Coherent sheaves, superconnections, and Riemann-Roch-Grothendieck / Jean-Michel Bismut, Shu Shen, Zhaoting Wei.
Bismut, Jean-Michel.; Shen, Shu.; Wei, Zhaoting.
2023
QA613.618 .B57 2023
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Title Coherent sheaves, superconnections, and Riemann-Roch-Grothendieck / Jean-Michel Bismut, Shu Shen, Zhaoting Wei.
Author Bismut, Jean-Michel.
ISBN 9783031272349 electronic book
303127234X electronic book
9783031272332
3031272331
DOI https://doi.org/10.1007/978-3-031-27234-9
Published Cham : Birkhauser, 2023.
Language English
Description 1 online resource (x, 184 pages) : illustrations.
Item Number 10.1007/978-3-031-27234-9 doi
Call Number QA613.618 .B57 2023
Dewey Decimal Classification 516/.183
Summary 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 for many researchers in geometry, analysis, and mathematical physics.
Bibliography, etc. Note Includes bibliographical references and indexes.
Access Note Access limited to authorized users.
Added Author Shen, Shu.
Wei, Zhaoting.
Series Progress in mathematics (Boston, Mass.) ; v. 347.
Available in Other Form Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck
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Introduction
Bott-Chern Cohomology and Characteristic Classes
The Derived Category ${\mathrm{D^{b}_{\mathrm{coh}}}}$
Preliminaries on Linear Algebra and Differential Geometry
The Antiholomorphic Superconnections of Block
An Equivalence of Categories
Antiholomorphic Superconnections and Generalized Metrics
Generalized Metrics and Chern Character Forms
The Case of Embeddings
Submersions and Elliptic Superconnections
Elliptic Superconnection Forms and Direct Images
A Proof of Theorem 10-1 when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.
The Hypoelliptic Superconnections
The Hypoelliptic Superconnection Forms
The Hypoelliptic Superconnection Forms when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$
Exotic Superconnections and Riemann-Roch-Grothendieck
Subject Index
Index of Notation
Bibliography.

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