The p-adic Simpson Correspondence and Hodge-Tate Local Systems
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Beschreibung
This book delves into the p-adic Simpson correspondence, its construction, and development. Offering fresh and innovative perspectives on this important topic in algebraic geometry, the text serves a dual purpose: it describes an important tool in p-adic Hodge theory, which has recently attracted significant interest, and also provides a comprehensive resource for researchers. Unique among the books in the existing literature in this field, it combines theoretical advances, novel constructions, and connections to Hodge-Tate local systems.
This exposition builds upon the foundation laid by Faltings, the collaborative efforts of the two authors with T. Tsuji, and contributions from other researchers. Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence, whose construction has been taken up in several different ways. Following the approach they initiated with T. Tsuji, the authors develop new features of the p-adic Simpson correspondence, inspired by their construction of the relative Hodge-Tate spectral sequence. First, they address the connection to Hodge-Tate local systems. Then they establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, they expand the scope of their original construction.
The book targets a specialist audience interested in the intricate world of p-adic Hodge theory and its applications, algebraic geometry and related areas. Graduate students can use it as a reference or for in-depth study. Mathematicians exploring connections between complex and p-adic geometry will also find it valuable.
This book delves into the p-adic Simpson correspondence, its construction, and development. Offering fresh and innovative perspectives on this important topic in algebraic geometry, the text serves a dual purpose: it describes an important tool in p-adic Hodge theory, which has recently attracted significant interest, and also provides a comprehensive resource for researchers. Unique among the books in the existing literature in this field, it combines theoretical advances, novel constructions, and connections to Hodge-Tate local systems.
This exposition builds upon the foundation laid by Faltings, the collaborative efforts of the two authors with T. Tsuji, and contributions from other researchers. Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence, whose construction has been taken up in several different ways. Following the approach they initiated with T. Tsuji, the authors develop new features of the p-adic Simpson correspondence, inspired by their construction of the relative Hodge-Tate spectral sequence. First, they address the connection to Hodge-Tate local systems. Then they establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, they expand the scope of their original construction.
The book targets a specialist audience interested in the intricate world of p-adic Hodge theory and its applications, algebraic geometry and related areas. Graduate students can use it as a reference or for in-depth study. Mathematicians exploring connections between complex and p-adic geometry will also find it valuable.
Makes key contributions to the foundations of the p-adic Simpson correspondence Establishes the functoriality of the p-adic Simpson Correspondence by proper direct images Provides an exceptionally rigorous and thorough study
Autor*in
Ahmed Abbes
Themen in »The p-adic Simpson Correspondence and Hodge-Tate Local Systems«
p-adic Simpson correspondence p-adic Hodge theory Dolbeault modules Higgs bundles Hodge-Tate local systems Katz-Oda Higgs field Faltings topos Hodge-Tate spectral sequence p-adic geometry Langlands program Shimura varieties
Stimmen zu »The p-adic Simpson Correspondence and Hodge-Tate Local Systems«
“The structure of the book lends itself well to both experts and non-experts alike. ... the book is pleasantly self-contained and accessible to anyone with the required homological background ... . this text now can also serve as an entry point for less expert researchers looking to deepen their understanding and gain mastery over this important and powerful tool.” (Lance Edward Miller, Mathematical Reviews, April, 2026)
“The Simpson correspondence ... establishes a correspondence between Higgs bundles and local systems on a compact Kähler manifold ... . The correspondence enjoys a number of functoriality properties and it has found numerous applications in complex Hodge theory. ... Specifically, the present book studies two main themes: (1) Hodge-Tate local systems ... and (2) functoriality of the correspondence under proper higher direct images.” (Shishir Agrawal, zbMATH 1547.14002, 2024)
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