniques employed cover a wide range of methods borrowed from the heart of mathematics: elliptic PDE theory, complex differential geometry, algebraic geometry incharacteristic $p$, cohomological and sheaf-theoretic methods, deformation theory of complex varieties, Calabi-Yau manifolds, singularity theory, etc. A special effort has been made to approach the various themes from their most na The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.
Book Synopsis Hodge Cycles, Motives, and Shimura Varieties by : Pierre Deligne
Download or read book Hodge Cycles, Motives, and Shimura Varieties written by Pierre Deligne and published by Springer. This book was released on 2009-03-20 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Monodromy Group by : Henryk Zoladek
Download or read book The Monodromy Group written by Henryk Zoladek and published by Springer Science & Business Media. This book was released on 2006-08-10 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: In singularity theory and algebraic geometry, the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. There is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. In covering these and other topics, this book underlines the unifying role of the monogropy group.
Book Synopsis Hodge Theory and Complex Algebraic Geometry I: by : Claire Voisin
Download or read book Hodge Theory and Complex Algebraic Geometry I: written by Claire Voisin and published by Cambridge University Press. This book was released on 2007-12-20 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions.
Book Synopsis Hodge Theory by : Eduardo H.C. Cattani
Download or read book Hodge Theory written by Eduardo H.C. Cattani and published by Springer. This book was released on 2006-11-15 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past 2O years classical Hodge theory has undergone several generalizations of great interest in algebraic geometry. The papers in this volume reflect the recent developments in the areas of: mixed Hodge theory on the cohomology of singular and open varieties, on the rational homotopy of algebraic varieties, on the cohomology of a link, and on the vanishing cycles; L -realization of the intersection cohomology for the cases of singular varieties and smooth varieties with degenerating coefficients; applications of cubical hyperresolutions and of iterated integrals; asymptotic behavior of degenerating variations of Hodge structure; the geometric realization of maximal variations; and variations of mixed Hodge structure. N
