The p-adic Simpson Correspondence and Hodge-Tate Local Systems

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.

Preface.- An overview.- Preliminaries.- Local Study.- Global Study.- Relative cohomologies of Higgs-Tate algebras.Local Study.- Relative cohomology of Dolbeault modules.- References.- Index..

Ahmed Abbes' research primarily focuses on the study of geometric and cohomological properties of sheaves on varieties over perfect fields of characteristic p>0 or p-adic fields, and their applications to arithmetic and algebraic geometry. He is the author/co-author of three books, including a seminal work on the p-adic Simpson Correspondence in collaboration with M. Gros and T. Tsuji.

Michel Gros' research interests revolve around recent advancements in cohomology theories, both in characteristic p and in p-adic contexts. He has co-authored two books, including a seminal work on the p-adic Simpson Correspondence in collaboration with A. Abbes and T. Tsuji.