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Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

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  • © 2023

Overview

Authors:
  1. Jean-Michel Bismut

    1. Institut de Mathématique d'Orsay, University of Paris-Saclay, Orsay, France

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  2. Shu Shen

    1. Institut de Mathématiques de Jussieu, Paris, France

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    You can also search for this author in PubMed   Google Scholar

  3. Zhaoting Wei

    1. Mathematics, Texas A&M University – Commerce, Commerce, USA

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  • Presents a solution to a long-standing problem in complex algebraic geometry
  • Proves an RRG theorem for coherent sheaves of a compact complex manifold
  • Offers a valuable resource for many researchers in geometry, analysis, and mathematical physics

Part of the book series: Progress in Mathematics (PM, volume 347)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 1-6

  3. Bott-Chern cohomology and characteristic classes

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 7-13

  4. The derived category \({\text{D}}_{{{\text{coh}}}}^{{\text{b}}} (X)\)

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 15-17

  5. Preliminaries on linear algebra and differential geometry

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 19-28

  6. The antiholomorphic superconnections of Block

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 29-42

  7. An equivalence of categories

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 43-59

  8. Antiholomorphic superconnections and generalized metrics

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 61-65

  9. Generalized metrics and Chern character forms

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 67-84

  10. The case of embeddings

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 85-93

  11. Submersions and elliptic superconnection forms

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 95-107

  12. Elliptic superconnection forms and direct images

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 109-115

  13. A proof of Theorem 10.1.1 when \(\overline{\partial}^{X} \partial^{X} \omega^{X} = 0\)

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 117-123

  14. The hypoelliptic superconnections

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 125-143

  15. The hypoelliptic superconnection forms

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 145-147

  16. The hypoelliptic superconnection forms when \(\overline{\partial}^{X}\partial^{X}\omega^{X}=0\)

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 149-161

  17. Exotic superconnections and Riemann-Roch-Grothendieck

    • Jean-Michel Bismut, Shu Shen, Zhaoting Wei
    Pages 163-173

  18. Back Matter

    Pages 175-184
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Keywords

About this book

This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck  theorem.  One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections.  Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian.


Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource formany researchers in geometry, analysis, and mathematical physics. 

Authors and Affiliations

  • Institut de Mathématique d'Orsay, University of Paris-Saclay, Orsay, France

    Jean-Michel Bismut

  • Institut de Mathématiques de Jussieu, Paris, France

    Shu Shen

  • Mathematics, Texas A&M University – Commerce, Commerce, USA

    Zhaoting Wei

About the authors

​Jean-Michel Bismut is a French mathematician who is a professor in the Mathematics Department in Orsay. He is known for his contributions to index theory, geometric analysis and probability theory. Together with Gilles Lebeau, he has developed the theory of the hypoelliptic Laplacian, to which he found applications in various fields of mathematics. He shared the Shaw Prize in Mathematical Sciences 2021 with Jeff Cheeger. 

Shu Shen is a maître de conférences at Sorbonne University in Paris. His research focuses on the fields of analysis, geometry, and representation theory.


Zhaoting Wei is an assistant professor in mathematics at Texas A&M University-Commerce, USA. His research interests include noncommutative geometry and higher category theory.

Bibliographic Information

  • Book Title: Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

  • Authors: Jean-Michel Bismut, Shu Shen, Zhaoting Wei

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-27234-9

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-27233-2Published: 14 November 2023

  • Softcover ISBN: 978-3-031-27236-3Due: 15 December 2023

  • eBook ISBN: 978-3-031-27234-9Published: 13 November 2023

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: X, 184

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Category Theory, Homological Algebra, K-Theory, Analysis, Differential Geometry

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