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Books on the topic 'Hodge bundle'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Mochizuki, Takuro. Asymptotic behaviour of tame harmonic bundles and an application to pure twister D-modules. Providence, RI: American Mathematical Society, 2007.

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2

Gerverdinck, Tjebbe. Wetenschappelijk bijdragen: Bundel ter gelegenheid van het 35-jarig bestaan van het wetenschappelijk bureau van de Hoge Raad der Nederlanden. Den Haag: Boom Juridische uitgevers, 2014.

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3

Charles, Franc¸ois, and Christian Schnell. Notes on Absolute Hodge Classes. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0011.

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This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.

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4

Noether-lefschetz Problems For Degeneracy Loci. American Mathematical Society, 2003.

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5

Cattani, Eduardo. Introduction to Variations of Hodge Structure. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0007.

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This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.

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6

Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001.

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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

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7

Faltings, Gerd. Facsimile : A p-adic Simpson correspondence. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.003.0007.

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This chapter presents the facsimile of Gerd Faltings' article entitled “A p-adic Simpson Correspondence,” reprinted from Advances in Mathematics 198(2), 2005. In this article, an equivalence between the category of Higgs bundles and that of “generalized representations” of the étale fundamental group is constructed for curves over a p-adic field. The definition of “generalized representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. The method used in the proofs is the theory of almost étale extensions. A nonabelian Hodge–Tate theory is also developed.

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8

Abbes, Ahmed, Michel Gros, and Takeshi Tsuji. The p-adic Simpson Correspondence (AM-193). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.001.0001.

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The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.

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9

Courbes et Fibres Vectoriels en Theorie de Hodge $p$-Adique. American Mathematical Society, 2018.

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10

Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $D$-Modules, Part 1 (Memoirs of the American Mathematical Society) (Memoirs of the American Mathematical Society). American Mathematical Society, 2006.

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11

Mochizuki, Takuro. Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $D$-Modules, Part 2 (Memoirs of the American Mathematical Society) (Memoirs of the American Mathematical Society). American Mathematical Society, 2006.

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12

Gray, Loretta, and Cheryl Glenn. Bundle: Hodges Harbrace Handbook, 2016 MLA Update, 19th + MindTap English, 2 Terms Printed Access Card. Wadsworth, 2017.

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13

Gray, Loretta, and Cheryl Glenn. Bundle: Hodges Harbrace Handbook, 2016 MLA Update, 19th + MindTap English 1 Term Printed Access Card. Wadsworth, 2017.

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14

Gray, Loretta, and Cheryl Glenn. Bundle: The Hodges Harbrace Handbook, 19th + 2016 MLA Update Card + MindTap English, 2 Terms Printed Access Card. Cengage Learning, 2016.

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15

Gray, Loretta, and Cheryl Glenn. Bundle: The Hodges Harbrace Handbook, 19th + 2016 MLA Update Card + MindTap English 1 Term Printed Access Card. Cengage Learning, 2016.

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16

Gray, Loretta, and Cheryl Glenn. Bundle: Hodges Harbrace Handbook, 2016 MLA Update, Loose-Leaf Version, 19th + MindTap English, 2 Terms Printed Access Card. Wadsworth, 2017.

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17

Gray, Loretta, and Cheryl Glenn. Bundle: Hodges Harbrace Handbook, 2016 MLA Update, Loose-Leaf Version, 19th + MindTap English 1 Term Printed Access Card. Wadsworth, 2017.

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18

Gray, Loretta, and Cheryl Glenn. Bundle: The Hodges Harbrace Handbook, Loose-Leaf Version, 19th + 2016 MLA Update Card + MindTap English, 2 Terms Printed Access Card. Cengage Learning, 2016.

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19

Gray, Loretta, and Cheryl Glenn. Bundle: The Hodges Harbrace Handbook, Loose-Leaf Version, 19th + 2016 MLA Update Card + MindTap English 1 Term Printed Access Card. Cengage Learning, 2016.

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20

Surveys on Recent Developments in Algebraic Geometry. American Mathematical Society, 2017.

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